# Nonlinear Optical Microscopy of Interface Layers of Epitaxial Gamet Films
## 4 Discussion  
For the case of noncentrosymmetric magnetic media, nonlinear polarization at the second or third harmonics' wavelengths is given by a vector sum of the two types of terms of the electric field at the corresponding wavelengths, the so called _even_ and _odd_ components  
Figure 4: **(a)** Scheme of the light propagation in the THG microscopy measurements; **(b)** the dependence of the averaged THG intensity of the coordinate \(z\) along the normal to the film. THG intensity patterns measured with the polarization plane of the fundamental radiation oriented in the vertical direction in the figure, the THG polarization is horizontal **(c)** and tilted to about 5\({}^{\circ}\) from the horizontal direction **(d)**. Fundamental wavelength of 1500 nm was used in these measurements. Figure 3: **(a)** Scheme of the light beam propagation in the SHG microscopy measurements of the \((LuBi)_{3}F_{56}O_{12}/GGG\) interface; **(b)** dependence of the SHG intensity on the coordinate within the garnet film, z = 0 corresponds to the garnet/GGG interface. **(c,d)** SHG intensity patterns of the \((LuBi)_{3}F_{56}O_{12}/GGG\) interface layer, the polarization plane of the fundamental radiation is oriented in the vertical direction on the figure, the polarization of the generated SHG is vertical **(e)** and horizontal **(d)**. with respect to magnetization: \(\mathbf{P}_{2\omega,3\omega}\propto\mathbf{E}_{2\omega,3\omega}^{even}+\mathbf{E}_{2 \omega,3\omega}^{odd}\). In turn, these field components are determined by the effective _even_ and _odd_ components of the nonlinear susceptibilities \(\chi_{2\omega,3\omega}^{even}(M)\)=\(\chi_{2\omega,3\omega}^{even}(-\) M) and \(\chi_{2\omega,3\omega}^{odd}(\)M)=\(-\chi_{2\omega,3\omega}^{odd}(-\)M), the latter changes its sign under the magnetization reversal. In this simplified description, we restrict ourselves linearly in \(\mathbf{M}\) terms of the magnetization-induced nonlinear optical response at the SHG or THG wavelengths, as they are the strongest in the case of a magnetized sample or in a medium with spontaneous magnetic structure (i.e., in a domain-like one) [35]. Interference of these components of the harmonics' fields provides linear in magnetization modulation of the SHG (THG) intensity, which can be characterized by the corresponding magnetic contrast \(\rho_{2\omega,3\omega}\propto\frac{\chi_{2\omega,3\omega}^{odd}}{\chi_{2 \omega,3\omega}^{even}}\)[25] and allows the visualization of the magnetic domains with different orientation of the average magnetization. At this stage, magnetization-induced effects in the nonlinear optical response are qualitatively similar to their linear optical analogues, i.e., magnetooptical Kerr and Faraday effects, while the values of the SHG magnetic contrast \(\rho_{2\omega}\) is typically one to two orders of magnitude larger. Evidently, the contrast in the nonlinear optical response from different types of magnetic domains required for their imaging appears only if both _odd_ and _even_ in \(\mathbf{M}\) effective components of the nonlinear susceptibility exist. The sets of the nonzero components of the nonlinear susceptibility tensors are governed by the crystallographic and magnetic symmetry of a medium. As garnet belongs to the centrosymmetric cubic (\(m3m\)-symmetric) class, the bulky SHG is suppressed by symmetry arguments and thus the second harmonic (in the electric dipole approximation) should be generated by the interfaces with broken inversion symmetry. At the same time, the bulk SHG effect was registered in substituted garnets with the structure distorted by the presence of large ions such as bismuth or induced by the lattice mismatch at the garnet/GGG interface [43], which seems to be our case. Meanwhile, third harmonic generation is allowed in the bulk of the materials of any symmetry class, including garnets.