pc5271AY2526wiki - User contributions [en]

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-24T03:42:23Z

Junxiang: /* Sample Preparation */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|400px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|700px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

In this specific experiment, measuring the Kerr angle was performed using S-polarized light, where the incident EM wave direction is perpendicular (electric) and parallel (magnetic) to the incident plane. To obtain the magnetic field amplitude ratios of the reflected and refracted waves to the incident wave, two boundary conditions are applied where the tangential components of the electric and magnetic fields are continuous on the surface.
<math display="block">
E +E' = E'', H\cos\theta - H'\cos\theta = H''\cos\theta
</math>
where <math> E </math> and <math> H </math> are the electric and magnetic field amplitude of the incident wave, <math> E' </math> and <math> H' </math> the electric and magnetic field amplitude of reflected wave, and <math> E'' </math> and <math> H'' </math> are the electric and magnetic field amplitudes of the refracted wave. Assuming a linear medium,
<math display="block">
E = \sqrt{\frac{\mu _{1}}{\varepsilon _{1}}}H, \quad E' = \sqrt{\frac{\mu _{1}}{\varepsilon _{1}}} H', \quad E'' = \sqrt{\frac{\mu _{2}}{\varepsilon _{2}}} H''
</math>
From the three equations above, we can obtain the magnetic field amplitude of the reflected wave <math> H'</math> with
<math display="block">
H \cos\theta - H'\cos\theta' = \sqrt{ \frac{\varepsilon_{2}\mu_{1}}{\varepsilon_{1}\mu_{2}} } \; (H + H'){\mathrm{cos}}\theta ''
</math>
By the law of refraction, where <math> \frac{\sin\theta}{\sin\theta''} = \sqrt{\frac{\mu_2 \varepsilon_2}{\mu_1 \varepsilon_1}} </math>, we obtain the following result for <math> H'</math>:
<math display="block">
H' = \frac{\sqrt{\frac{\mu_2}{\varepsilon_2}}\cos\theta - \sqrt{\frac{\mu_1}{\varepsilon_1}}\sqrt{1 - \frac{\mu_1 \varepsilon_1}{\mu_2 \varepsilon_2} \sin^{2} \theta}}{\sqrt{\frac{\mu_2}{\varepsilon_2}} \cos\theta + \sqrt{\frac{\mu_1}{\varepsilon_1}} \sqrt{1 - \frac{\mu_1 \varepsilon_1}{\mu_2 \varepsilon_2} \sin^{2}\theta}} H
</math>
When <math> \vec{M}_0</math> is along the longitudinal and polar directions, the Kerr angle is zero, due to the reflected magnetic field being parallel to the incident plane. Only when <math> \vec{M}_0</math> is along the transverse direction to we see a Kerr angle, where reflected wave’s total magnetic field is given by <math>\vec{H'}_{\text{tot}} = -H' \cos \theta \cdot \vec{i} + M_{0} \cdot \vec{j} - H' \sin \theta \cdot \vec{k}</math>. The Kerr rotation angle is thus given by
<math display="block">
\theta _{K}=\text{arctan} \frac{\left(\sqrt{\frac{\mu_2}{\varepsilon_2}}\cos\theta + \sqrt{\frac{\mu_1}{\varepsilon_1}}\sqrt{1 - \frac{\mu_1 \varepsilon_1}{\mu_2\varepsilon_2}}\sin^{2}\theta\right) M_0}{\left(\sqrt{\frac{\mu_{2}}{\varepsilon_{2}}}\cos\theta - \sqrt{\frac{\mu_{1}}{\varepsilon_{1}}}\sqrt{1 - \frac{\mu_{1}\varepsilon_{1}}{\mu_{2}\varepsilon_{2}}} \sin^{2}\theta \right) H}
</math>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). MOKE also has possible applications in industry, where it may be used to further characterize the development of magneto-optical characteristics of data storage devices (Huang et al, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|700px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current MOKE experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|700px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A GIF file demonstrating the operation of the imaging system is provided at the below.
[[File:Moke_sample_imagegif.gif|700px|center]]
<div style="text-align:center;">
'''Gif 1''' MOKE sample image
</div>

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|700px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|700px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: a Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer and rinsed with acetone, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

Each sample was mounted on carbon tape and set onto an optical mirror mount. For more accurate measurement of the hysteresis loop, the Fe thin film was mounted onto a glass slide and then mounted onto the mirror mount to reduce errors from the uneven and non-rigid carbon tape support.
[[File:FEfilm2.jpeg|500px|center|]]
<div style="text-align:center;">
'''Figure 9''' Fe Film mounted on mirror mount carbon tape system
</div>

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE Second Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment, sample magnetization, or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a more stable power supply capable of delivering greater power and a more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

A minor improvement would be the construction of a specialized holder for the sample. The sensitive of the MOKE measurements, where a minute disturbance to the focused laser beam and the incident angle would affect readings, indicates that a more rigid holder, constraining any possible movement of the thin film samples during measurement would be beneficial for more accurate measurements. As mentioned in the article, an initial holder using carbon tape was employed and eventually replaced with a glass slide holder. This is because as a ductile material, wear and tear eventually caused the carbon tape to move, changing the focus of the laser beam and affecting measurements with only small movements along the optical setup.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==References==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

13. Huang, T., Dong, J., Huang, T., Mei, T., Hu, L., Lin, C. H., Chen, F., Chu, D., Cheng, Z., & Li, Z. (2026). Advancing Characterization for Magnetic Materials via Magneto-Optical Kerr Effect Microscopy. Small (Weinheim an der Bergstrasse, Germany), 22(5), e10608. https://doi.org/10.1002/smll.202510608

==Appendix==
===Python Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:57:42Z

Junxiang: /* Second Experiment */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|400px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|700px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|700px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|700px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A GIF file demonstrating the operation of the imaging system is provided at the below.
[[File:Moke_sample_imagegif.gif|700px|center]]
<div style="text-align:center;">
'''Gif 1''' MOKE sample image
</div>

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|700px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|700px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE Second Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Python Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:54:25Z

Junxiang: /* Image System */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|400px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|700px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|700px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|700px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A GIF file demonstrating the operation of the imaging system is provided at the below.
[[File:Moke_sample_imagegif.gif|700px|center]]
<div style="text-align:center;">
'''Gif 1''' MOKE sample image
</div>

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|700px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|700px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Python Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:53:32Z

Junxiang: /* Image System */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|400px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|700px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|700px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|700px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the below.
[[File:Moke_sample_imagegif.gif|700px|thumb|center|MOKE sample]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|700px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|700px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Python Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

File:Moke sample imagegif.gif

2026-04-22T17:51:38Z

Junxiang:

File:Moke sample image 1.gif

2026-04-22T17:48:33Z

Junxiang:

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:45:51Z

Junxiang: /* Image System */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|400px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|700px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|700px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|700px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the below.
[[File:Moke_sample_image.gif|700px|thumb|center|MOKE sample]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|700px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|700px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
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8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

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10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

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==Appendix==
===Python Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:42:11Z

Junxiang: /* Code for Reading Data from SR860 Lock-in Amplifier */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|400px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|700px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|700px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|700px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|700px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|700px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Python Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:40:31Z

Junxiang: /* Physical Principle */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|400px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|700px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|700px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|700px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|700px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|700px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:40:22Z

Junxiang: /* Physical Principle */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|700px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|700px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|700px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|700px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|700px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|700px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:40:12Z

Junxiang: /* Three Geometries of MOKE */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|700px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|700px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|700px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|700px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|700px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:40:02Z

Junxiang: /* Schematic Figure */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|700px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|700px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|700px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|700px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:39:51Z

Junxiang: /* Mainstream Optical Path */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|700px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|700px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|700px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|700px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:39:38Z

Junxiang: /* Image System */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|700px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|700px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|700px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:39:28Z

Junxiang: /* Magnetic Field and Measuring */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|700px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|700px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

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7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

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12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:39:17Z

Junxiang: /* Measuring Method */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|700px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:39:02Z

Junxiang: /* Third Experiment: Hysteresis Loop of Fe Film */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|700px|center|]]
<div style="text-align:center;">
'''Figure 10''' Fe Film Hysteresis Loop
</div>
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:37:47Z

Junxiang: /* Second Experiment */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|700px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:37:19Z

Junxiang: /* Second Experiment */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|500px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:37:00Z

Junxiang: /* Second Experiment */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|center]]
<div style="text-align:center;">
'''Figure 9''' MOKE First Measurement
</div>
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:36:12Z

Junxiang: /* Measuring Method */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Analyzer Offset Method
</div>
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:35:33Z

Junxiang: /* Image System */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:34:56Z

Junxiang: /* Image System */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
<video width="800" controls>
<source src="/path/to/Moke_sample_image.mp4" type="video/mp4">
</video>

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

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4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

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6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

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12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:34:34Z

Junxiang: /* Experimental Setup */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|800px|thumb|center|MOKE sample video]]
=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:33:08Z

Junxiang: /* Experimental Setup */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 3''' MOKE Experimental Setup Scheme
</div>
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 4''' MOKE Setup Upstream
</div>
[[File:Moke_downstream.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 5''' MOKE Setup Downstream
</div>
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 6''' MOKE Setup sample image system
</div>
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px||center|]]
<div style="text-align:center;">
'''Figure 7''' MOKE Setup Sample Image System
</div>
=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|center|]]
<div style="text-align:center;">
'''Figure 8''' MOKE Magnetic Field and Measuring
</div>
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:29:40Z

Junxiang: /* Three Geometries of MOKE */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:29:06Z

Junxiang: /* Three Geometries of MOKE */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

Polar MOKE: The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

Longitudinal MOKE: The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

Transverse MOKE: The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:28:34Z

Junxiang: /* Three Geometries of MOKE */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.

'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.

'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:28:24Z

Junxiang: /* Three Geometries of MOKE */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

'''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
'''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
'''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:27:56Z

Junxiang: /* Introduction */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|center]]
<div style="text-align:center;">
'''Figure 2''' Three Geometries of MOKE
</div>
=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:26:47Z

Junxiang: /* Physical Principle */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

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12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:26:25Z

Junxiang: /* Physical Principle */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center]]
<div style="text-align:center;">
'''Figure 1''' Coordinate System with Corresponding Media in the MOKE
</div>
Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:23:04Z

Junxiang: /* MOKE theory */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:21:06Z

Junxiang: /* Physical Principle */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
<math display="block">
E_{1t} = E_{2t}, \quad B_{2t} - B_{1t} = \mu_0(M_{2t} - M_{1t}), \quad D_{2n} = D_{1n}, \quad B_{2n} = B_{1n}
</math>

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

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12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:20:04Z

Junxiang: /* Physical Principle */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math display="block">
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:19:42Z

Junxiang: /* Physical Principle */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math>
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0
</math>

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:19:09Z

Junxiang: /* Physical Principle */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

<math>\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \left(\frac{\vec{B}}{\mu_0} - \vec{M}\right) = \vec{J} + \frac{\partial \vec{D}}{\partial t}, \quad \nabla \cdot \vec{D} = \rho, \quad \nabla \cdot \vec{B} = 0</math>

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
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2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

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4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

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7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

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12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:16:29Z

Junxiang: /* Limitations and Suggested Improvements */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

[[File:Maxwellequations.jpeg|400px|thumb|center]]

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion Between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:16:13Z

Junxiang: /* Limitations and Suggested Improvements */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

[[File:Maxwellequations.jpeg|400px|thumb|center]]

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:
<div style="text-align:center;">
'''Table 4''' Detailed Comparsion between Single and Balanced Detection
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:14:30Z

Junxiang: /* Third Experiment: Hysteresis Loop of Fe Film */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

[[File:Maxwellequations.jpeg|400px|thumb|center]]

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
<div style="text-align:center;">
'''Table 3''' Hysteresis Loop of Fe Film Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:

{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
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2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

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7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

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12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:12:23Z

Junxiang: /* Second Experiment */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

[[File:Maxwellequations.jpeg|400px|thumb|center]]

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:
<div style="text-align:center;">
'''Table 2''' Second Experiment Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:

{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:11:20Z

Junxiang: /* Initial Test */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

[[File:Maxwellequations.jpeg|400px|thumb|center]]

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
<div style="text-align:center;">
'''Table 1''' Initial Test Measurement Data
</div>
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:

{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:

{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:08:51Z

Junxiang: /* Third Experiment: Hysteresis Loop of Fe Film */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

[[File:Maxwellequations.jpeg|400px|thumb|center]]

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:

{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film Hysteresis Loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:

{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:06:08Z

Junxiang: /* Reference */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

[[File:Maxwellequations.jpeg|400px|thumb|center]]

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:

{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film hysteresis loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:

{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).

2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).

3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).

4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).

5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).

6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).

7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).

8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).

9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).

11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).

12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:05:52Z

Junxiang: /* Reference */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

[[File:Maxwellequations.jpeg|400px|thumb|center]]

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:

{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film hysteresis loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:

{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. ''J. Phys. D: Appl. Phys.'' 48, 333001 (2015).
2. Erskine, J.L., Stern, E.A. Magneto-optic Kerr effects in gadolinium. ''Phys. Rev. B'' 8(3), 1239 (1973).
3. Bader, S.D. Surface magneto-optic Kerr effect (SMOKE). ''J. Magn. Magn. Mater.'' 200(1–3), 664–678 (1999).
4. Haider, T. A Review of Magneto-Optic Effects and Its Application. ''Int. J. Electromagn. Appl.'' 7(1), 17–24 (2017).
5. Kato, T. et al. Fundamentals of Magneto-Optical Spectroscopy. ''Front. Phys.'' 10, 946515 (2022).
6. Qiu, Z.Q., Bader, S.D. Surface magneto-optic Kerr effect. ''Rev. Sci. Instrum.'' 71(3), 1243–1255 (2000).
7. Stobiecki, T. et al. Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Rev. Sci. Instrum.'' 82, 083703 (2011).
8. Suzuki, D.H., Beach, G.S.D. Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''J. Appl. Phys.'' 135, 063901 (2024).
9. Wikipedia contributors. Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia'' (2026). https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect
10. Zhang, H. et al. Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv'' 2201.11744 (2022).
11. Oppeneer, P.M. Magneto-optical Kerr spectra. In Buschow, K.H.J. (Ed.), ''Handb. Magn. Mater.'' Vol. 13, pp. 229–422. Elsevier (2001).
12. Stanford Research Systems. ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf (2016).

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T17:02:40Z

Junxiang: /* Limitations and Suggested Improvements */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

[[File:Maxwellequations.jpeg|400px|thumb|center]]

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:

{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film hysteresis loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

An additional source of systematic error is the possible pre-magnetization of the Fe film prior to measurement. As a ferromagnetic material, Fe retains a remanent magnetization in the absence of an applied field. If the sample was not demagnetized before the experiment, the initial magnetic state of the film may have been non-zero, meaning the hysteresis loop was not started from a magnetically neutral state. This could account for the offset observed in the remanent Kerr angle between the two branches, as well as the asymmetry in the positive and negative saturation values. To mitigate this in future experiments, the sample should be fully demagnetized by cycling the applied field with decreasing amplitude before commencing the measurement.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:

{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. J. Phys. D: Appl. Phys. 48, 333001 (2015).

2. Erskine J L, Stern E A. Magneto-optic Kerr effects in gadolinium[J]. Physical Review B, 1973, 8(3): 1239

3. Bader, S.D. (1999). Surface magneto-optic Kerr effect (SMOKE). ''Journal of Magnetism and Magnetic Materials'', 200(1–3), 664–678. https://doi.org/10.1016/S0304-8853(99)00311-X

4. Haider, T. (2017). A Review of Magneto-Optic Effects and Its Application. ''International Journal of Electromagnetics and Applications'', 7(1), 17–24. https://doi.org/10.5923/j.ijea.20170701.03

5. Kato, T. et al. (2022). Fundamentals of Magneto-Optical Spectroscopy. ''Frontiers in Physics'', 10, 946515. https://doi.org/10.3389/fphy.2022.946515

6. Qiu, Z.Q. and Bader, S.D. (2000). Surface magneto-optic Kerr effect. ''Review of Scientific Instruments'', 71(3), 1243–1255. https://doi.org/10.1063/1.1150496

7. Stobiecki, T. et al. (2011). Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Review of Scientific Instruments'', 82, 083703. https://doi.org/10.1063/1.3610967

8. Suzuki, D.H. and Beach, G.S.D. (2024). Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''Journal of Applied Physics'', 135, 063901. https://doi.org/10.1063/5.0185341

9. Wikipedia contributors. (2026). Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia''. https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. (2022). Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv preprint'', arXiv:2201.11744. https://arxiv.org/abs/2201.11744

11. Oppeneer, P.M. (2001). Magneto-optical Kerr spectra. In K.H.J. Buschow (Ed.), ''Handbook of Magnetic Materials'', Vol. 13, pp. 229–422. Elsevier.

12. Stanford Research Systems. (2016). ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T16:56:25Z

Junxiang: /* Initial Test */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

[[File:Maxwellequations.jpeg|400px|thumb|center]]

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:

{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film hysteresis loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:

{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. J. Phys. D: Appl. Phys. 48, 333001 (2015).

2. Erskine J L, Stern E A. Magneto-optic Kerr effects in gadolinium[J]. Physical Review B, 1973, 8(3): 1239

3. Bader, S.D. (1999). Surface magneto-optic Kerr effect (SMOKE). ''Journal of Magnetism and Magnetic Materials'', 200(1–3), 664–678. https://doi.org/10.1016/S0304-8853(99)00311-X

4. Haider, T. (2017). A Review of Magneto-Optic Effects and Its Application. ''International Journal of Electromagnetics and Applications'', 7(1), 17–24. https://doi.org/10.5923/j.ijea.20170701.03

5. Kato, T. et al. (2022). Fundamentals of Magneto-Optical Spectroscopy. ''Frontiers in Physics'', 10, 946515. https://doi.org/10.3389/fphy.2022.946515

6. Qiu, Z.Q. and Bader, S.D. (2000). Surface magneto-optic Kerr effect. ''Review of Scientific Instruments'', 71(3), 1243–1255. https://doi.org/10.1063/1.1150496

7. Stobiecki, T. et al. (2011). Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Review of Scientific Instruments'', 82, 083703. https://doi.org/10.1063/1.3610967

8. Suzuki, D.H. and Beach, G.S.D. (2024). Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''Journal of Applied Physics'', 135, 063901. https://doi.org/10.1063/5.0185341

9. Wikipedia contributors. (2026). Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia''. https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. (2022). Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv preprint'', arXiv:2201.11744. https://arxiv.org/abs/2201.11744

11. Oppeneer, P.M. (2001). Magneto-optical Kerr spectra. In K.H.J. Buschow (Ed.), ''Handbook of Magnetic Materials'', Vol. 13, pp. 229–422. Elsevier.

12. Stanford Research Systems. (2016). ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T16:56:12Z

Junxiang: /* Sample Preparation */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

[[File:Maxwellequations.jpeg|400px|thumb|center]]

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:

{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film hysteresis loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:

{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. J. Phys. D: Appl. Phys. 48, 333001 (2015).

2. Erskine J L, Stern E A. Magneto-optic Kerr effects in gadolinium[J]. Physical Review B, 1973, 8(3): 1239

3. Bader, S.D. (1999). Surface magneto-optic Kerr effect (SMOKE). ''Journal of Magnetism and Magnetic Materials'', 200(1–3), 664–678. https://doi.org/10.1016/S0304-8853(99)00311-X

4. Haider, T. (2017). A Review of Magneto-Optic Effects and Its Application. ''International Journal of Electromagnetics and Applications'', 7(1), 17–24. https://doi.org/10.5923/j.ijea.20170701.03

5. Kato, T. et al. (2022). Fundamentals of Magneto-Optical Spectroscopy. ''Frontiers in Physics'', 10, 946515. https://doi.org/10.3389/fphy.2022.946515

6. Qiu, Z.Q. and Bader, S.D. (2000). Surface magneto-optic Kerr effect. ''Review of Scientific Instruments'', 71(3), 1243–1255. https://doi.org/10.1063/1.1150496

7. Stobiecki, T. et al. (2011). Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Review of Scientific Instruments'', 82, 083703. https://doi.org/10.1063/1.3610967

8. Suzuki, D.H. and Beach, G.S.D. (2024). Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''Journal of Applied Physics'', 135, 063901. https://doi.org/10.1063/5.0185341

9. Wikipedia contributors. (2026). Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia''. https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. (2022). Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv preprint'', arXiv:2201.11744. https://arxiv.org/abs/2201.11744

11. Oppeneer, P.M. (2001). Magneto-optical Kerr spectra. In K.H.J. Buschow (Ed.), ''Handbook of Magnetic Materials'', Vol. 13, pp. 229–422. Elsevier.

12. Stanford Research Systems. (2016). ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T16:56:04Z

Junxiang: /* Sample Preparation */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

[[File:Maxwellequations.jpeg|400px|thumb|center]]

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

The initial test yielded a Kerr rotation of 1.91 mrad for the Fe film compared to 0.6428 mrad for the Si(111) reference, demonstrating a clear and measurable contrast between the magnetic and non-magnetic samples. This confirms that the optical setup is functioning correctly and is sensitive to the magneto-optical response of the Fe film. The measured Kerr angle for Fe is consistent with values reported in the literature for thin ferromagnetic films, validating the reliability of our system.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:

{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film hysteresis loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:

{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. J. Phys. D: Appl. Phys. 48, 333001 (2015).

2. Erskine J L, Stern E A. Magneto-optic Kerr effects in gadolinium[J]. Physical Review B, 1973, 8(3): 1239

3. Bader, S.D. (1999). Surface magneto-optic Kerr effect (SMOKE). ''Journal of Magnetism and Magnetic Materials'', 200(1–3), 664–678. https://doi.org/10.1016/S0304-8853(99)00311-X

4. Haider, T. (2017). A Review of Magneto-Optic Effects and Its Application. ''International Journal of Electromagnetics and Applications'', 7(1), 17–24. https://doi.org/10.5923/j.ijea.20170701.03

5. Kato, T. et al. (2022). Fundamentals of Magneto-Optical Spectroscopy. ''Frontiers in Physics'', 10, 946515. https://doi.org/10.3389/fphy.2022.946515

6. Qiu, Z.Q. and Bader, S.D. (2000). Surface magneto-optic Kerr effect. ''Review of Scientific Instruments'', 71(3), 1243–1255. https://doi.org/10.1063/1.1150496

7. Stobiecki, T. et al. (2011). Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Review of Scientific Instruments'', 82, 083703. https://doi.org/10.1063/1.3610967

8. Suzuki, D.H. and Beach, G.S.D. (2024). Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''Journal of Applied Physics'', 135, 063901. https://doi.org/10.1063/5.0185341

9. Wikipedia contributors. (2026). Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia''. https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. (2022). Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv preprint'', arXiv:2201.11744. https://arxiv.org/abs/2201.11744

11. Oppeneer, P.M. (2001). Magneto-optical Kerr spectra. In K.H.J. Buschow (Ed.), ''Handbook of Magnetic Materials'', Vol. 13, pp. 229–422. Elsevier.

12. Stanford Research Systems. (2016). ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>

Optical Sensing of Magnetic Dynamics: A Lock-in Detected Single Spot MOKE Magnetometer

2026-04-22T16:55:54Z

Junxiang: /* Second Experiment */

==Team Members==
LI Junxiang E1127462@u.nus.edu

Patricia Breanne Tan SY pb.sy82@u.nus.edu

==Idea==
We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

== Introduction ==

=== The Kerr Effect ===

In 1875, physicist John Kerr discovered the Kerr Effect, a phenomenon wherein the refractive index of a material varies with the application of an electric field. The change in refractive index is described to be directly proportional to the square of the electric field, and may occur either from the initial application of an electric field (Kerr electro-optic effect) or from an electric field proceeding from an incident ray of light (Optical Kerr effect). In addition to these types of Kerr effect, a third exists: the magneto-optical Kerr effect, or MOKE. The magneto-optic Kerr effect was discovered in 1877 by John Kerr when he examined the polarization of light reflected from a polished electromagnet pole (Bader, 1999).

=== Physical Principle ===

In the MOKE, a magnetized surface causes reflected light beams to vary in its polarization and reflected intensity. We may describe this phenomenon with the use of Maxwell's equations from classical electromagnetic theory:

[[File:Maxwellequations.jpeg|400px|thumb|center]]

with the following corresponding boundary conditions:
[[File:Boundaryconditions.jpeg|400px|thumb|center]]

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with <math>\vec B = \mu \vec H +\mu_{0} \vec M_{0}</math> and a homogeneous linear medium, following the diagram below. With the magnetization <math> \vec M_{0} </math> taken as a constant vector, Equation 1 describes the hysteresis loop of a the ferromagnetic medium, which is simply the sum of a permanent magnet and linear magnetic medium.

[[File:MOKE Diagram EMtheory.png|200px|thumb|center|Fig 1.Coordinate System with Corresponding Media in the MOKE]]

Both the MOKE and the Faraday effect arise from the off-diagonal components of the dielectric tensor <math>\varepsilon</math>, which give the magneto-optic material an anisotropic permittivity. The physical origin of this anisotropy lies in the spin-orbit coupling and exchange splitting of the electronic band structure in magnetic materials (Haider, 2017).

=== Three Geometries of MOKE ===

Depending on whether <math> \vec M_{0} </math> is along the polar, longitudinal, or transverse directions, the effects and rotation angles when linearly polarized plane waves (LPPW) of light will vary. The Kerr angle <math> \theta _{K}</math> is the angle by which LPPW rotates after being incident on the sample, and is proportional to the dot product between light propagation and the magnetization <math> \vec M_{0} </math> (Qiu and Bader, 2000). The three standard MOKE geometries are (Kato et al., 2022):

* '''Polar MOKE''': The magnetization <math>\vec{M}</math> is perpendicular to the sample surface and parallel to the plane of incidence. The polar Kerr effect is strongest at near-normal incidence and is most commonly used for measuring out-of-plane magnetization, such as in thin ferromagnetic films.
* '''Longitudinal MOKE''': The magnetization <math>\vec{M}</math> is parallel to both the sample surface and the plane of incidence. This geometry requires oblique incidence and is used to probe in-plane magnetization components along the direction of the incident beam.
* '''Transverse MOKE''': The magnetization <math>\vec{M}</math> is parallel to the sample surface but perpendicular to the plane of incidence. Unlike the polar and longitudinal cases, the transverse MOKE manifests as a change in reflected intensity rather than a rotation of polarization.

[[File:MOKEgeometries.png|400px|thumb|center|Fig 2.Three Geometries of MOKE]]

=== Methods of Measuring the Kerr Angle ===

Two principal methods are used to measure the Kerr rotation angle <math>\theta_K</math> in practice:

'''Analyzer offset (single-detector) method''': In this approach, the analyzer is rotated by a small offset angle <math>\varphi</math> away from the extinction position. The reflected intensity then becomes sensitive to the Kerr rotation, and the signal measured by a single photodetector is proportional to <math>\theta_K</math>. By recording the intensity at <math>+\varphi</math> and <math>-\varphi</math> positions and taking the normalized difference, the Kerr angle can be extracted as (Suzuki and Beach, 2024):

<math display="block">
\theta_K = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}
</math>

This method is simple and low-cost but is susceptible to laser intensity fluctuations between the two sequential measurements.

'''Balanced detection method''': In this approach, the reflected beam is split by a polarizing beamsplitter (such as a Wollaston prism) into two orthogonally polarized components, which are simultaneously detected by a matched pair of photodetectors. The Kerr signal is extracted from the differential output <math>I_+ - I_-</math>, which cancels common-mode intensity noise. This scheme offers significantly improved signal-to-noise ratio and is widely adopted in high-sensitivity MOKE setups (Stobiecki et al., 2011). A sensitivity as high as <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math> has been demonstrated using balanced detection with a stabilized laser source (Zhang et al., 2022).

=== Applications of MOKE ===

Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications ranging from materials characterization to Kerr microscopy, where differences in the magnetization on a surface are imaged by passing the reflected beam through an analyzer before a standard optical microscope (Wikipedia, 2026). The technique is particularly powerful for studying thin magnetic films, where it offers sensitivity down to a single monolayer, earning it the designation of surface MOKE (SMOKE) when applied to ultrathin film systems (Bader, 1999).

==Experimental Setup==
=== Schematic Figure ===
[[File:Moke setup.png|1000px|thumb|center|Fig 3.MOKE Experimental Setup Scheme]]
We employed a 633 nm red continuous-wave laser as the light source with a maximum output power of 1 mW. The beam first passes through a polarizer to define the initial polarization state, followed by the sample stage surrounded by an electromagnet for applying the external magnetic field. A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference signal for the lock-in amplifier. Upon reflection from the sample, the beam passes through an analyzer set at a small offset angle <math>\pm\varphi = \pm 3^\circ</math> from the extinction (crossed) position. The reflected intensity is then detected by a Si photodetector.

=== Mainstream Optical Path ===
[[File:Moke_upstream.png|1000px|thumb|center|Fig 4.MOKE Setup Upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|Fig 5.MOKE Setup Downstream]]
The figures above illustrate our current Magneto-Optic Kerr Effect (MOKE) experimental setup. The light source is a 633 nm laser, whose output first passes through a quarter-wave plate to convert the polarization state to circular. A continuously variable circular neutral density (ND) filter is placed downstream for precise intensity control, followed by a series of irises for beam alignment and aperture definition. The beam then passes through a mechanical chopper operating at 400 Hz, which serves as the external reference signal for the SR860 lock-in amplifier to improve the signal-to-noise ratio. A polarizing beam splitter (PBS) cube is subsequently used to select pure s-polarized light, with the rejected p-polarized component directed into a beam block. The s-polarized beam is focused onto the sample by a lens with a focal length of 25 mm. The reflected signal is redirected by a D-shaped mirror into a Glan–Taylor calcite polarizer with an extinction ratio of 100,000:1. This polarizer is initially set to a crossed-polarization configuration relative to the incident beam, calibrated in the absence of the sample. The use of this high-extinction polarizer enables the isolation of the weak polarization rotation signal arising from the material's magnetization. The transmitted optical signal is finally detected by a silicon photodetector.

=== Image System ===
[[File:Moke_sample_image_system.png|1000px|thumb|center|Fig 6.MOKE Setup sample image system]]
The sample imaging system uses a white LED as a uniform illumination source. Light from the LED is collimated by lens L1 and directed toward the sample via two 50:50 beam splitters (BS1 and BS2). Lens L2 focuses the white light onto the sample surface, and the back-reflected signal propagates along the primary optical path before being imaged onto a CCD camera by lens L3. A video demonstrating the operation of the imaging system is provided at the link below.
[[File:Moke_sample_image.mp4|1000px|thumb|center|Fig 7.MOKE Setup Sample Image System]]

=== Magnetic Field and Measuring ===
The MOKE signal was measured using a controllable permanent magnet, with the applied magnetic field strength monitored in situ by a calibrated Gauss meter placed at the surface of the Fe film:
[[File:Moke_gaussmeter_and_magnet.png|1000px|thumb|center|Fig 7.MOKE Magnetic Field and Measuring]]
The permanent magnet was positioned at an oblique angle behind the sample. By varying its distance from the sample while simultaneously reading the Gauss meter, a range of applied magnetic field strengths could be selected.

==Methods==

===MOKE theory===
The permittivity of a magnetic material can be expressed as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\
\epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\
\epsilon_{31} & \epsilon_{32} & \epsilon_{33}
\end{pmatrix} + i \begin{pmatrix}
0 & \epsilon_{12} & \epsilon_{13} \\
-\epsilon_{21} & 0 & \epsilon_{23} \\
-\epsilon_{31} & -\epsilon_{32} & 0
\end{pmatrix}
</math>
Then the permittivity tensor can be simplified as:
<math display="block">
\epsilon = \begin{pmatrix}
\epsilon_x & i\sigma & 0 \\
-i\sigma & \epsilon_y & 0 \\
0 & 0 & \epsilon_z
\end{pmatrix}
</math>
From electrodynamics, we have:
<math>\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}</math>, then we combine it with Faraday's law and Ampère's law:
<math display="block">
\begin{cases}
\mathbf{\nabla} \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \\
\mathbf{\nabla} \times \mathbf{H} = \epsilon_0 \epsilon \frac{\partial \mathbf{E}}{\partial t}
\end{cases}
</math>
Assuming the incident laser beam is a plane wave, then:
<math display="block">
\begin{cases}
\mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{E}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)} \\
\mathbf{H} = \mathbf{H}_0 e^{i(\mathbf{k}\mathbf{r} - \omega t)} = \mathbf{H}_0 e^{i(\frac{n}{c}\mathbf{s} \cdot \mathbf{r} - t)}
\end{cases}
</math>
Therefore,
<math display="block">
\begin{cases}
\frac{n}{c} \mathbf{E} \times \mathbf{S} = -\mu_0 \mathbf{H} \\
\frac{n}{c} \mathbf{H} \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
\end{cases}
</math>
Combining our previous equations yields:
<math display="block">
\frac{n}{c} \left( -\frac{1}{\mu_0} \frac{n}{c} \mathbf{E} \times \mathbf{S} \right) \times \mathbf{S} = \epsilon_0 \epsilon \mathbf{E}
</math>
Solving this equation, we obtain:
<math display="block">
\epsilon \mathbf{E} = n^2 [\mathbf{E} - \mathbf{S}(\mathbf{S} \cdot \mathbf{E})]
</math>
Then, by substituting the simplified permittivity tensor, we have:
<math display="block">
\begin{cases}
(\epsilon - n)^2 \mathbf{E}_x + i \delta \mathbf{E}_y = 0 \\
-i \delta \mathbf{E}_x + (\epsilon - n)^2\mathbf{E}_y = 0 \\
\epsilon \mathbf{E}_z = 0
\end{cases}
</math>
For non-trivial solutions, the determinant of the coefficients must vanish:
<math display="block">
\begin{vmatrix}
(\epsilon - n)^2 & i \delta & 0 \\
-i \delta & (\epsilon - n)^2 & 0 \\
0 & 0 & \epsilon
\end{vmatrix} = 0
</math>
Solving this characteristic equation yields <math>\quad n_{\pm}^2 = \epsilon \pm \delta</math>. Substituting these eigenvalues back into the linear equations gives:
<math display="block">
\mathbf{E}_y = \mp i \mathbf{E}_x \quad
\begin{cases}
\mathbf{E}_y = -i \mathbf{E}_x & n_+ \\
\mathbf{E}_y = i \mathbf{E}_x & n_-
\end{cases}
</math>
It is clear that the refractive indices for left- and right-circularly polarized light are different .
Next, we define the reflection coefficients for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">r_{\pm} = \frac{n_{\pm} - 1}{n_{\pm} + 1}</math>
Using these defined coefficients, we rewrite the reflected components for <math>\mathbf{E}_x + i \mathbf{E}_y</math> and <math>\mathbf{E}_x - i \mathbf{E}_y</math>:
<math display="block">
\begin{cases}
\mathbf{E}'_x + i \mathbf{E}'_y = r_+ (\mathbf{E}_x + i \mathbf{E}_y) \\
\mathbf{E}'_x - i \mathbf{E}'_y = r_- (\mathbf{E}_x - i \mathbf{E}_y)
\end{cases}
</math>
This can be rearranged into the following form:
<math display="block">
\begin{aligned}
\mathbf{E}'_x &= \frac{r_+ + r_-}{2} \mathbf{E}_x + \frac{i(r_+ - r_-)}{2} \mathbf{E}_y \\
\mathbf{E}'_y &= -\frac{i(r_+ - r_-)}{2} \mathbf{E}_x + \frac{r_+ + r_-}{2} \mathbf{E}_y
\end{aligned}
</math>
In matrix form, this is expressed as:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
\mathbf{E}_y
\end{bmatrix}
</math>
For incident light that is linearly polarized along the x-axis:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{E}_x \\
0
\end{bmatrix}
</math>
Evaluating the matrix multiplication gives:
<math display="block">
\begin{bmatrix}
\mathbf{E}'_x \\
\mathbf{E}'_y
\end{bmatrix} =
\begin{bmatrix}
r_{11} \mathbf{E}_x \\
r_{21} \mathbf{E}_x
\end{bmatrix}
</math>
We can then determine the small polarization change by defining the complex Kerr rotation angle:
<math display="block">\phi_k = \frac{\mathbf{E}'_y}{\mathbf{E}'_x} = \theta_k + i \epsilon_k = \frac{r_{21}}{r_{11}}</math>
Finally, we obtain the final expression:
<math display="block">\theta_k = \text{Im} \frac{n_+ - n_-}{1 - n_+ n_-} = \text{Im} \frac{\delta}{n(1 - \epsilon)}</math>

===Measuring Method===
As shown in the figure below, our setup employs the analyzer offset method (Suzuki and Beach, 2024).
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|Fig 8.MOKE Analyzer Offset Method]]
To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume <math>\theta \ll \varphi \ll \pi</math>. The intensity recorded by the detector, where <math>I_0 \equiv (E_0)^2</math> and <math>I_{\varphi} \equiv (E_0 \varphi)^2</math>, can then be written as:
<math display="block">
\begin{aligned}
I = (E_0)^2 |\theta + \varphi|^2 \\
= (E_0)^2 (\cancel{\theta^2} + 2\theta\varphi + \varphi^2) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= (E_0 \varphi)^2 (2\theta / \varphi + 1) \\
= I_{\varphi} (2\theta / \varphi + 1)
\end{aligned}
</math>
The differential intensity at the detector is then:
<math display="block">
\begin{aligned}
\Delta I = \Delta [I_0 |\theta + \varphi|^2] \\
= \Delta [I_0 (2\theta\varphi + \varphi^2)] \\
= (\Delta I_0)(\cancel{2\theta\varphi} + \varphi^2) + I_0 2(\Delta \theta)\varphi
\end{aligned}
</math>
This yields:
<math display="block">
\begin{aligned}
\frac{\Delta I}{I_{\varphi}} = \frac{\Delta I}{I_0 \varphi^2} = \frac{\Delta I_0}{I_0} + \color{red}{\frac{2(\Delta \theta)}{\varphi}} \\
\color{red}{\frac{2(\Delta \theta)}{\varphi}} = \frac{\Delta I}{I_{\varphi}} - \frac{\Delta I_0}{I_0}
\end{aligned}
</math>
For our configuration, the detected intensity at analyzer offset <math>\varphi</math> is:
<math display="block">
\begin{aligned}
\ I = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Two measurements, <math>I_+</math> and <math>I_-</math>, are taken with the analyzer set at <math>+\varphi</math> and <math>-\varphi</math> from the crossed position, respectively:
<math display="block">
\begin{aligned}
\ I_+ = I_{\varphi}\left(\frac{2\theta}{\varphi}+1\right)
\end{aligned}
</math>
<math display="block">
\begin{aligned}
\ I_- = I_{\varphi}\left(\frac{-2\theta}{\varphi}+1\right)
\end{aligned}
</math>
Flipping the sign of <math>\varphi</math> reverses the sign of the <math>\frac{2\theta}{\varphi}</math> term, while <math>I_\varphi=(E_0\varphi)^2</math> remains unchanged. Computing the sum and difference gives:
<math display="block">
\begin{aligned}
\ I_++I_- = 2I_\varphi \\
\ I_+-I_- = I_\varphi \times \frac{4\theta}{\varphi}
\end{aligned}
</math>
Taking their ratio:
<math display="block">
\begin{aligned}
\frac{I_+-I_-}{I_++I_-}=\frac{I_\varphi\times\frac{4\theta}{\varphi}}{2I_\varphi}=\frac{2\theta}{\varphi}
\end{aligned}
</math>
<math display="block">
\color{red}{\theta = \frac{\varphi}{2} \cdot \frac{I_+ - I_-}{I_+ + I_-}}
</math>
The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities <math>I_{+\varphi}</math> and <math>I_{-\varphi}</math> to determine the Kerr angle. The signal was collected using a Si photodetector connected to the A-channel input of an SR860 lock-in amplifier (Stanford Research Systems, 2016). A mechanical chopper operating at 400 Hz was inserted into the optical path as an external reference to improve the signal-to-noise ratio. Measurements were performed using a sensitivity of 30 mV and a time constant of 100 ms; 50 data points were recorded and averaged from the R-channel output following an auto-phase operation.

==Results==
===Sample Preparation===

Two samples were used in this experiment: an Fe thin film as the primary ferromagnetic sample, and a Si(111) wafer as a non-magnetic reference. Prior to measurement, the surface of the Fe film was mechanically polished using 8000-grit sandpaper to remove the native oxide layer, ensuring a clean metallic surface with minimal optical scattering and reliable magneto-optical response.

===Initial Test===

Fe was chosen as the primary sample due to its well-established ferromagnetic properties, making it an ideal candidate for MOKE hysteresis loop measurements. To verify the functionality of the optical setup before proceeding to the full field-sweep measurement, an initial test was performed by placing a permanent magnet beneath each sample and recording the Kerr signal. The Si(111) wafer, being non-magnetic, serves as a control reference: any signal detected from Si reflects the background noise floor of the system, while a measurably larger signal from Fe would confirm that the setup is correctly sensitive to magnetic contrast. The results of this initial test are presented in the table below:
{| class="wikitable" style="text-align: center; margin: auto;"
! Material !! I₊ (mV) !! I₋ (mV) !! Kerr Angle
|-
| Si(111) || 0.4694 || 0.4469 || 0.6428 mrad
|-
| Fe || 0.4627 || 0.5455 || 1.91 mrad
|}
===Second Experiment===
Having confirmed the system performance, we proceeded to measure the Fe film under a variable applied magnetic field using an electromagnet. The analyzer was set sequentially at <math>+\varphi</math> and <math>-\varphi</math> offset positions, and the Kerr signal was recorded at both positive and negative applied fields. The results are presented in the table and figure below:

{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| −68.24 || 5.37 || 3.5904 || 5.1995
|-
| −75.18 || 6.0926 || 3.1328 || 8.3994
|-
| −53.12 || 4.9763 || 3.0107 || 6.4429
|-
| −41.03 || 5.2575 || 4.0589 || 3.3682
|-
| −31.54 || 6.2806 || 4.9 || 3.2327
|-
| −20.566 || 4.4975 || 3.6979 || 2.5543
|-
| −10.208 || 4.9851 || 4.251 || 2.0808
|-
| 90.9 || 4.6897 || 4.9118 || −0.6056
|-
| 67.4 || 4.475 || 4.5594 || −0.2446
|-
| 54.7 || 4.9048 || 4.6664 || 0.6521
|-
| 42.58 || 5.0926 || 4.6235 || 1.2640
|}
[[File:Moke_measurement.png|1000px|thumb|center|Fig 9.MOKE First Measurement]]
The data represent a single branch field sweep from negative to positive applied field. The green star indicates the estimated remanence center at B ≈ 16.2 mT, defined as the midpoint between the two nearest measurement points flanking B = 0. The orange diamond marks the interpolated zero crossing of the Kerr angle at B ≈ 63.9 mT, which provides an estimate of the effective coercive field of the sample. The non-monotonic behavior observed on the negative field branch is attributed to measurement instability near the magnetization switching region. A complete hysteresis loop, obtained by sweeping the field symmetrically in both directions, would be required to precisely determine the coercive field and saturation Kerr angle.

===Third Experiment: Hysteresis Loop of Fe Film===
Then we tried to measure the hysteresis loop for the Fe film.
The analyzer was offset by <math>\varphi = 3^\circ</math> from the extinction position, and the photodetector output was connected to the SR860 lock-in amplifier with a time constant of 100 ms and sensitivity of 30 mV. Auto-phase was performed at remanence to zero the phase offset. The magnet was then ramped from <math>+B_\text{max}</math> to <math>-B_\text{max}</math> (descending branch) and back to <math>+B_\text{max}</math> (ascending branch). At each field step, the lock-in signal was recorded at both <math>+\varphi</math> and <math>-\varphi</math> analyzer positions to obtain <math>I_+</math> and <math>I_-</math> respectively, after waiting at least 5 time constants for the signal to stabilize.
Data points showing sign flips inconsistent with the branch trend or anomalous jumps were removed as outliers. The hysteresis loop was obtained by plotting <math>\theta</math> against <math>B</math>.
The measured data has shown below:
{| class="wikitable" style="text-align: center; margin: auto;"
! B (mT) !! I₊ (mV) !! I₋ (mV) !! Kerr Angle (mrad)
|-
| 6.08 || 3.1749 || 2.3326 || 4.00
|-
| 27.55 || 3.0486 || 2.3823 || 3.36
|-
| 38.78 || 2.8432 || 2.3599 || 2.53
|-
| 49.69 || 2.8973 || 1.9646 || 4.97
|-
| 57.39 || 3.1678 || 2.0887 || 4.91
|-
| 65.84 || 3.2119 || 1.9343 || 6.06
|-
| 76.98 || 3.2703 || 2.3975 || 4.06
|-
| 84.03 || 3.2657 || 2.2307 || 4.72
|-
| 89.75 || 3.3513 || 2.2199 || 5.01
|-
| 98.13 || 3.6786 || 2.313 || 5.62
|-
| 88.7 || 3.4073 || 2.1704 || 5.57
|-
| 75.72 || 3.0149 || 2.4514 || 2.75
|-
| 66.63 || 2.8241 || 2.366 || 2.27
|-
| 51.2 || 2.9293 || 2.4842 || 2.11
|-
| 42.21 || 2.8023 || 2.3409 || 2.30
|-
| 31.72 || 2.8866 || 2.2818 || 2.87
|-
| 17.87 || 3.103 || 2.4852 || 2.84
|-
| 7.98 || 3.0568 || 2.6102 || 1.98
|-
| −5.76 || 2.7932 || 2.3823 || 2.00
|-
| −15.08 || 2.8344 || 2.5606 || 1.28
|-
| −25.17 || 2.4213 || 2.5288 || −0.55
|-
| −47.76 || 2.3783 || 2.8969 || −2.53
|-
| −61.79 || 2.4034 || 2.9073 || −2.44
|-
| −75.38 || 2.4825 || 3.0117 || −2.49
|-
| −90.81 || 2.5461 || 3.4386 || −3.99
|-
| −94.01 || 2.3552 || 3.0396 || −3.33
|-
| −91.54 || 2.6405 || 3.488 || −3.78
|-
| −59.49 || 2.4391 || 2.9959 || −2.62
|-
| −38.79 || 2.3562 || 2.8498 || −2.46
|-
| −19.52 || 2.3611 || 2.8997 || −2.63
|-
| −6.65 || 2.4117 || 2.7919 || −1.85
|-
| 4.89 || 2.852 || 3.1563 || −1.46
|-
| 14.79 || 2.8426 || 3.0405 || −0.99
|-
| 23.52 || 3.0519 || 2.9247 || 0.62
|-
| 37.73 || 3.2323 || 2.8062 || 2.08
|-
| 50.65 || 2.9864 || 2.4205 || 2.67
|-
| 72.35 || 3.1982 || 2.6516 || 2.46
|-
| 87.52 || 3.5163 || 2.7766 || 3.28
|-
| 93.26 || 4.0844 || 3.2435 || 3.76
|}
Then our task is to draw the hysteresis loop. In MOKE, the measured Kerr rotation θ is linearly proportional to the sample's magnetization component along the probing direction. For polar MOKE (out-of-plane magnetization):
<math display="block">
\begin{aligned}
\theta_k=\frac{Q \cdot M_z}{M_s} \\
\theta_K \propto M
\end{aligned}
</math>
where <math>Q</math>is the magneto-optical constant (Voigt parameter) of the material, <math>M_z</math> is the local out-of-plane magnetization, and <math>M_s</math> is the saturation magnetization. Since <math>Q</math> and <math>M_s</math> are material constants, <math>\theta_K</math> is a direct linear proxy for the normalized magnetization <math>\frac{M}{M_s}</math>.
So we can substitute the magnetization with the <math>\theta_K</math>. The figure of the hysteresis loop is shown below:
[[File:Moke_hysteresis_loop.png|1000px|thumb|center|Fig 10.Fe Film hysteresis loop]]
The hysteresis loop is formed by two complementary field sweep branches. The descending branch (orange) was recorded from positive saturation at B ≈ +98 mT down to negative saturation at B ≈ −94 mT, while the return ascending branch (green) was recorded from negative saturation back toward positive fields up to B ≈ +93 mT. The two branches together form a well-defined closed hysteresis loop, confirming the ferromagnetic nature of the Fe film. At large positive applied fields, the Kerr angle saturates at approximately +6.0 mrad, while at large negative fields it saturates at approximately −3.9 mrad. The slight asymmetry between the positive and negative saturation values is likely attributable to a non-zero background offset in the optical alignment or a small systematic imbalance in the analyzer offset calibration. The coercive fields, determined by linear interpolation of the zero-crossings of each branch, are Hc⁻ ≈ −22.1 mT for the descending branch and Hc⁺ ≈ +20.1 mT for the return ascending branch, yielding an average coercive field of Hc ≈ 21.1 mT. The near-symmetric coercive fields indicate negligible exchange bias in the film. The finite remanent Kerr angle of approximately ±2.5 mrad at B = 0 on the two branches reflects a substantial remanent magnetization, with a squareness ratio Mr/Ms ≈ 0.42, suggesting the Fe film possesses moderate but not fully uniaxial magnetic anisotropy. The initial ascending sweep (blue) is included for reference but does not form part of the closed loop, as it represents a non-equilibrium first-magnetization curve recorded prior to full magnetic saturation being established. The overall loop shape, with its gradual rather than abrupt switching flanks, is consistent with a polycrystalline Fe film undergoing domain-wall mediated reversal across a distribution of local pinning fields rather than coherent single-domain rotation.

==Conclusion and Discussion==
=== Magnetic Properties of Fe Film ===

In this experiment, the magneto-optical Kerr effect (MOKE) was successfully employed to characterize the magnetic properties of an Fe thin film. The initial test confirmed the functionality of the optical setup, yielding a Kerr rotation of 1.91 mrad for Fe compared to 0.6428 mrad for the non-magnetic Si(111) reference, demonstrating clear contrast between magnetic and non-magnetic samples.

The subsequent hysteresis loop measurement revealed the ferromagnetic nature of the Fe film, with saturation Kerr angles of approximately +6.0 mrad and −3.9 mrad at positive and negative applied fields respectively. The coercive field was determined to be <math>H_c \approx 21.1</math> mT, extracted from the average of the two zero-crossing fields <math>H_c^- \approx -22.1</math> mT and <math>H_c^+ \approx +20.1</math> mT on the descending and ascending branches respectively. The near-symmetric coercive fields indicate negligible exchange bias in the film. The remanent Kerr angle of approximately ±2.5 mrad yields a squareness ratio of <math>M_r/M_s \approx 0.42</math>, suggesting moderate but not fully uniaxial magnetic anisotropy. The gradual switching flanks observed in the loop are consistent with domain-wall mediated magnetization reversal across a distribution of local pinning fields in a polycrystalline film, rather than coherent single-domain rotation.

=== Limitations and Suggested Improvements ===

However, several limitations of the current setup were identified. The laser source used in this experiment had a maximum output power of only 1 mW, which resulted in relatively weak photodetector signals and a reduced signal-to-noise ratio. Since the Kerr rotation signal is proportional to the incident intensity, a low-power source directly limits the measurable signal level. Furthermore, the slight asymmetry between the positive and negative saturation Kerr angles is likely attributable to intensity instability of the laser, which introduces a systematic drift in the recorded <math>I_+</math> and <math>I_-</math> values. Better signal quality would therefore require a higher-power and more stable laser source.

In terms of detection scheme, the current single-detector setup measures <math>I_+</math> and <math>I_-</math> sequentially by rotating the analyzer, which makes the measurement susceptible to laser intensity fluctuations between the two measurements. A balanced detection scheme, in which the beam is split by a polarizing beamsplitter and both orthogonal polarization components are measured simultaneously by two matched photodetectors, would offer significant advantages. The comparison is summarized below:

{| class="wikitable" style="text-align: center; margin: auto;"
! Feature !! Single Detector (current) !! Balanced Detection
|-
| Intensity noise rejection || Poor — fluctuations affect each measurement independently || Excellent — common-mode noise cancels in <math>I_+ - I_-</math>
|-
| Sensitivity || Limited by laser noise floor || Up to <math>\sqrt{2}</math> improvement in SNR
|-
| Measurement speed || Sequential <math>I_+</math> and <math>I_-</math> acquisition || Simultaneous acquisition of both channels
|-
| Systematic offset || Susceptible to drift between measurements || Drift largely cancelled by differential measurement
|-
| Complexity || Simple, low cost || Requires matched detector pair and beamsplitter
|}

Overall, upgrading to a higher-power stable laser and adopting a balanced detection scheme would substantially improve the signal quality and measurement reliability, enabling more precise determination of the coercive field, remanence, and saturation Kerr angle in future experiments.

==Reference==
1. McCord, J. Progress in magnetic domain observation by advanced magneto-optical microscopy. J. Phys. D: Appl. Phys. 48, 333001 (2015).

2. Erskine J L, Stern E A. Magneto-optic Kerr effects in gadolinium[J]. Physical Review B, 1973, 8(3): 1239

3. Bader, S.D. (1999). Surface magneto-optic Kerr effect (SMOKE). ''Journal of Magnetism and Magnetic Materials'', 200(1–3), 664–678. https://doi.org/10.1016/S0304-8853(99)00311-X

4. Haider, T. (2017). A Review of Magneto-Optic Effects and Its Application. ''International Journal of Electromagnetics and Applications'', 7(1), 17–24. https://doi.org/10.5923/j.ijea.20170701.03

5. Kato, T. et al. (2022). Fundamentals of Magneto-Optical Spectroscopy. ''Frontiers in Physics'', 10, 946515. https://doi.org/10.3389/fphy.2022.946515

6. Qiu, Z.Q. and Bader, S.D. (2000). Surface magneto-optic Kerr effect. ''Review of Scientific Instruments'', 71(3), 1243–1255. https://doi.org/10.1063/1.1150496

7. Stobiecki, T. et al. (2011). Scanning magneto-optical Kerr microscope with auto-balanced detection scheme. ''Review of Scientific Instruments'', 82, 083703. https://doi.org/10.1063/1.3610967

8. Suzuki, D.H. and Beach, G.S.D. (2024). Measurement of Kerr rotation and ellipticity in magnetic thin films by MOKE magnetometry. ''Journal of Applied Physics'', 135, 063901. https://doi.org/10.1063/5.0185341

9. Wikipedia contributors. (2026). Magneto-optic Kerr effect. ''Wikipedia, The Free Encyclopedia''. https://en.wikipedia.org/wiki/Magneto-optic_Kerr_effect

10. Zhang, H. et al. (2022). Measurement of DC Magneto-Optical Kerr Effect with Sensitivity of <math>10^{-7}</math> rad/<math>\sqrt{\text{Hz}}</math>. ''arXiv preprint'', arXiv:2201.11744. https://arxiv.org/abs/2201.11744

11. Oppeneer, P.M. (2001). Magneto-optical Kerr spectra. In K.H.J. Buschow (Ed.), ''Handbook of Magnetic Materials'', Vol. 13, pp. 229–422. Elsevier.

12. Stanford Research Systems. (2016). ''SR860 Lock-In Amplifier User Manual''. https://www.thinksrs.com/downloads/pdfs/manuals/SR860m.pdf

==Appendix==
===Code for Reading Data from SR860 Lock-in Amplifier===
<pre>
from srsinst.sr860 import SR860
import time
import numpy as np

# Connect
lockin = SR860('visa', 'USB0::0xB506::0x2000::006011::INSTR')

# Configure for external reference (chopper)
lockin.ref.reference_source = 'external'
lockin.signal.input = 'A'
lockin.signal.sensitivity = 30e-3 # 30 mV
lockin.signal.time_constant = 100e-3 # 100 ms

# Check detected frequency
print(f"Detected frequency: {lockin.data.value['ext. freq.']:.2f} Hz")

# Wait for signal to stabilize
time.sleep(0.5)

# Auto phase
lockin.ref.auto_phase()
print(f"Auto phase done. Phase = {lockin.ref.phase:.4f} deg")

# Wait for lock-in to settle after auto phase (5 time constants)
time.sleep(5 * 0.1)

# Collect 50 points
N = 50
X_arr = np.zeros(N)
Y_arr = np.zeros(N)
R_arr = np.zeros(N)
theta_arr = np.zeros(N)

print(f"\nCollecting {N} points...")
for i in range(N):
X_arr[i] = lockin.data.value['X']
Y_arr[i] = lockin.data.value['Y']
R_arr[i] = lockin.data.value['R']
theta_arr[i] = lockin.data.value['Theta']
time.sleep(0.1)
if (i+1) % 10 == 0:
print(f" {i+1}/{N} points collected")

# Average
X_mean = np.mean(X_arr)
Y_mean = np.mean(Y_arr)
R_mean = np.mean(R_arr)
theta_mean = np.mean(theta_arr)

X_std = np.std(X_arr)
Y_std = np.std(Y_arr)
R_std = np.std(R_arr)

print(f"\n--- Averaged Results ({N} points) ---")
print(f"X = {X_mean*1e3:.4f} ± {X_std*1e3:.4f} mV")
print(f"Y = {Y_mean*1e3:.4f} ± {Y_std*1e3:.4f} mV")
print(f"R = {R_mean*1e3:.4f} ± {R_std*1e3:.4f} mV")
print(f"Theta = {theta_mean:.4f} deg")
</pre>