## 3 Experiments
### Pump-Probe Nonlinear Magneto-Optical Spectroscopy  
#### 3.2.1 Magnetization-Induced Second Harmonic Generation  
Static and photoexcited magnetic properties of Eu\({}_{1-x}\)Gd\({}_{x}\)O films were investigated by using nonlinear optical techniques, especially optical second harmonic generation (SHG). Nonlinear optical effects are associated with higher order optical susceptibilities and SHG, the frequency doubling of a light wave in a material, is one of the simplest second-order nonlinear optical effects. In the majority of cases, SHG is applied to non-centrosymmetric materials or to the surface or interface of centrosymmetric materials where the space inversion symmetry is locally broken and the strongest interaction by the electric-dipole transitions is expected [43; 44; 45]. An electromagnetic light field \(\mathbf{E}\) at frequency \(\omega\) is incident on a crystal, inducing an oscillation of electric polarization \(\mathbf{P}\) at frequency \(2\omega\), which acts as a source of a frequency-doubled light wave. This is expressed by:  
\[P_{i}(2\omega)=\varepsilon_{0}\chi^{(2)}_{ijk}E_{j}(\omega)E_{k}(\omega), \tag{1}\]  
with \(\chi\) as the second-order nonlinear susceptibility and \(I^{\mathrm{SHG}}\propto\left|\mathbf{P}(2\omega)\right|^{2}\) as intensity of the emitted SHG wave. Following the Neumann principle, symmetry determines the set of tensor components \(\chi^{(2)}_{ijk}\
eq 0\)[46]. In centrosymmetric systems such as EuO, SHG is only allowed if higher-order multipole contributions in the expansion of the electromagnetic field such as magnetic-dipole or electric-quadrupole contributions are involved [47]. In magnetic semiconductor Eu\(X\), this leads to [24; 25; 48; 49; 50]:  
\[P_{i}(2\omega)=\varepsilon_{0}\Big{(}\chi^{(i)}_{ijk}+\chi^{(c)}_{ijk}\Big{)}E _{j}(\omega)H_{k}(\omega), \tag{2}\]  
where the time-invariant (\(i\)-type) tensor \(\chi^{(i)}_{ijk}\) and the time-noninvariant (\(c\)-type) tensor \(\chi^{(c)}_{ijk}\) are the second-order nonlinear susceptibilities related to, in the leading order, crystallographic SHG and magnetization-induced SHG (MSHG), respectively. \(E_{j}(\omega)\) and \(H_{k}(\omega)\) in Equation (2) denote the \(j\)-polarized electric field and the \(k\)-polarized magnetic field of the fundamental light, respectively, that induce the \(i\)-polarized SHG wave \(P_{i}(2\omega)\). For the SHG measurements in this study, the orientation of the samples was chosen such that crystallographic contributions (\(\chi^{(i)}_{ijk}\) tensors) to the SHG signal are suppressed so that the coupling to the magnetic order occurs free of background [24; 46]. For this purpose, the Eu\({}_{1-x}\)Gd\({}_{x}\)O films are probed with light incident along one of the principal axes (\(k\parallel z\)) with the spontaneous magnetization parallel to the \(x\) (\(y\)) direction. In this configuration, any \(i\)-type magnetic-dipole SHG are forbidden so that it is expected that the SHG signal has a purely magnetic origin. In this case, the MSHG intensity \(I^{\mathrm{MSHG}}\) is written by:  
\[I^{\mathrm{MSHG}}\propto(J_{\mathrm{ex}}\cdot M)^{2}, \tag{3}\]  
which is proportional to the square of the exchange interaction \(J_{\mathrm{ex}}\) and the magnetization \(M\)[13]. The MSHG signal was measured in a normal-incidence transmission geometry under an in-plane magnetic field of 50 mT, which was applied to sustain a single-domain magnetic state, and separated from the incident fundamental light by short-pass filters and a monochromator and detected with a photomultiplier tube. It is known that EuO shows a large temperature-dependent spectral shift of the absorption due to the exchange splitting between the spin-up and spin-down states of the 5\(d\) orbitals [51]. Previous experiments revealed [24] that at 2\(\hbar\omega=2.60\) eV, the interference by temperature-dependent linear absorption effects is avoided. We therefore chose it as the MSHG probe energy.