## 2 Results and Discussions
### Time Resolved Faraday Rotation
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Analyzer** & **Brewster Angle** & \(R_{s}\)**(\%)** & \(R_{p}\)**(\%)** & \(P\) \\ \hline ML for 150 eV & 45\({}^{\circ}\) & 7.0 & 0.021 & 0.994 \\ Au mirror for 56 eV & 49\({}^{\circ}\) & 5.5 & 0.28 & 0.903 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reflectances, \(R_{s,p}\), and polarizing power, \(P=(R_{s}-R_{p})/(R_{s}+R_{p})\), of the analyzer mirrors. The values for the ML mirror were determined at the Metrology Light source of PTB, Berlin, and the values for the Au mirror were calculated with a tabulated index of refraction at 56 eV [43]. The Brewster angles are measured in a grazing incidence geometry. with fixed undulator gaps at FLASH. The spectral bandwidth of the FEL radiation on the order of 1-2% also results in an effectively smaller value of \(\theta_{F}\). At 3 ps after optical excitation, we completely quench the magnetization and, consequently, the two measurements for opposite magnetization directions lie on top of each other and are centered around \(\alpha=0^{\circ}\). In Figure 3b, we show the magnetic asymmetry calculated as \(A=(I_{+}-I_{-})/(I_{+}+I_{-})\). Before time delay zero, we observe a very large asymmetry of approximately 40 % at a rotation angle of \(\alpha=\pm 4^{\circ}\), while 3 ps after optical excitation, the asymmetry is almost completely zero for all angles \(\alpha\). In Figure 4a, we show the intensity \(I_{\pm}\) measured at an angle of \(\alpha=4^{\circ}\) as a function of the time delay between the optical excitation and the XUV probe pulses. Upon excitation, we observe an ultrafast drop or increase in the intensity depending on the magnetization direction of the sample. For the approximation that \(\alpha\pm\theta_{F}\ll 1\) rad and for \(P_{\mathrm{Lin}}\approx 1\), we can show that [37]:  
\[I_{+}-I_{-}\propto\theta_{F}\propto\Delta\delta\propto M. \tag{5}\]  
We normalize our observable, proportional to \(M(t)\), to the average value before optical excitation, \(M_{0}\), and show the corresponding relative changes in the magnetization as a function of time delay in Figure 4b. A nonlinear least square fit of the data with a mono-exponential function convolved with the temporal resolution of the experiment yields a demagnetization amplitude of the Gd sublattice by approximately 80% with a corresponding time constant of \(\tau=(233\pm 16)\) fs. Analogously measured time resolved data \(M(t)/M_{0}\) at the Fe \(M_{2,3}\) edge are shown in Figure 4c. While the amplitude of the demagnetization after 1 ps is comparable between the Fe and Gd sublattices, in Fe, we observe a significantly faster time constant of only \(\tau=(94\pm 16)\) fs. Distinct demagnetization times of the Gd and Fe sublattices in an FeGd alloy have been observed before in time resolved magnetic circular dichroism experiments. The microscopic origin has been explained by an ultrafast reduction in the interatomic exchange interaction due to the rapid heating of the electronic system close to, or even above the Curie  
Figure 3: **(a)** Detected intensity after reflection off the analyzer mirror as a function of rotation angle, \(\alpha\), for \(E\approx 149\) eV, resonant at the \(N_{4,5}\) edge of Gd. The two measurements before optical excitation at \(t=-10\) ps for two opposite directions of the sample magnetization, \(M_{\pm}\), are shifted by the Faraday angle \(2\theta_{F}=4^{\circ}\). After optical excitation at \(t=3\) ps, the Faraday angle is zero, \(\theta_{F}=0\), and the measurements are centered around \(\alpha=0^{\circ}\), independent of the applied magnetic field. The lines are non-linear least square fits according to Equation (4). **(b)** Magnetic asymmetry before and after optical excitation is shown as a function of the analyzer angle. The maximum value reaches very large values of approximately \(\pm 40\%\) for \(\pm\alpha=\pm 2\theta_{F}=\pm 4^{\circ}\). temperature [10].