# Measurement of the Resonant Magneto-Optical Kerr Effect Using a Free Electron Laser
## 4 MOKE Measurement Scheme
### Rmoke  
In the soft X-ray region, the method described above cannot be applied because of a lack of appropriate optical components, such as Faraday cells and transmission-type polarizers. This is because of the extremely low transmission or the high extinction coefficient \(k\) of the optical constant in this photon energy range. Additionally, the refractive index \(n\) of the optical constant of a soft X-ray is nearly one, and is thus close to that of air, which means that the intensity of the reflected beam is also very low. Under these severe experimental conditions, researchers have tended to choose reflection-type optical components because there are very few candidates for the transparent materials for soft X-ray light. To improve the reflectivity of these components, the constructive interference effects from multilayer structures have been useful in guaranteeing sizable reflectivity in the soft X-ray range. Multilayer mirrors can reflect light within a specific energy range that is roughly defined by [PERSON]'s law (\(\lambda\) = 2 dsin\(\theta\), where d is the thickness of a single period of the layer and \(\theta\) is the angle of incidence  
Figure 4: Measurement system for polar MOKE in the visible range using a polarization plane modulation technique. A Faraday cell is used to compensate for the Kerr angle that originated from the sample. Figure 5: Polar Kerr hysteresis loop measured at a wavelength of 700 nm. The sample is a thin-film structure composed of Ta (2 nm)/Gd\({}_{21}\)(Fe\({}_{90}\)Co\({}_{10}\))\({}_{79}\) (20 nm)/Ta (10 nm)/SiO\({}_{2}\). The measurements were taken at room temperature. with respect to the surface normal). The constituents of the multilayer structures are materials with low \(k\) values and large differences in \(n\). In the soft X-ray region, the first material is selected to have an absorption edge at a higher energy than a specific target energy, and a counterpart material with low \(k\) and \(n\) in the energy region is then chosen. If light is incident on the multilayer mirror at the Brewster angle, which is defined by the angle where the \(p\)-component of the reflected light is suppressed when compared with the \(s\)-component, the multilayer mirror can then be used as a polarizer. In the \(L\)-edge region, the value of \(n\) is almost one, which means that the Brewster angle is approximately \(45^{\circ}\). On the other hand, in the \(M\)-edge region, \(n\) deviates from one, so that the Brewster angle is not around \(45^{\circ}\). In this case, the polarizance, which is defined as the ratio between the \(s\)- and \(p\)-reflected intensities \(r_{s}/r_{p}\), under \(45^{\circ}\) incident angle is of the order of 10-100 using Mo/Si multilayer mirrors. In the \(L\)-edge range, the polarizance is around 1000-100,000 with the same reflection geometry, although the multilayer combination is different (e.g., W (tungsten)/B\({}_{4}\)C (boron carbide), W/C (carbon)). Figure 6 shows a collection of polarizers for photons at wavelengths ranging from visible light to hard X-rays [69]. Even with the order of the polarizance in the EUV range, it is able to determine a principal axis of reflected polarization for extracting Kerr angles. When the multilayer mirror described above is used, the Kerr rotation angle \(\theta_{k}\) and the Kerr ellipticity \(\eta_{k}\) can be determined experimentally using the rotating analyzer ellipsometry (RAE) technique, which is shown in Figure 7. In RMOKE measurements using this setup, the intensity of the light that is reflected by the analyzer is monitored at the detector as a function of \(\chi\), as shown in Figure 7. The incident light impinges on the sample, and the reflected light is then transferred to the ellipsometry unit, which is used to determine the Kerr rotation angles. After passing through two pinholes that are positioned to ensure accurate alignment of the reflected light, the light reaches an analyzer with an angle of incidence that is roughly equal to the Brewster angle.