Significance The quasi-phase matching theory proposed by Bloembergen provides an effective method for phase mismatch compensation and conversion efficiency improvement in the process of nonlinear optical interaction. In order to fulfil the quasi-phase matching conditions, nonlinear photonic crystals, domain modulated LiNbO₃, LiTaO₃ and other ferroelectric materials, have been extensively studied in the past few decades. The quasi-phase matching technology has gained in-depth research and made a great progress at the one-dimensional and two-dimensional levels, and it is used for efficient frequency conversion in nonlinear optics. Due to the existence of collinear and non-collinear reciprocal vectors, many interesting phenomena have been discovered, including nonlinear optical frequency conversion, nonlinear ?erenkov radiation, conical second-harmonic generation, and nonlinear Talbot self-imaging.

Three-dimensional nonlinear photonic crystals can realize a variety of new nonlinear optical interaction processes, such as synchronous quasi-phase matching of different nonlinear processes, volume nonlinear holography, nonlinear beam shaping, etc. However, because the traditional periodic structure preparation techniques such as the electric field polarization method are difficult to achieve three-dimensional control of nonlinear coefficients, the preparation of three-dimensional nonlinear photonic crystals has not made a major breakthrough, which is the bottleneck of the experimental research of the current three-dimensional quasi-phase matching process.

Progress Lithium niobate crystal, one of the most popular nonlinear photonic crystal materials (Table 1), can be experimentally verified and easily obtained by researchers working in the field of nonlinear optics, and the experiment is compatible with the current nonlinear optical modulation technology. The requirements of laser direct writing are relatively easy to meet, especially along the depth direction (Figs. 5 and 6) and can be easily extended to various nonlinear crystals including LiTaO₃ and KTP crystals. In addition, this laser direct writing method can be effectively used to manufacture more complex nonlinear photonic structures to perform precise three-dimensional processing of nonlinear light waves. It has advantages in nonlinear beam shaping, nonlinear imaging, and three-dimensional nonlinear holography, and has a wide range of application prospects. In addition, a laser erasing method is used to prepare a three-dimensional nonlinear structure in a lithium niobate waveguide, and the waveguide core can be designed into two or four parts to achieve parallel multi-wavelength frequency conversion (Fig. 9), thereby achieving a compact design.

A three-dimensional nonlinear photonic crystal is produced by processing three-dimensional ferroelectric domains with tightly focused femtosecond laser pulses in a ferroelectric barium calcium titanate crystal, which can compensate for the phase mismatch of the second-order nonlinear optical process in any directions (Figs.15 and 16). The optical polarization method used here is fully compatible with other existing optical manufacturing technologies, including the common femtosecond laser writing for refractive index structures. This achievement is a three-dimensional nonlinear integrated photonic device that realizes next-generation optical communication and on-chip signal processing. The monolithic manufacturing has paved the way.

As an engineering material with modulated second-order nonlinear polarization, potassium tantalate niobate crystals can be widely used in many scientific and industrial fields that need to generate and control new frequency light. It breaks the strict restrictions on the incident light polarization and crystal direction, and can achieve quasi-phase matching conditions in a wide spectral range (Fig. 28). Naturally formed potassium tantalum niobate shows abundant reciprocal lattice vectors, which breaks the strict requirements of traditional polarized nonlinear photonic crystals on the polarization direction of incident light and crystal direction. It is easily compatible with laser writing technology, which means that it is possible to create layered nonlinear optical modulation. Therefore, the three-dimensional nonlinear photonic crystal in this perovskite ferroelectric should find a wide range of applications in optical communications, nonlinear imaging, and on-chip signal processing.

Conclusion and Prospect This article focuses on three-dimensional quasi-phase matching theory and experimental verification. Three-dimensional quasi-phase matching is achieved in lithium niobate, barium calcium titanate, and potassium tantalate niobate crystals, and effective frequency-doubled light output is obtained. Three-dimensional quasi-phase matching technology provides a new feasible solution for nonlinear optical interaction, and has obtained more applications, including cascaded QPM for different nonlinear processes, nonlinear Talbot imaging, on-chip entangled light source, terahertz radiation, three-dimensional nonlinear holography, and beam shaping. In recent years, there have been some recent reports on the development of NPC in holography, such as large-capacity nonlinear holography technology using photon orbital angular momentum coding and three-dimensional nonlinear volume holography technology. Abundant coherent light sources can also be applied to basic atomic, molecular and optical physics, especially advanced scientific instruments with wide spectrometers and high resolution. This article reviews the research progress of several three-dimensional nonlinear photonic crystals, including artificially made three-dimensional nonlinear photonic crystals made of lithium niobate and barium calcium titanate, integrated three-dimensional quasi-phase-matched waveguide structures made of lithium niobate, and spontaneous barium calcium titanate and potassium tantalate niobate crystals with three-dimensional nonlinear photonic crystal structure.