# Measurement of the Resonant Magneto-Optical Kerr Effect Using a Free Electron Laser
## 4 MOKE Measurement Scheme
### Rmoke
The reflected light from the analyzer is then detected by a detector represented by the microchannel plate (MCP). The MCP is used in current detection mode. During extraction of the Kerr rotation angle, the section that is indicated by the rectangle of broken yellow lines in Figure 7 is rotated with respect to the axis of the light reflected from the sample using a rotary flange. The reflected light intensity from the magnetic sample is dependent on the ellipticity angle \(\eta_{K}\) and the azimuthal angle of the major axis of the polarization ellipse \(\theta_{K}\) of the beam. It should be noted that the RMOKE measurement system does not require the lock-in amplification that was adopted in the visible MOKE setup shown in Figure 4.  
Figure 6: Methods for polarization analysis for wavelengths ranging from visible light to hard X-rays. Typical materials that are used as polarizers are shown for each energy region. We describe the resulting intensity that is obtained by rotating the ellipsometry unit indicated by the yellow dashed rectangle in Figure 7 using a Mueller formalism [70; 71], in which the polarization states are expressed using the Stokes vector. The Stokes vector consists of four elements, \(S_{0}\), \(S_{1}\), \(S_{2}\) and \(S_{3}\), which express the sum of the vertical and horizontal components of the polarization (= total intensity), the difference between the vertical and horizontal components, the difference between the \(\pi/4\) and \(-\pi/4\) components and the difference between the right-handed and left-handed components, respectively. It is given by:  
\[S= \begin{bmatrix}S_{0}\\ S_{1}\\ S_{2}\\ S_{3}\end{bmatrix}  \tag{2}\]  
The degree of polarization, \(V\), can be expressed as follows:  
\[V=\frac{\sqrt{S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}}{S_{0}} \tag{3}\]  
The absolute intensity is less important than the polarization state in most real cases, so it is convenient to define the normalized Stokes parameter that is divided by \(S_{0}\), which means that \(S_{0}=1\). The resulting normalized Stokes parameters can then be expressed as follows. \[S_{0} = 1 \tag{4}\] \[S_{1} = V\text{cos}2\eta_{K}\text{cos}2\theta_{K}\] (5) \[S_{2} = V\text{cos}2\eta_{K}\text{sin}2\theta_{K}\] (6) \[S_{3} = V\text{sin}2\eta_{K} \tag{7}\]  
In the Mueller scheme, the optical components are expressed by the Mueller matrix. Using \(A_{R,}\) which is the analyzer, and \(R(\chi)\), which is the coordinate rotation, the Stokes vector \(S^{\prime}(\chi)\) of the light that is reflected by the analyzer and reaches the detector is a function of \(\chi\). It is expressed using the Mueller matrices as:  
\[S^{\prime}(\chi)=R(\chi)\cdot A_{R}\cdot R(-\chi)\cdot S \tag{8}\]  
where \(S^{\prime}\) is the Stokes vector of the light that reaches the detector after reflection from the analyzer. Figure 7: Measurement system for polar resonant MOKE that uses a rotating analyzer ellipsometry (RAE) technique. The section enclosed by the dotted rectangular line is rotated together. The RAE unit is shown in the inset photograph. The intensity I(\(\chi\)) of the reflected light that reaches the detector is derived from Equation (8):  
\[I(\chi)=S_{0}^{\prime}(\chi)\frac{r_{p}^{2}}{2}\{S_{0}(\alpha^{2}+1)+S_{1}( \alpha^{2}-1)\text{cos}2\chi+S_{2}(\alpha^{2}-1)\text{sin}2\chi\} \tag{9}\]  
where \(\alpha\) is the ratio of the reflectance amplitudes, \(r_{s}/r_{p}\), for the \(s\)- and \(p\)-components. Using the relationship between the Stokes vector and \(V\), \(\eta_{K}\), and \(\theta_{K}\) (from Equation (7)), the intensity can be written as:  
\[I(\chi)=\frac{r_{p}^{2}}{2}[2V(\alpha^{2}-1)\cos 2\eta_{K}\cdot\cos^{2}( \theta_{K}-\chi)+\alpha^{2}+1-V(\alpha^{2}-1)\cos 2\eta_{K}] \tag{10}\]  
The phase-shift in \(I(\chi)\) corresponds to \(\theta_{K}\). The value of \(V\) is unity if the light is perfectly polarized. The dependence of the cosine square function on \(\chi\) is Malus' law [72; 73].