Tutorial: Magneto-optical Kerr Effect¶

Magneto-optical Kerr Effect (MOKE) microscopy is an optical technique that can be used to image the magnetisation structures of samples. This technique uses the change in polarisation in light through a magnetic media in order to detect the magnetisation.

The current version of MOKE microscopy is under development and currently the angle of incidence is approximated to be the same throughout the sample. This is not correct but allows roughly behaviour to be seen and we are looking to correct this in future releases. Please feel free to contact us or raise an issue if you have any comments on this technique.

MOKE reference frame¶

In mag2exp the coordinate system is defined with the beam in the yz plane with $\theta$ defined as the angle between the beam direction and the z direction.

The micromagnetic simulation¶

A micromagnetic simulation can be set up using Ubermag to obtain a 3-dimensional magntic structure.

Plot the initial magnetisation:

Relax the system and plot its magnetisation.

From the magnetisation, we can compute the MOKE related quantities.

Computing MOKE¶

Calculation of the magneto-optical Kerr effect is based off an incident wave $E^i$ described by \begin{equation} E^i = \begin{pmatrix} E^i_s \\ E^i_p \end{pmatrix}, \end{equation} where $E^i_s$ and $E^i_p$ are the electric field components of incident linear s and p polarisation modes.

i.e. \begin{equation} E^i = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \end{equation} is s polarised,

\begin{equation} E^i = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \end{equation}

is p polarised,

and \begin{equation} E^i = \begin{pmatrix} 1 \\ 1j \end{pmatrix}, \end{equation} is circularly polarised.

The reflected wave can be obtained by multiplication by the reflection matrix \begin{equation} \begin{pmatrix} E^r_s \\ E^r_p \end{pmatrix} = \begin{pmatrix} r_{ss} & r_{sp} \\ r_{ps} & r_{pp} \end{pmatrix} \begin{pmatrix} E^i_s \\ E^i_p \end{pmatrix} \end{equation}. Similarly, the transmitted wave can be obtained by multiplication by the reflection matrix \begin{equation} \begin{pmatrix} E^r_s \\ E^r_p \end{pmatrix} = \begin{pmatrix} t_{ss} & t_{sp} \\ t_{ps} & t_{pp} \end{pmatrix} \begin{pmatrix} E^i_s \\ E^i_p \end{pmatrix} \end{equation}.

These matrices can be obtained from the 4-by-4 product matrix $M$ which can be descried as four 2-by-2 matrices using \begin{align} M &= \begin{pmatrix} G & H \\ I & J \end{pmatrix}, \\ G^{-1} &= \begin{pmatrix} t_{ss} & t_{sp} \\ t_{ps} & t_{pp} \end{pmatrix},\\ I^{-1} &= \begin{pmatrix} r_{ss} & r_{sp} \\ r_{ps} & r_{pp} \end{pmatrix}. \end{align}

The product matrix $M$ is calculated based of the the system being studied by dividing the system up into layers of magnetisation in the $z$ direction. Each of these layers have a thickness and boundaries, in order to describe this we can use the boundary matrix of the $j$th layer $A_j$ and the propagation matrix of the layer $D_{j}$. This leads to a description of a single layer as $A_j D_j A^{-1}_j$. $A_{j}$ and $D_{j}$ are both 4-by-4 matrices.

If we assume are material is surrounded by free space, the whole system can thus be described by the product matrix $M$ withe the following form \begin{equation} M = A_f^{-1} \prod_{j} A_{j} D_j A_j^{-1} A_f, \end{equation} where $A_j$ is the boundary matrix for the $j$th layer, $D_j$ is the propagation matrix for the $j$th layer, and $A_f$ is the boundary layer matrix for free space.

The boundary matrix can be written as \begin{equation} A_{j} = \begin{pmatrix} 1 & 0 & 1 & 0 \\ \frac{iQ\alpha^2_{yi}}{2} \left(m_y \frac{1+\alpha^2_{zi}} {\alpha_{yi}\alpha_{zi}} - m_z\right) & \alpha_{zi} & -\frac{iQ\alpha^2_{yi}}{2} \left(m_y \frac{1+\alpha^2_{zi}} {\alpha_{yi}\alpha_{zi}} + m_z\right) & -\alpha_{zi} \\ -\frac{in_jQ}{2} \left(m_y \alpha_{yi} + m_z \alpha_{zi} \right) & -n_j & -\frac{in_jQ}{2} \left(m_y \alpha_{yi} - m_z \alpha_{zi} \right) & -n_j \\ n_j\alpha_{zj} & -\frac{in_jQ}{2} \left(m_y \frac{\alpha_{yi}} {\alpha_{zi}} - m_z \right) & -n_j \alpha_{zj} & \frac{in_jQ}{2} \left(m_y \frac{\alpha_{yi}} {\alpha_{zi}} + m_z \right) \end{pmatrix} \end{equation} where $\alpha_{yi}=\sin\theta_j, \alpha_{zi}=\cos\theta_j$, $\theta_j$ is the complex refractive angle, $Q$ is the Voight parameter and $n_j$ is the refractive index of the $j$th layer. The angle is measure with respect to the $z$ axis and is calculated using Snell's law. $m_x$, $m_y$ and $m_z$ are the normalized magnetisation.

The propagation matrix can be written as \begin{equation} D_{j} = \begin{pmatrix} U\cos\delta^i & U\sin\delta^i & 0 & 0 \\ -U\sin\delta^i & U\cos\delta^i & 0 & 0 \\ 0 & 0 & U^{-1}\cos\delta^r & U^{-1}\sin\delta^r \\ 0 & 0 & -U^{-1}\sin\delta^r & U^{-1}\cos\delta^r \end{pmatrix} \end{equation}

where

\begin{align} U &= \exp\left(\frac{-i2\pi n_j \alpha_{zj} d_j}{\lambda} \right)\\ \delta^i &= -\frac{\pi n_j Q d_j g^i}{\lambda \alpha_{zj}} \\ \delta^r &= -\frac{\pi n_j Q d_j g^r}{\lambda \alpha_{zj}} \\ g^i &= m_z \alpha_{zj} + m_y \alpha_{yj} \\ g^r &= m_z \alpha_{zj} - m_y \alpha_{yj} \end{align}

where $\alpha_{yi}=\sin\theta_j, \alpha_{zi}=\cos\theta_j$, $\theta_j$ is the complex refractive angle, $Q$ is the Voight parameter, $n_j$ is the refractive index, and $d_j$ is the thickness of the $j$th layer. The angle is measure with respect to the $z$ axis and is calculated using Snell's law. $m_x$, $m_y$ and $m_z$ are the normalized magnetisation. $\lambda$ is the wavelength of light.

Please be aware that the current version of MOKE is still under development and does not deal with the refractive index correctly

The function mag2exp.moke.e_field is used to calculate the relected or transmitted electric field based on an incident field E_i.

The reflected field is discretisedfield objects with components s and p for each of the linear polarisations. These can be accessed using

The intensity of the MOKE image is \begin{equation} I_K = \lvert E^r_s \rvert ^2 + \lvert E^r_p \rvert ^2. \end{equation} The function mag2exp.moke.intensity can be used to calculate this quantity.

As the intensity is a discretisedfield object, the built in plotting functions can be used to view it.

Similarly the transmitted intensity can be calculated.

This can also be done for circularly polarised light.

Incident at an angle

The Kerr angle of the reflected light can be obtained using \begin{align} \Phi_s &= \phi_s' + i \phi_s'' \\ &= \frac{r_{ps}}{r_{ss}} \\ \Phi_p &= \phi_p' + i \phi_p'' \\ &= -\text{Real}\left(\frac{r_{sp}}{r_{pp}}\right) + \text{Imag}\left(\frac{r_{sp}}{r_{pp}}\right) i \end{align} where $\phi'$ is the Kerr rotation and $\phi''$ is the ellipticity for the s and p polarisations.

The function mag2exp.moke.kerr_angle enables the rotation and ellipticity to be obtained.

The s polarisation Kerr rotation

The s polarisation ellipticity

The p polarisation Kerr rotation

The p polarisation ellipticity

Often experiments can be limited by their resolution. In order to take this into account when computing images the kerr_angle, e_field, and intensity functions has the ability to convolute the output with a 2-dimensional Gaussian to view different spatial resolutions. A value for the Full Width Half Maximum (FWHM) of the Gaussian can be specified (in meters) for each dimension. For example a convolution of a 2 dimensional Gaussian with the intensity image is shown below.