# Nonlinear Optical Microscopy of Interface Layers of Epitaxial Gamet Films
## 4 Discussion
If taking into consideration the transparency of garnet films at the fundamental wavelength, transmission coefficients at the second and third harmonic's wavelengths and localization of the pump beam in the laser beam waist, the intensity of the SHG (THG) transmitted through the garnet film can be expressed as  
\[I_{m\omega}(\pm M,z,r)=\int_{V}T_{m\omega}(z)(\chi_{m\omega}^{even}\pm\chi_{ m\omega}^{odd}(\pm M))^{2}L(z,r)^{m}dzdr \tag{1}\]  
where \(m=2\), \(3\) corresponds to the number of harmonic (second or third), \(L(z,r)\) is the dependence of the fundamental field intensity on the transversal radius-vector \(\mathbf{r}\) and on the coordinate along the direction of the beam propagation, \(z\). Integration over \(z\) can be restricted by the waist length according to the expression \(L(z,r)=I_{0}/(1+(\frac{z}{z_{R}})^{2})\), where \(z_{R}\) is the Rayleigh length of the focusing objective and \(d\) is the film's thickness, \(I_{0}\) is the input intensity of the fundamental laser beam, \(T_{m\omega}(z)\propto\exp[-\alpha_{m\omega}(d-z)]\) are the intensity transmission functions at the SHG and THG wavelengths, and the integral is taken over the excited volume \(V\) of the \((LuBi)_{3}Fe_{5}O_{12}\) film. Here we assume that the effective nonlinear susceptibilities \(\chi_{m\omega}\) are uniformly distributed in the garnet film and the nonlinear contribution of the GGG substrate is negligibly small, as stems from our experiments. Thus the SHG or THG intensity maxima of the \(I_{m\omega}(z)\) dependencies (see Figures 3b and 4b) are attained within the garnet film close to its exit boundary. It is worth noting that the resolution of the THG microscopy technique is higher as compared to the SHG one due to the third-order dependence of its intensity on \(L(z,r)\), while it is quadratic for the SHG case. The analysis of the SHG and THG intensity patterns shown in Figures 3 and 4 is a complicated task, as the epitaxial garnet films are (111) symmetric and the tensors \(\hat{\chi}^{(2)}\) and \(\hat{\chi}^{(3)}\) contain many nonzero components, both crystallographic (_even_) and magnetization-induced (_odd_) [35;44]. Still, based on the analysis of the SHG intensity patterns measured for the parallel (Figures 2b and 3c) and orthogonal (Figures 2d and 3d) combinations of polarization of the fundamental and SHG beams, one can judge the orientation of the magnetization in domains similarly to those discussed recently in [42] for the garnet/air interface. Namely, the method reveals only the in-plane M components orthogonal to the polarization of the SHG wave, i.e., \"vertical\" in the first case and \"horizontal\" in the second. It means that the spontaneous magnetization of stripe domains of the garnet film is tilted with respect to the film's normal, which may be due to the (111) film's symmetry. Similar conclusion on the direction of the magnetization in such films has been proposed based on the analysis of the MFM image of the cleavage (end face) of the garnet film [42]. This provides a large (up to 60%) contrast in the second harmonic intensity generated by the neighbour stripe domains. Shorter-period modulation of the SHG intensity in the direction along the stripe domains observed at the interfaces of the \((LuBi)_{3}Fe_{5}O_{12}\) layer with air (Figure 2) and with the GGG substrate (Figure 3) is associated with the nonlinear response of the block (closure) domains; the modulation is better seen for the crossed polarizations of the SHG and pump radiation (panels \(d\) in Figures 2 and 3). In that case, the maximum contrast of the SHG intensity is attained for the adjacent closure domains placed along the horizontal line in Figures 2d or 3d, which correspond to the opposite signs of the \"vertical\" \(\mathbf{M}\) component. If taking into account the analogous analysis of the SHG intensity pattern for the parallel polarizations of the SHG and pump beams, we have to conclude that the in-plane magnetization of the closure domains forms a nonzero angle with the magnetic stripes.