## 2 Results and Discussions  
### Static Magneto-Optical Functions  
Before we discuss the time resolved experiments, we briefly review details on the complex magneto-optical function in the XUV spectral range, which allows us to probe the element-specific response of multi-component magnetic systems. Linearly polarized electromagnetic radiation can be decomposed into two circularly polarized waves with opposite helicity and equal amplitude. Its propagation through a magnetized material is then characterized by the complex index of refraction:  
\[n_{\pm}(E)=1-\left(\delta(E)\pm\Delta\delta(E)\right)+i\left(\beta(E)\pm\Delta \beta(E)\right). \tag{1}\]  
\(\delta(E)\) and \(\beta(E)\) are the dispersive and absorptive parts of the electro-optical functions valid at the photon energy, \(E\), and describe the interaction with non-magnetic materials. The effect of the magnetization is given by the magneto-optical functions \(\Delta\beta(E)\) and \(\Delta\delta(E)\). The positive and negative sign refers to either parallel or antiparallel alignment of the magnetization with respect to the wave vector of the incident radiation. Equation (1) directly implies that the element specific magnetization can be accessed either via the dispersive or absorptive part of the magneto-optical function, leading to two possible experimental configurations in a transmission geometry, MCD and Faraday rotation. The _absorptive_ magnetic contribution, \(\Delta\beta\), leads to a difference in transmitted intensity, \(I_{\pm}\), between left and right circularly polarized light and is generally referred to as MCD. It is defined for a sample with thickness, \(d\), as:  
\[\Delta\beta(E)=-\frac{\hbar c}{4 Ed}ln\left(\frac{I_{+}}{I_{-}}\right). \tag{2}\]  
The _dispersive_ magnetic contribution, \(\Delta\delta\), leads to a difference in the phase of the two oppositely polarized modes and rotates the plane of polarization of the incoming radiation. The corresponding rotation angle, \(\theta_{F}\), is called Faraday angle and can be expressed as:  
\[\theta_{F}(E)=\frac{Ed}{\hbar c}\Delta\delta. \tag{3}\]  
This equation is valid for a normal incident geometry and an out-of-plane magnetization, such that refraction of the radiation at the sample interfaces does not need to be considered [24; 35; 36; 37]. In Figure 1, we show the magneto-optical functions of an Fe\({}_{26}\)Gd\({}_{74}\) (15 nm thickness) alloy at the Gd \(N_{4,5}\) resonance (\(4d\to 4f\)) around 150 eV and at the Fe \(M_{2,3}\) (\(3p\to 3d\)) resonance around 55 eV. The measurements were performed at the UE112 beamline of the BESSY II synchrotron light source, HZB, Germany. We recorded \(I_{\pm}\) and calculated the values of \(\Delta\beta\) according to Equation (2). The corresponding dispersive part, \(\Delta\delta\), is determined via the Kramers-Kronig relation. Note that, because the magneto-optical functions are equal to zero away from the resonance, the connecting real or imaginary part of the complex response function \(n_{\pm}(E)\) can be computed with high accuracy [38]. We also show the deduced Faraday rotation, which reaches a maximum of \(\approx\) 0.36 deg/nm for Gdaround 149 eV and \(\approx\) 0.07 deg/nm for Fe around 56 eV. Note that this is about an order of magnitude larger than in the visible spectral range. Co exhibits a magneto-optical function around its \(M_{2,3}\) resonance with a similar shape and magnitude compared to Fe: the absorptive part reaches a maximum of \(\Delta\beta\approx-0.014\) at 60.3 eV. In a CoPt alloy, Co atoms induce a finite magnetic moment at their nearest neighbor Pt atoms with parallel aligned magnetic moments (cf. Figure 2e). In the spectral range between 45 and 72.5 eV, we can probe the local magnetization of Pt via the Pt \(O_{2,3}\) and Pt \(N_{7}\) resonances at 51.7, 65.3 and 71.2 eV, respectively [24; 39]. For more details on the magneto-optical functions of the transition metals, as well as their _ab initio_ description in a static and transient state, we refer to our recent works [PERSON] et al. [38] and [PERSON] et al. [40].