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:

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 want to utilize a 658 nm HL6501 red light CW (continuous wavelength) laser (according to the datasheet). Then the laser beam passes through a ND filter to decrease its intensity the first time. Then it will go through a polarizer and a half-wave plate set to make it intensity continuously adjustable and be initially polarized to S polarized or P polarized. Then the incident laser light is focused on the sample by using a lens/objective. Then the reflected signal is detected by using a Wollaston prism as an analyzer to first, splits the incident signal light beam into two orthogonal, linearly polarized beams that diverge from each other. Then use two detectors (balanced detectors) to detect two orthogonal, linear polarized beam intensities. The small Kerr rotation of the polarization by the material's magnetic properties can be calculated by making a substract of two intensities read from the two detecters.

MOKE Setup upstream

The figure shows our current MOKE setup. We temporarily utilize a 532nm green light laser, then the laser beam pass through a quarter wave to reconstruct the laser polarization state to circular polarized. Then a rotate round shape ND filter was put into optical path so that we can control the light intensity continuously. Then we utilized several iris sets to collimate the optical path. Then we use a PBS(polarizing beamsplitter cube) to get pure S polarized light. The other P polarized light was blocked by a beam blocker. Then the beam was focused on the sample by a lens which f=20mm, then we used a D shape mirror to directed the reflected signal in to a Glan-Taylor calcite Polarizers which extinction ratio for output beam is 100 000:1. The Glan-Tylar was originally set cross polarized to the laser beam without sample. By using Glan-Tylar calcite polarizers, we can capture the slight signals caused by the magnetism of the material. Then the signal was detected by the Si based detector.

Methods

1. MOKE TheoryThe permittivity of a magnetic material can be expressed by a complex tensor. In general, it consists of a symmetric part representing the isotropic properties and an antisymmetric part representing the magneto-optic effects:Failed to parse (unknown function "\begin{pmatrix}"): {\displaystyle \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}} For a material magnetized along the $z$-axis, the permittivity tensor can be simplified as:Failed to parse (unknown function "\begin{pmatrix}"): {\displaystyle \epsilon = \begin{pmatrix}\epsilon_x & i\delta & 0 \-i\delta & \epsilon_y & 0 \0 & 0 & \epsilon_z\end{pmatrix}} 1.1 Wave Equation in Magneto-optic MediaFrom electrodynamics, the constitutive relation is given by $\mathbf{D} = \epsilon_0 \epsilon \mathbf{E}$. Combining this with Faraday's Law and Ampere's Law:$$\{\begin{array}{l}\mathrm{\nabla }\times \mathbf{𝐄}=-{\mu }_0\frac{\partial \mathbf{𝐇}}{\partial t}\\ nabla\times \mathbf{𝐇}={ϵ}_0ϵ\frac{\partial \mathbf{𝐄}}{\partial t}\end{array}$$Assuming the incident laser beam is a monochromatic plane wave:$$\{\begin{array}{l}\mathbf{𝐄}={\mathbf{𝐄}}_0{e}^{i(\mathbf{𝐤}\cdot \mathbf{𝐫}-\omega t)}={\mathbf{𝐄}}_0{e}^{i\omega (\frac{n}{c}\mathbf{𝐬}\cdot \mathbf{𝐫}-t)}\\ mathbfH={\mathbf{𝐇}}_0{e}^{i(\mathbf{𝐤}\cdot \mathbf{𝐫}-\omega t)}={\mathbf{𝐇}}_0{e}^{i\omega (\frac{n}{c}\mathbf{𝐬}\cdot \mathbf{𝐫}-t)}\end{array}$$Substituting these plane wave solutions into Maxwell's equations yields:$$\{\begin{array}{l}\frac{n}{c}\mathbf{𝐬}\times \mathbf{𝐄}={\mu }_0\mathbf{𝐇}\\ fracnc\mathbf{𝐬}\times \mathbf{𝐇}=-{ϵ}_0ϵ\mathbf{𝐄}\end{array}$$By eliminating $\mathbf{H}$ from the equations above, we obtain the wave equation for the electric field:$$\frac{n^2}{c^2{\mu }_0{ϵ}_0}\mathbf{𝐬}\times (\mathbf{𝐬}\times \mathbf{𝐄})=-ϵ\mathbf{𝐄}$$Since $c^2 = 1/(\mu_0 \epsilon_0)$, this simplifies to the eigen equation:$$ϵ\mathbf{𝐄}=n^2[\mathbf{𝐄}-\mathbf{𝐬}(\mathbf{𝐬}\cdot \mathbf{𝐄})]$$1.2 Eigenvalues and EigenmodesUsing the simplified permittivity tensor for a wave propagating along the $z$-axis ($\mathbf{s} = \mathbf{e}_z$), the system of linear equations for the field components is:Failed to parse (unknown function "\begin{cases}"): {\displaystyle \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_z \mathbf{E}_z = 0\end{cases}} For non-trivial solutions, the determinant of the coefficients must vanish:Failed to parse (unknown function "\begin{vmatrix}"): {\displaystyle \begin{vmatrix}\epsilon - n^2 & i \delta & 0 \-i \delta & \epsilon - n^2 & 0 \0 & 0 & \epsilon_z\end{vmatrix} = 0} Solving this characteristic equation yields the eigenvalues for the refractive indices:$$n_{\pm }^2=ϵ\pm \delta$$Substituting these back into the linear equations, we find the corresponding eigenmodes, which represent Left and Right Circularly Polarized (LCP/RCP) light:$${\mathbf{𝐄}}_y=\mp i{\mathbf{𝐄}}_x⟹\{\begin{array}{ll}{\mathbf{𝐄}}_y=-i\mathbf{𝐄}x& (n+)\\ mathbf{E}_y=i\mathbf{𝐄}x& (n-)\end{array}$$1.3 Reflection and Jones MatrixThe complex reflection coefficients $r_{\pm}$ for the two circular modes at the interface are given by the Fresnel equations:$$r_{\pm }=\frac{n_{\pm }-1}{n_{\pm }+1}$$We can express the reflected field components $\mathbf{E}'$ in terms of the circular basis:$$\{\begin{array}{l}\mathbf{𝐄}{\text{'}}_x+i{\mathbf{𝐄}}^{\prime }y=r+({\mathbf{𝐄}}_x+i{\mathbf{𝐄}}_y)\\ mathbfE{\text{'}}_x-i{\mathbf{𝐄}}^{\prime }y=r-({\mathbf{𝐄}}_x-i{\mathbf{𝐄}}_y)\end{array}$$Solving for the Cartesian components $\mathbf{E}'_x$ and $\mathbf{E}'_y$:Failed to parse (unknown function "\begin{aligned}"): {\displaystyle \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}} This transformation can be compactly written using the Jones Matrix $\mathbf{R}$:Failed to parse (unknown function "\begin{bmatrix}"): {\displaystyle \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}} 1.4 Complex Kerr RotationFor an incident beam that is linearly polarized along the $x$-axis ($\mathbf{E}_y = 0$):Failed to parse (unknown function "\begin{bmatrix}"): {\displaystyle \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} = \begin{bmatrix} r{11} \mathbf{E}x \ r{21} \mathbf{E}_x\end{bmatrix}} The polarization change is characterized by the complex Kerr rotation $\phi_k$, where the real part $\theta_k$ is the rotation angle and the imaginary part $\epsilon_k$ is the ellipticity:$${\varphi }_k=\frac{{\mathbf{𝐄}}^{\prime }y}{{\mathbf{𝐄}}^{\prime }x}={\theta }_k+i{ϵ}_k=\frac{r21}{r11}$$By substituting the expressions for $r_{\pm}$ and $n_{\pm}$, we arrive at the final analytical expression for the Kerr rotation:$${\theta }_k=\text{Im}\left(\frac{n_{+}-n_{-}}{1-n_{+}{n}_{-}}\right)=\text{Im}\left(\frac{\delta }{n(1-ϵ)}\right)$$

Reference

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