# Measurement of the Resonant Magneto-Optical Kerr Effect Using a Free Electron Laser
## 3 Techniques for Magnetization Dynamics Capture
#### Characteristics of the TR-RMOKE Technique
However, the reflectivity drops off dramatically with decreasing angle of incidence \(\theta\) (with respect to the sample normal) and \(R\sim 10^{-10}-10^{-11}\) at \(\theta=45^{\circ}\)[53]. To use the reflection setup in the \(L\)-edge region, grazing incidence is required to enhance the reflectivity, and this reduces the magnetic contrast in the out-of-plane magnetization components. TR-XMCD is usually used in the \(L\)-edge range and is detected using a reflection geometry with grazing incidence [54]. There is also the possibility that out-of-plane magnetization dynamics may be detected using TR-XMCD in the \(L\)-edge range via transmission [3] and total fluorescence yield (TFY) [55] modes. However, for the transmission measurements, the sample thickness must be thin enough to allow high transmission, and the detection system for the TFY measurements must be carefully constructed to prevent the pump laser from entering the detectors and collecting a low photon flux. In contrast, around the \(M\) absorption edge (50\(\sim\)70 eV), the light is less absorbed by the materials than it is in the \(L\)-edge range. Therefore, sizable reflectivity can be obtained at all angles of incidence. In the polar MOKE geometry, which is used to detect out-of-plane magnetization, the effect is maximized at normal incidence. However, even under this condition, there is a detectable reflectance that ensures that a better signal-to-noise ratio is obtained in the \(M\)-edge range than in the \(L\)-edge range. These facts mean that the freedom of the experimental setup is greater in the \(M\)-edge range than in the \(L\)-edge range for the detection of out-of-plane magnetization dynamics on a sub-picosecond timescale. When these advantages are used, it is expected that it will be possible to measure the depth-dependent RMOKE signal by varying the angle of incidence. In addition to the high degree of measurement freedom, the magnitude of the RMOKE for the \(M\)-edge is of the same order as that for the \(L\)-edge that was reported in earlier studies [56; 57], whereas the magnitude of the \(M\)-edge XMCD is much smaller than that of the \(L\)-edge XMCD. In addition, the \(M\)-edge XMCD lacks the advantage that exists in the \(L\)-edge range, in which the spin and the orbital magnetic moment can be extracted using the sum-rule. Because the spin-orbit splitting in the \(M\)-edge range is smaller than that in the \(L\)-edge range, the assumption made for formalization of the sum-rule is not fully met in the \(M\)-edge range. This causes a discrepancy between the spin/orbital magnetic moments that were extracted from the \(M\)-edge and \(L\)-edge XMCD measurements [58]. (2) Measurement of both rotation and ellipticity:  
In principle, it is possible to measure both rotation and ellipticity in MOKE measurements, whereas XMCD measures only the ellipticity. Through simple calculations, the non-diagonal component of the permittivity tensor can be determined based on the rotation and the ellipticity [10]. In earlier studies, the determination of the permittivity over the EUV to soft X-ray energy range was conducted indirectly with a certain number of errors. However, by performing RMOKE measurements in this energy range, it is possible to determine the permittivity component that carries magnetic information directly. There is another advantage to be obtained from measuring both the rotation and the ellipticity. In the femtomagnetism field, the magnetic response in transient magneto-optical signals is not trivial, and this has led to controversial discussions in earlier studies. There are both optical and magnetic contributions to the magneto-optical response. A recent theoretical study suggested that analysis of time-dependent magneto-optical signals in terms of both their rotation and ellipticity is important for the extraction of the intrinsic magnetization dynamics on a sub-picosecond timescale [59]. In TR-RMOKE techniques for use in a transverse geometry, the transient reflected intensity is measured.