Second-order nonlinear frequency conversion in nonlinear photonic crystals (NPCs) is an important research branch of nonlinear optics, which is of great practical importance to create coherent light sources at new frequency bands. It is governed by the phase-matching requirements due to the reciprocal lattice of these crystals. Therefore, the modulation of the nonlinear susceptibility enables to engineer the spatial and spectral response in these processes.

As for χ(2) processes, there are various phenomena of second harmonic wave (SH) generated in one-dimensional (1D) and two-dimensional (2D) NPCs by different types of phase match, such as birefringence phase match (BPM) , quasi-phase match (QPM) , non-colinear phase match for nonlinear Raman-Nath diffraction (NRND) , nonlinear Čerenkov radiation (NCR) , as well as nonlinear Bragg diffraction (NBD) which is a degeneration of NRND and NCR. Besides the forward emitting phenomena, there are also backward emitting cases with low conversion efficiency like backward QPM, which might have been ignored in common cases compared with forward ones. They are simple in structure and have been well researched. If the structures become more complicated with 1D and 2D NPCs extended to three-dimensional (3D) NPCs, SH that satisfying different phase-matching conditions shall emit simultaneously and mix together.

However, the research of χ(2) processes in 3D NPCs is merely reported both in theory and experiment due to the limitation of existing manufacture techniques. Thoroughly theoretical analysis helps to discriminate different types of phase match. Moreover, the interaction of different types of phase match also needs to be further studied.

Furthermore, making appropriate artificial χ(2) structures in NPCs would allow us to control the behavior of harmonic generation more efficiently, which strongly extends potential applications in optical control, harmonic generation, as well as in photonics studies.