5. Conclusion

We have investigated the nonlinear optical interactions that are strongly dependent on the existence of ferroelectric domain walls, i.e., the spatial variation of the second-order nonlinear coefficient In particular, we have discussed the so-called nonlinear Čerenkov radiation focusing on two special cases including the signal generation from a single and multiple ferroelectric domain wall(s). We have shown that the localized spatial change of nonlinearity constitutes a sufficient condition for strong Čerenkov second harmonic generation. The emitted Čerenkov signals arising from multiple walls can interfere with each other, resulting in the strong dependence of the strength of the overall Čerenkov beams on the wavelengths. Furthermore, the emission from regular periodic domain pattern gives rise to another type of nonlinear interaction, namely, the nonlinear Raman-Nath diffraction. We have derived analytical formulas that govern the emission process and discussed factors that influence the strength of the nonlinear diffraction, including the duty cycle, thickness of the crystal, randomness in domain size, as well as the beam width and wavelength of the fundamental wave. We also utilized the effect of Čerenkov second harmonic generation from a single-domain wall for direct 3D imaging of the antiparallel domains in ferroelectric crystals with sub-diffraction limit resolution. Our studies are important for a better understanding of nonlinear diffraction from ferroelectric domain structures. The nonlinear optical microscopy forms a very powerful tool that will further inspire the design and development of new and sophisticated ferroelectric domain structures for advanced photonic applications.
