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

with the following corresponding boundary conditions:

We assume that no free charges or free currents exist at the interface between the two mediums: an ideal ferromagnetic medium with $\overrightarrow{B}=\mu \overrightarrow{H}+{\mu }_0{\overrightarrow{M}}_0$ (Eqn. 1) and a homogeneous linear medium, following the diagram below. With the magnetization ${\overrightarrow{M}}_0$ 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.

Coordinate System with Corresponding Media in the MOKE

Depending on whether ${\overrightarrow{M}}_0$ 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 ${\theta }_k$ 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 ${\overrightarrow{M}}_0$. 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.

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

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.

MOKE Setup upstream

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.

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 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:

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: $$ϵ=\left(\begin{array}{ccc}{ϵ}_{11}& {ϵ}_{12}& {ϵ}_{13}\\ {ϵ}_{21}& {ϵ}_{22}& {ϵ}_{23}\\ {ϵ}_{31}& {ϵ}_{32}& {ϵ}_{33}\end{array}\right)+i\left(\begin{array}{ccc}0& {ϵ}_{12}& {ϵ}_{13}\\ -{ϵ}_{21}& 0& {ϵ}_{23}\\ -{ϵ}_{31}& -{ϵ}_{32}& 0\end{array}\right)$$ Then the permittivity tensor can be simplified as:

$$ϵ=\left(\begin{array}{ccc}{ϵ}_x& i\sigma & 0\\ -i\sigma & {ϵ}_y& 0\\ 0& 0& {ϵ}_z\end{array}\right)$$

From electrodynamics, we have: $\mathbf{𝐃}={ϵ}_0ϵ\mathbf{𝐄}$, then we combine it with Faraday's law and Ampère's law: $$\{\begin{array}{l}\mathrm{\nabla }\times \mathbf{𝐄}=-{\mu }_0\frac{\partial \mathbf{𝐇}}{\partial t}\\ \mathrm{\nabla }\times \mathbf{𝐇}={ϵ}_0ϵ\frac{\partial \mathbf{𝐄}}{\partial t}\end{array}$$ Assuming the incident laser beam is a plane wave, then: $$\{\begin{array}{l}\mathbf{𝐄}={\mathbf{𝐄}}_0{e}^{i(\mathbf{𝐤}\mathbf{𝐫}-\omega t)}={\mathbf{𝐄}}_0{e}^{i(\frac{n}{c}\mathbf{𝐬}\cdot \mathbf{𝐫}-t)}\\ \mathbf{𝐇}={\mathbf{𝐇}}_0{e}^{i(\mathbf{𝐤}\mathbf{𝐫}-\omega t)}={\mathbf{𝐇}}_0{e}^{i(\frac{n}{c}\mathbf{𝐬}\cdot \mathbf{𝐫}-t)}\end{array}$$ Therefore, $$\{\begin{array}{l}\frac{n}{c}\mathbf{𝐄}\times \mathbf{𝐒}=-{\mu }_0\mathbf{𝐇}\\ \frac{n}{c}\mathbf{𝐇}\times \mathbf{𝐒}={ϵ}_0ϵ\mathbf{𝐄}\end{array}$$ Combining our previous equations yields: $$\frac{n}{c}(-\frac{1}{{\mu }_0}\frac{n}{c}\mathbf{𝐄}\times \mathbf{𝐒})\times \mathbf{𝐒}={ϵ}_0ϵ\mathbf{𝐄}$$ Solving this equation, we obtain: $$ϵ\mathbf{𝐄}=n^2[\mathbf{𝐄}-\mathbf{𝐒}(\mathbf{𝐒}\cdot \mathbf{𝐄})]$$ Then, by substituting the simplified permittivity tensor, we have: $$\{\begin{array}{l}(ϵ-n{)}^2{\mathbf{𝐄}}_x+i\delta {\mathbf{𝐄}}_y=0\\ -i\delta {\mathbf{𝐄}}_x+(ϵ-n{)}^2{\mathbf{𝐄}}_y=0\\ ϵ{\mathbf{𝐄}}_z=0\end{array}$$ For non-trivial solutions, the determinant of the coefficients must vanish: $$\left|\begin{array}{ccc}(ϵ-n{)}^2& i\delta & 0\\ -i\delta & (ϵ-n{)}^2& 0\\ 0& 0& ϵ\end{array}\right|=0$$ Solving this characteristic equation yields $n_{\pm }^2=ϵ\pm \delta$. Substituting these eigenvalues back into the linear equations gives: $${\mathbf{𝐄}}_y=\mp i{\mathbf{𝐄}}_x\{\begin{array}{ll}{\mathbf{𝐄}}_y=-i{\mathbf{𝐄}}_x& n_{+}\\ {\mathbf{𝐄}}_y=i{\mathbf{𝐄}}_x& n_{-}\end{array}$$ It is clear that the refractive indices for left- and right-circularly polarized light are different. Next, we define the reflection coefficients for ${\mathbf{𝐄}}_x+i{\mathbf{𝐄}}_y$ and ${\mathbf{𝐄}}_x-i{\mathbf{𝐄}}_y$: $$r_{\pm }=\frac{n_{\pm }-1}{n_{\pm }+1}$$ Using these defined coefficients, we rewrite the reflected components for ${\mathbf{𝐄}}_x+i{\mathbf{𝐄}}_y$ and ${\mathbf{𝐄}}_x-i{\mathbf{𝐄}}_y$: $$\{\begin{array}{l}\mathbf{𝐄}{\text{'}}_x+i\mathbf{𝐄}{\text{'}}_y=r_{+}({\mathbf{𝐄}}_x+i{\mathbf{𝐄}}_y)\\ \mathbf{𝐄}{\text{'}}_x-i\mathbf{𝐄}{\text{'}}_y=r_{-}({\mathbf{𝐄}}_x-i{\mathbf{𝐄}}_y)\end{array}$$ This can be rearranged into the following form: $$\begin{array}{rl}\mathbf{𝐄}{\text{'}}_x& =\frac{r_{+}+r_{-}}{2}{\mathbf{𝐄}}_x+\frac{i(r_{+}-r_{-})}{2}{\mathbf{𝐄}}_y\\ \mathbf{𝐄}{\text{'}}_y& =-\frac{i(r_{+}-r_{-})}{2}{\mathbf{𝐄}}_x+\frac{r_{+}+r_{-}}{2}{\mathbf{𝐄}}_y\end{array}$$ In matrix form, this is expressed as: $$\left[\begin{array}{c}\mathbf{𝐄}{\text{'}}_x\\ \mathbf{𝐄}{\text{'}}_y\end{array}\right]=\left[\begin{array}{cc}{r}_{11}& r_{12}\\ r_{21}& r_{22}\end{array}\right]\left[\begin{array}{c}{\mathbf{𝐄}}_x\\ {\mathbf{𝐄}}_y\end{array}\right]$$ For incident light that is linearly polarized along the x-axis: $$\left[\begin{array}{c}\mathbf{𝐄}{\text{'}}_x\\ \mathbf{𝐄}{\text{'}}_y\end{array}\right]=\left[\begin{array}{cc}{r}_{11}& r_{12}\\ r_{21}& r_{22}\end{array}\right]\left[\begin{array}{c}{\mathbf{𝐄}}_x\\ 0\end{array}\right]$$ Evaluating the matrix multiplication gives: $$\left[\begin{array}{c}\mathbf{𝐄}{\text{'}}_x\\ \mathbf{𝐄}{\text{'}}_y\end{array}\right]=\left[\begin{array}{c}{r}_{11}{\mathbf{𝐄}}_x\\ r_{21}{\mathbf{𝐄}}_x\end{array}\right]$$ We can then determine the small polarization change by defining the complex Kerr rotation angle: $${\varphi }_k=\frac{\mathbf{𝐄}{\text{'}}_y}{\mathbf{𝐄}{\text{'}}_x}={\theta }_k+i{ϵ}_k=\frac{r_{21}}{r_{11}}$$ Finally, we obtain the final expression: $${\theta }_k=\text{Im}\frac{n_{+}-n_{-}}{1-n_{+}{n}_{-}}=\text{Im}\frac{\delta }{n(1-ϵ)}$$

Measuring Method

As shown in the figure below, our setup employs the analyzer offset method.

MOKE analyzer offset method

To extract the Kerr rotation using a small angular offset from the analyzer's crossed position, we assume $\theta \ll \phi \ll \pi$. The intensity recorded by the detector, where $I_0\equiv (E_0{)}^2$ and $I_{\phi }\equiv (E_0\phi {)}^2$, can then be written as: $$\begin{array}{r}I=(E_0{)}^2|\theta +\phi {|}^2\\ =(E_0{)}^2(\cancel{{\theta }^2}+2\theta \phi +{\phi }^2)\\ =(E_0\phi {)}^2(2\theta /\phi +1)\\ =(E_0\phi {)}^2(2\theta /\phi +1)\\ =I_{\phi }(2\theta /\phi +1)\end{array}$$ The differential intensity at the detector is then: $$\begin{array}{r}\mathrm{\Delta }I=\mathrm{\Delta }[I_0|\theta +\phi {|}^2]\\ =\mathrm{\Delta }[I_0(2\theta \phi +{\phi }^2)]\\ =(\mathrm{\Delta }{I}_0)(\cancel{2\theta \phi }+{\phi }^2)+I_{0}2(\mathrm{\Delta }\theta )\phi \end{array}$$ This yields: $$\begin{array}{r}\frac{\mathrm{\Delta }I}{I_{\phi }}=\frac{\mathrm{\Delta }I}{I_0{\phi }^2}=\frac{\mathrm{\Delta }{I}_0}{I_0}+\frac{2(\mathrm{\Delta }\theta )}{\phi }\\ \frac{2(\mathrm{\Delta }\theta )}{\phi }=\frac{\mathrm{\Delta }I}{I_{\phi }}-\frac{\mathrm{\Delta }{I}_0}{I_0}\end{array}$$ For our configuration, the detected intensity at analyzer offset $\phi$ is: $$\begin{array}{r}I=I_{\phi }(\frac{2\theta }{\phi }+1)\end{array}$$ Two measurements, $I_{+}$ and $I_{-}$, are taken with the analyzer set at $+\phi$ and $-\phi$ from the crossed position, respectively: $$\begin{array}{r}{I}_{+}=I_{\phi }(\frac{2\theta }{\phi }+1)\end{array}$$ $$\begin{array}{r}{I}_{-}=I_{\phi }(\frac{-2\theta }{\phi }+1)\end{array}$$ Flipping the sign of $\phi$ reverses the sign of the $\frac{2\theta }{\phi }$ term, while $I_{\phi }=(E_0\phi {)}^2$ remains unchanged. Computing the sum and difference gives: $$\begin{array}{r}{I}_{+}+I_{-}=2I_{\phi }\\ I_{+}-I_{-}=I_{\phi }\times \frac{4\theta }{\phi }\end{array}$$ Taking their ratio: $$\begin{array}{r}\frac{I_{+}-I_{-}}{I_{+}+I_{-}}=\frac{I_{\phi }\times \frac{4\theta }{\phi }}{2I_{\phi }}=\frac{2\theta }{\phi }\end{array}$$ $$\theta =\frac{\phi }{2}\cdot \frac{I_{+}-I_{-}}{I_{+}+I_{-}}$$ The experimental procedure therefore consists of first locating the extinction position of the analyzer, and subsequently measuring the intensities $I_{+\phi }$ and $I_{-\phi }$ 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:

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 the Kerr angle between the Fe film (should have magnetism) and the Si film (shouldn't have magnetism), we suppose that our system is successfully working. The rotated Kerr angle for Fe (1.91 mrad) is in a reasonable range. Then we measured the Fe film at an adjustable magnetic field. The following figure shows the MOKE signal at positive and negative magnetic fields:

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

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

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")