# 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
Here, we refer to such characterization as polarizer measurements. In general, the polar axis of the investigated sample can point along an unknown direction. This orientation can be accessed by fixing both the incoming and detected light polarization while rotating the sample along the azimuthal angle defined by the sample surface. Equivalently, this in-plane polar anisotropy measurement can also be performed with a simultaneous rotation of the polarization of both probing and detected light while keeping the sample position fixed. Because the SHG technique only probes polar distortions projected on a plane perpendicular to the k-vector of the incident light wave, it may be necessary to introduce a tilt of the angle of incidence (denoted by \(\theta\) in Figure 5). This changes the direction of the k-vector with respect to the crystal and provides access to certain domain orientations. SHG is a well-established technique to non-invasively probe bulk materials where domain dimensions are expected to be larger than the spatial resolution limit (about a micrometer). Indeed, it has been used to image ferroelectric domains [83], antiferromagnetic domains [84; 85] and more recently multiferroic domains in bulk systems [86; 87], as can be seen in Figure 6.  
Figure 6: Imaging ferroic domains in bulk using SHG. (**a**) Ferroelectric domain structure in BaTiO\({}_{3}\) (BTO). The scale bar is 8 \(\upmu\)m. (**b**) Observation of orthorhombic twinning in a BTO crystal. The scale bar is 12 \(\upmu\)m. Reprinted by permission from [83], Copyright Macmillan Publishers Ltd.: Nature Communications, 2014. (**c**) Antiferromagnetic domains in Cr\({}_{2}\)O\({}_{3}\), reprinted with permission from [84] Copyright AIP Publishing, 1995. Scale bar is 1 mm. (**d**) Multiferroic domains in TbMnO\({}_{3}\), Reprinted with permission from [86], Copyright AAAS, 2015. Local symmetry mapping is performed by selectively coupling to SHG tensor components which are domain orientation dependent. Spatially-resolved SHG, using a charge-coupled device (CCD) camera detection system, then shows domains with 90\({}^{\circ}\) relative orientation in different brightness; see Figure 6a. Since each phase of a material corresponds to an individual set of non-linear tensor components, phase coexistence can also be mapped using optical SHG. Figure 6b demonstrates the presence of orthorhombic twins in a tetragonal BaTiO\({}_{3}\) (BTO) crystal. Furthermore, domains with opposite order parameter, i.e., \(\pm P\) in the ferroelectric case, emit SHG light waves with a relative 180\({}^{\circ}\) phase shift. Superimposing these light waves with a reference light, one can achieve domain contrast and observe oppositely oriented domains with different brightness [79; 88]. Moreover, the emergence of SHG signal as a function of temperature is a powerful method to identify the critical temperature of phase transitions in the ferroic compound under investigation. This is particularly useful when standard characterization techniques are unable to characterize the emergence of the ferroic order [86]. For exemplifying the relation that connects a particular pattern of ferroic domains to a specific polarization-dependent SHG signal we consider the case of a single-domain ferroelectric state in the multiferroic model system BFO. The point symmetry of ferroelectric BFO allows a set of independent non-zero SHG susceptibilities. These are described as complex values so that they possess an amplitude \(\chi_{ijk}\) and a relative phase \(\Psi_{ijk}\). The resulting ferroelectric SHG signal is then given by [88]:  
\[\left|P_{i}(2\omega)\right|^{2}\propto\left|\sum_{j,k}\chi_{ijk}e^{i\Psi_{ijk}} E_{j}(2\omega)E_{k}(2\omega)\right|^{2} \tag{2}\]  
With \(z\) as surface normal, we find \(\chi_{xxx}\), \(\chi_{xyy}\), \(\chi_{yyy}\), \(\chi_{yyx}\), \(\chi_{yxx}\), \(\chi_{zzz}\), \(\chi_{yzz}\), \(\chi_{zzx}\), \(\chi_{zzy}\) as non-zero independent components for BFO (and accordingly for \(\Psi_{ijk}\)) in accordance with its point symmetry group \(m\).