# Measurement of the Resonant Magneto-Optical Kerr Effect Using a Free Electron Laser
## 2 MOKE Phenomena  
The MOKE can be characterized based on its rotation and ellipticity properties. Linearly-polarized light can be decomposed into left and right circularly-polarized components. Each of these components has the same phase velocity and amplitude. When linearly-polarized light interacts with magnetized materials, the polarization state of the light will change in two ways. The first way involves the rotation of the polarization axis, and the second is a change from linearly-polarized to elliptically-polarized light that is characterized by the ellipticity \(\eta_{k}\). Figure 1 shows an example of the (polar) MOKE measurement. The phase (amplitude) variations between the right and left circularly-polarized light components is responsible for the rotation, denoted by \(\theta_{K}\) (ellipticity, \(\eta_{K}\)). The measurement geometry is shown in Figure 2. There are two main schemes that are used for MOKE measurements. One scheme is based on polarization analysis, because the polarization of the reflected light rotates, and its ellipticity changes (see Figure 2a,b). The other scheme involves intensity measurements and is based on the fact that the polarization states of the reflected light do not change from the corresponding states of the incident light (see Figure 2c) This is called the transverse MOKE, or T-MOKE. Schemes in the first group are classified based on the magnetization direction \(\mathbf{M}\). When \(\mathbf{M}\) is perpendicular to the sample surface, the method is called polar MOKE (P-MOKE; see Figure 2a). If \(\mathbf{M}\) is parallel to the sample surface and the reflection plane defined by the incident and reflected light beams, it is called longitudinal MOKE (L-MOKE; see Figure 2b). Historically, the schemes shown in Figure 2a,b are recognized separately, but the MOKE signals that involve polarization analysis in the cases of Figure 2a,b both originate from \(\mathbf{k}\cdot\mathbf{M}\), and there is thus no intrinsic difference between the two geometries. These measurement geometries correspond to the measurement under applying a magnetic field to a specific direction with respect to the incident plane as shown in Figure 2. In the case of the arbitrary direction of the external magnetic field, magneto-optical effects observed in the reflected light are contributed from any of the three MO geometries. The phenomenological and analytical expressions in such cases can be found in the preceding studies [16; 17]. \(\theta_{k}\) and \(\eta_{k}\) are defined with Fresnel coefficients, which depend on the frequency of incident light (\(\omega\)) as follows [16]. \[ \begin{array}{rcl}\theta_{ks}&=&-\text{Re}\left(\frac{\widetilde{\tau}_{ps}( \omega)}{\widetilde{\tau}_{ps}(\omega)}\right)\\ \theta_{kp}&=&\text{Re}\left(\frac{\widetilde{\tau}_{xp}(\omega)}{\widetilde{ \tau}_{pp}(\omega)}\right)\\ \eta_{ks}&=&\text{Im}\left(\frac{\widetilde{\tau}_{pp}(\omega)}{\widetilde{ \tau}_{ss}(\omega)}\right)\text{Re}\left(\frac{\widetilde{\tau}_{ps}(\omega)}{ \widetilde{\tau}_{ss}(\omega)}\right)\\ \eta_{kp}&=&\frac{\text{Im}\left(\frac{\widetilde{\tau}_{pp}(\omega)}{\widetilde {\tau}_{pp}(\omega)}\right)}{\text{Re}\left(\frac{\widetilde{\tau}_{pp}( \omega)}{\widetilde{\tau}_{pp}(\omega)}\right)}\\ \omega&\approx&\omega_{\text{res}}\text{ (RMOKE definition)}\end{array}  \tag{1}\]  
\(\omega_{\text{res}}\) corresponds to an absorption edge of the target magnetic element. \(\theta_{ks}\) (\(\eta_{ks}\)) and \(\theta_{kp}\) (\(\eta_{ks}\)) are the Kerr rotation angle (ellipticity) for \(s\)- and \(p\)-polarized incident light, respectively. \(i\) and \(j\) in the \(\widetilde{\tau}_{ij}\) are the electric-field component of the reflected and of the incident light, respectively. The tilde shows that the quantity is complex. The specific derivatives depends on the MOKE configurations, which are found in the earlier study [16]. Figure 1: Illustration of the (polar) magneto-optical Kerr effect (MOKE).