# Measurement of the Resonant Magneto-Optical Kerr Effect Using a Free Electron Laser
## 5 Static RMOKE Measurement  
In the visible region, the MOKE measurements essentially probe the average magnetizations of target samples. In contrast, in the shorter wavelength range around the soft X-ray region, light  
Figure 8: Intensity variation with rotation angle \(\chi\). Red (blue) circles show the results for \(+\)H (\(-\)H). The orange and purple curves show fitted cosine functions. The sample is a thin-film structure of Ta (2 nm)/Gd\({}_{21}\)(Fe\({}_{90}\)Co\({}_{10}\))\({}_{79}\) (20 nm)/Ta (10 nm)/SiO\({}_{2}\). Measurements were taken at room temperature. The photon energy was tuned to 53 eV, which corresponds to the \(M\) edge of iron. interacts with the electronic states in the core levels of matter, such as the \(2p\) or \(3p\) absorption edges for \(3d\) transition metals. These core level states are relatively localized to each of the element atoms when compared with the states near the Fermi level for magnetic metals, and there is a resonant effect in the MOKE data. Therefore, the RMOKE measurements detect the magnetization of materials with element selectivity. In this section, we review the research on the various types of RMOKE, including transverse RMOKE (T-RMOKE), longitudinal RMOKE (L-RMOKE) and polar RMOKE (P-RMOKE). In the soft X-ray region, T-RMOKE has been used earlier and adopted more widely than the other RMOKE geometries. This seems to be because T-RMOKE requires only intensity measurements, as shown in Figure 2, and it is more technically feasible than the polarization analysis. The T-RMOKE measurements were initiated by measurement of a Ni sample at the Ni \(K\)-edge that led to observation of 0.2% as a peak-to-peak value of the asymmetry [74]. The research itself was motivated by the study of an interference effect between X-ray magnetic Bragg scattering and the electric scattering in the X-ray diffraction measurements [75]. Tuning of the photon energy at the \(K\)-edge means that the \(1\) s \(\to 4p\) dipole transition occurs and the magnetic signal, then mainly originates from \(p\)-\(d\) hybridization [76], which was similar to the visible MOKE case [77]. Subsequently, T-RMOKE measurements were carried out on a Fe sample at the \(L\)-edge [78]. In this case, the dipole transition is \(2p\to 3d\) and thus directly probed the itinerant \(3d\) bands. The asymmetry ratio increases by up to \(\sim\)20% as a peak-to-peak value, and the data obtained allow us to extract the total width of an excited state and the magnitude of the exchange splitting when combined with theoretical resonant scattering calculations [79]. Triggered by this research, T-RMOKE experiments were reported in [80; 81; 82; 83] for the Co \(M\)-edge, in [82; 83; 84], for the Co \(L\)-edge, in [85; 86; 87; 82], for the Fe \(M\)-edge, in [82; 88; 89; 90; 88; 87], for the Fe \(L\)-edge, in [82; 87; 87], for the Ni \(M\)-edge, in [82; 87], for the Ni \(L\)-edge, in [93], for the Pt \(L\)-edge, and in [94], for the Mn \(L\)-edge. Additionally, the \(L\)-edge T-MOKE experiment was applied to the investigation of the magneto-crystalline anisotropy energy of an ultra-thin transition metal (Co) film [84]. It should be noted that the T-RMOKE signal basically originates from pure-charge and pure-magnetic signals; however, when the angle of incidence is tuned to the Brewster angle, components of the charge scattering are suppressed, and a pure-magnetic signal can be obtained within this geometry [86; 89; 95]. Moving onto the L-RMOKE research, polarization analyses of L-RMOKE were reported in [11; 96]. The Kerr rotation angle was measured for a Fe/Cr multilayer structure at the individual photon energies of the Fe and Cr absorption edges. The Kerr rotation angles were two orders of magnitude higher than the corresponding values in the visible region. The L-RMOKE measurements were found to enable depth analysis by varying the photon energy, including analysis of the pre-edge region.