# Measurement of the Resonant Magneto-Optical Kerr Effect Using a Free Electron Laser
## 6 Demonstration of TR-RMOKE with a Soft X-ray FEL
The zero angle was set at the middle angle between the minima of the curves that were obtained under application of \(+\)H and \(-\)H. The position of the middle angle did not change with respect to the delay time to within 0.5 degrees, which was the resolution used to determine \(\theta_{K}\) in this experiment. We see that the apparent shifts in the entire curves increase the reliability and the accuracy of the change, even with the possible existence of fluctuations between the individual data points. The Kerr rotation angle at each delay time can be extracted from the phase difference between the two curves, 3.2\({}^{\circ}\), 2.5\({}^{\circ}\), \(-\)1.1\({}^{\circ}\) and \(-\)0.7\({}^{\circ}\) for the delay times of \(-\)100, 100, 200 and 600 fs, respectively. The Kerr rotation angle of 3.1\({}^{\circ}\), which was measured from the static RMOKE (see Figure 9), was reproduced in the time-resolved measurements by tuning the FEL energy to 53 eV at \(-\)100 fs, at which the Kerr rotation angle is 3.2\({}^{\circ}\). Because a polar geometry is used in the TR-RMOKE measurements, the Kerr rotation angle indicates the out-of-plane magnetic moment of Fe in the Gd\({}_{21}\)(Fe\({}_{90}\)Co\({}_{10}\))\({}_{79}\) structure. Figure 11 shows a schematic depiction of the magnetization dynamics with respect to the external field that result from the time-dependent Kerr rotation angle. The lengths of the arrows in the figure correspond to the Kerr rotation angle magnitude for each delay time. At 200 fs after the high-intensity laser irradiation, the changes in the sign of the Kerr rotation angle indicate reversal of the Fe magnetization. Because \(-\)1.1\({}^{\circ}\) at 200-fs delay time was larger than the resolution for determining Kerr angle in these measurements, the results showed magnetization reversal rather than the demagnetization. When the FERMI-FEL repetition rate of 10 Hz is considered, the Fe magnetic moment is recovered within at least 100 ms after pumping. This reversal mechanism is classified as a thermal process, unlike the non-thermal effect, such as the process involving the inverse Faraday effect that was observed in previous studies using circularly-polarized beams [109]. Because a linearly-polarized IR pump laser was used, there is no coupling in terms of the exchange of angular momentum between the photons and the spins in the material, although  
there is a possibility of coherent interaction between spin and photon field [110], which occurs on a much faster timescale than this experimental resolution. Therefore, the angular momentum transfer path is closed between the Gd and Fe sub-lattices [111]. In Figure 12, the time-dependent results that were obtained from this TR-RMOKE experiment with the FEL are shown. The magnetization reversal timescale of the Fe sublattice is the same as that in a previous study that used TR-XMCD with laser slicing [3]. Although the microscopic mechanism of the magnetization dynamics should be explored with various experimental conditions and materials, which is beyond the scope of this review paper, here we demonstrated that the TR-RMOKE scheme is useful for tracing of light-induced magnetization dynamics on a subpicosecond timescale. It should be noted that in time-resolved MOKE measurements, it has been argued that the MOKE signal is modified by a nonequilibrium state that is generated during the femtosecond pulse, meaning that the MOKE signal does not reflect the sample magnetization [112; 113]. While this effect is negligible for the time delays shown in Figures 11 and 12, this should be considered when the delay becomes close to the dephasing time of the coherent correlation between the photons and the spins. Figure 11: Intensity variation with rotation angle \(\chi\) at delay times of (**a**) \(-\)100 fs, (**b**) 100 fs, (**c**) 200 fs and (**d**) 600 fs under application of the external magnetic fields +H (red circles) and \(-\)H (blue circles). These characteristics were obtained using a photon energy of 53 eV.