Editing Optical Sensor of Magnetic Dynamics: A Balanced-Detection MOKE Magnetometer

==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==
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.

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> (Eqn. 1) 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|Coordinate System with Corresponding Media in the 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>. Consequently, the polar Kerr effect is seen most with light that is nearly perpendicularly incident on the material surface. This is also called S-polarization, where the incident electric field is nearly perpendicular to the surface while the incident magnetic field is nearly parallel. On the other hand, the longitudinal Kerr effect is most observed when light is nearly parallel to the material surface, or P-polarized, with the magnetic field perpendicular to the surface and the electric field nearly parallel.
[[File:MOKEgeometries.png|400px|thumb|center]]
Fundamentally, the MOKE may be used as a measure of how strongly a material is magnetized, with applications for the effect ranging from materials characterization to Kerr microscopy, where

==Experimental Setup==
[[File:Moke setup.png|1000px|thumb|center|MOKE Experimental Setup scheme]]
We employed a 658 nm HL6501 red continuous-wave (CW) laser as the light source. The beam first passes through a neutral density (ND) filter to attenuate its initial intensity, followed by a polarizer and a half-wave plate combination that enables continuous intensity adjustment while defining the initial polarization state as either s- or p-polarized. The beam is then focused onto the sample using a lens or objective. Upon reflection, the signal is analyzed using a Wollaston prism, which splits the reflected beam into two orthogonally and linearly polarized components that diverge from one another. These two beams are detected simultaneously by a pair of balanced photodetectors, and the small Kerr rotation induced by the material's magnetic properties is extracted by computing the difference between the two measured intensities.
[[File:Moke_upstream.png|1000px|thumb|center|MOKE Setup upstream]]
[[File:Moke_downstream.png|1000px|thumb|center|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.
[[File:Moke_sample_image_system.png|1000px|thumb|center|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|MOKE Setup sample image system]]
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|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.
[[File:Moke_analyzer_offset_method.png|1000px|thumb|center|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. 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==
Our measurements utilized Fe film and Si film(as a reference) to detect the Kerr angle. The ultimate goal is to measure the hysteresis loop of the Fe film, as Fe served as a good ferromagnetic material. Firstly, We measured the Fe film and the Si to test our system is working or not under specific magnetic field adding by a permanent magnet. The following is the result of the measuring for Si and Fe film testing:
{| 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
|}
As there shows clearly Kerr Angle between Fe film(should have magnetism) and Si film(shouldn't have magnetism), we supposed that our system is successfully working. Then we measured the Fe film at adjustable magnetic field. The following figure shows the MOKE signal at positive and negative magnetic field:

==Conclusion and Discussion==

==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

==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>