# Probing Ferroic States in Oxide Thin Films Using Optical Second Harmonic Generation
## 2 Optical Second Harmonic Generation for Investigation of Ferroic Materials
### SHG in Bulk Ferroic Systems  
Optical SHG refers to the frequency doubling of a light wave. The SHG emission in a crystal depends on its point group symmetry and SHG is therefore in general sensitive to any point symmetry violation in the material. Since the emergence of ferroic order is intimately linked to a lowering of the crystal point-group symmetry, such ordered states are aptly probed with SHG [16; 17; 18; 79]. More importantly, this technique can be used to probe _coexisting_ ferroic states, as in multiferroics. The source term for SHG is described macroscopically by the tensor Equation (1). \[P_{i}(2\omega)=\varepsilon_{0}\chi_{ijk}E_{j}(\omega)E_{k}(\omega) \tag{1}\]  
The tensor components \(\chi_{ijk}\) belong to the third-rank second-order susceptibility tensor \(\chi^{(2)}\), parameterizing the non-linear light-matter interaction. The indices \(i,j\) and \(k\) are defined with respect to the crystallographic axes of the crystal \(x,y\) and \(z\) and denote the directions of the respective oscillating fields. \(E\) is the electric field component of the incident light at frequency \(\omega\), and \(P\) is the polarization at frequency \(2\omega\) induced inside the material. This induced polarization acts as a source of an emitted, frequency-doubled, light wave with intensity \(I_{SHG}\propto 1\,P\,(2\omega)\,|\,^{2}\). The point-group symmetry of a compound determines its set of tensor components \(\chi_{ijk}\
eq 0\). The allowed tensor components for each symmetry group can be found tabulated in literature [80]. Experimentally, specific \(\chi_{ijk}\) components can be accessed through a careful choice of incident and detected light polarization [16; 17; 18]. The spectral dependence of the \(\chi^{(2)}\) tensor can be determined using a tunable wavelength light source and can provide in-depth information on specific electronic transitions and optimal SHG response [81; 82]. A schematic of a basic SHG set up in transmission is given in Figure 5. The direction of polarization of the probe beam is defined by the polarizer angle, and the polarization direction of the detected SHG light is projected along the analyzer angle, as indicated in Figure 5. Because the second harmonic light is spectrally distinct from the fundamental light, the SHG frequency can be simply monitored separately from the probe beam intensity and is as such a background-free characterization technique. Figure 5: Typical SHG measurement setup in transmission. The incident probe beam is linearly polarized by a Glan-Taylor prism (GT\({}_{1}\)). The polarization of the probe beam is then set to an arbitrary polarization state by a rotatable wave-plate. WP\({}_{1}\) can be a half-wave-plate for linear polarization states or a quarter-wave-plate to achieve circular polarization. The spot size on the sample is controlled by a focusing lens L\({}_{1}\). Low-pass filter F\({}_{1}\) removes any second harmonic light generated in the polarization optics before the sample. The fundamental beam and higher order harmonics are blocked by band/high-pass F\({}_{2}\) and low-pass F\({}_{3}\) filters. The SHG light is collected by L\({}_{2}\), and the polarization state is analyzed by projection on a second, rotatable, GT prism (GT\({}_{2}\)). The SHG intensity is integrated after passing a monochromator (MC) and photomultiplier tube (PMT). In the case of spatially-resolved SHG, L\({}_{2}\) is exchanged for a (microscope) objective lens and the detection unit is replaced with a CCD camera. The incident angle \(\theta\) can be changed by rotating the sample around the \(y\)-axis. In the specific case of ferroelectrics, inversion symmetry is broken by a polar distortion, allowing non-zero \(\chi_{ijk}\) components. To characterize the polar distortion along a certain crystallographic direction, and to be able to discriminate between excited tensor components, one can fix the detected polarization direction of the second harmonic light, i.e., the analyzer angle, while rotating the polarization of the probe beam.