Fig. 1. Principle diagram of birefringence phase matching based on negative uniaxial crystal^[21]

Fig. 2. Schematic of relationship between frequency doubling laser intensity and propagation distance under different phase matching cases^[31]. (a) Full phase matching; (b) first-order quasi-phase matching; (c) phase mismatching

Fig. 3. Schematic of nonlinear coefficient distribution for one-dimensional quasi-phase matching^[31]

Fig. 4. Structural diagram of three-dimensional nonlinear optical crystal with simple cubic lattice^[21]

Fig. 5. QPM mechanism in laser-engineered LN crystal^[48]. (a) Comparison among full phase matching,QPM based on poled LN crystal, QPM based on laser-engineered LN crystal and phase mismatching; (b) structural diagram of 3D NPC fabricated by femtosecond laser direct writing

Fig. 6. Sample characterization. (a) Image of 3D structure recorded using $\dot{\mathrm{C}}$erenkov-type second harmonic confocal microscopy^[61]; (b)second harmonic image in x-y plane taken through confocal second harmonic microscopic system^[62]; (c) intensity distribution along central black line in Fig.6(b)^[48]

Fig. 7. Demonstration of second harmonic generation process in 3D LN NPC^[48]. (a) 3D reciprocal lattice array and typical 3D reciprocal vectors; (b)-(e) measured and simulated 3D QPM second harmonic beam patterns for various input fundamental wavelengths with QPM modulation and corresponding reciprocal vectors shown in right column

Fig. 8. Dependence of second harmonic power on input parameters^[48]. (a) Dependence of second harmonic power on fundamental wavelength; (b) relationship between QPM second harmonic power and input pump power at fundamental wavelength of 829 nm

Fig. 9. Principle diagram of QPM waveguide^[63]. (a) Design diagram of QPM waveguide; schemes to modulate χ⁽²⁾nonlinearity inside QPM waveguide core: (b) one period (A), (c) two periods (A, B), and (d) four periods (A, B, C, D)

Fig. 10. Sample characterization^[21]. (a) SEM image of domain structure of c-direction BCT crystal after corrosion; (b) SEM image of domain structure of a-direction BCT crystal after corrosion; (c) statistical width distribution of random domains in BCT crystal; (d) structural diagram of random 3D stacking domain in BCT crystal

Fig. 11. 3D QPM process with random domain modulation in self-polarized BCT crystal^[21]

Fig. 12. Schematic of experimental setup for 3D QPM frequency doubling using BCT crystal^[21]

Fig. 13. Frequency-doubled laser spots obtained (left) by 3D QPM frequency doubling experiment using BCT crystal and (right) by theoretical simulation ^[21]

Fig. 14. Frequency-doubled laser power versus fundamental power with spectral curves of 1100 nm fundamental laser and its frequency-doubled laser shown in insets (1) and (2), respectively^[21]

Fig. 15. Schematic of experimental setup for femtosecond laser direct writing of 3D ferroelectric domain structure^[67]

Fig. 16. NPC and corresponding inverted lattice vectors^[67]. (a) Two-dimensional NPC diagram; (b) 3D NPC diagram; (c) Čerenkov second harmonic micrograph of 3D ferroelectric domain pattern fabricated in BCT crystal by femtosecond laser pulse

Fig. 17. Ferroelectric domain reversal in BCT crystal realized by using femtosecond laser^[67]. (a) Laser direct writing of ferroelectric domain structure in y-z plane with θ-like domain structure observed by using Čherenkov second harmonic microscope, in which background (weak random patterns around θ domain) originates from random submicron periodic domain structure inherent in BCT crystal; mechanism of formation of θ-shaped domain wall structure in BCT crystals: (b) temperature gradient induced by nonlinear absorption of near-infrared laser by BCT crystal (arrows pointing to high temperature region); (c) projection of temperature gradient along z-axis direction at different positions around focal point; (d) pair of oppositely oriented thermoelectric fields used to polarize (e) originally disordered random domains into (f) pair of head-to-head ferroelectric domains

Fig. 18. 3D QPM frequency-doubling phenomenon^[67]. (a) Far-field frequency-doubling spot pattern observed in simple tetragonal ferroelectric domain structure with frequency-doubling light wavelength of 800 nm; (b) schematic of 3D QPM of 3D ferroelectric domain structure modulation; (c) divergence outer angle of frequency-doubling loop with h=-1 versus fundamental wavelength and Λ_x(curve represents theoretical value, hollow point represents measured value); (d) experimental and (e) theoretical distributions of frequency-doubling bright spots on Čherenkov ring with h=-2; (f) relationship between fundamental frequency optical power and colinear frequency-doubling optical power modulated by 3D domain structure with Λ_x=25.6 μm (hollow point represents measured value, and solid curve represents quadratic fitting curve), and inset shows frequency doubling optical power as a function of wavelength after annealing (solid point represents experimental value, and curve represents theoretical calculation value)

Fig. 19. Erasable properties of domain structures in BCT crystals^[67]. (a) Čherenkov second harmonic microscopic image of BCT after annealing below Curie point; (b) Čherenkov second harmonic microscopic image of BCT after annealing above Curie point

Fig. 20. Sample characterization^[75]. (a) Phase diagram of KTN crystal structure; (b) photoelectron peaks of Ta 4f and Nb 3d in KTN crystal and their corresponding fitting curves; (c) electrical hysteresis loop of KTN crystal; (d) endothermic and exothermic effect curves of KTN crystal measured by differential scanning calorimetry (DSC)

Fig. 21. Sample characterization^[75]. (a) Image of ferroelectric domain structure on a-b plane of KTN crystal under polarized light microscope; (b) statistical distribution result of period length for KTN crystal 180° random domain

Fig. 22. Crystal structure and ferroelectric domain distributions of KTN^[75]. (a) Schematic of cubic-like domain of natural Rubik's cube structure in KTN crystal; (b) structure model of x-y plane ferroelectric domain (out-of-plane) of KTN crystal with × and · indicating internal and external polarization; (c) structure model of x-y plane ferroelectric domain (in-plane) of KTN crystal; (d) vertical PFM amplitude image and (e) transverse PFM phase image of KTN x-y plane in same area with scale bar of 2 μm

Fig. 23. Linear optical response of KTN crystal^[75]. (a) Bragg diffraction diagram of KTN supercell; visible laser Bragg diffraction spots of (b) a cut and (c) c cut KTN crystal samples at room temperature

Fig. 24. Nonlinear optical response of KTN crystal^[75]. (a) Schematic of near-field second harmonic spot imaging of KTN supercell; second harmonic microscopy images of (b) b-c and (c) a-b planes in KTN crystal with scale bar of 25 μm

Fig. 25. Statistical distributions of supercells^[75]. (a)(b) Statistical distributions of supercells in x-direction KTN crystal; (c)(d) statistical distributions of supercells in z-direction KTN crystal

Fig. 26. Second harmonic generation based on 3D KTN crystal generated by experiment and simulation^[75]. (a) Schematic of 3D inverse lattice vector of KTN crystal supercell; theoretical fitting of each QPM process in KTN supercell and comparison with experimental results: (b) two-dimensional nonlinear Bragg diffraction spot; (c) second harmonic spot generated by collinear QPM; (d)(e) second harmonic spots generated by interaction of fundamental frequency diffracted laser with two-dimensional inverse lattice vector in vertical plane of incident laser; (f)-(h) second harmonic spots produced by interaction of fundamental frequency diffracted laser with inverse lattice vector in direction of incident laser; (i) comparison of second harmonic results obtained by experiment and theoretical fitting

Fig. 27. Second harmonic intensity distribution images for different polarizations^[75]. (a) Schematic of experimental setup for 3D QPM second harmonic generation; (b) second harmonic spot distribution when incident fundamental laser is polarized along b-axis; (c) second harmonic spot distribution when incident fundamental laser is polarized along c-axis; (d) relative intensity of second harmonic laser at different polarization states when incident fundamental laser is polarized along b-axis; (e) relative intensity of second harmonic laser at different polarization states when incident fundamental laser is polarized along c-axis; (f) quadratic relationship curve between second harmonic laser power and fundamental laser power

Fig. 28. Experimental results^[75]. (a) Broadband second harmonic spectra of a-direction KTN crystal; second harmonic power versus incident power for fundamental laser wavelengths of (b) 900 nm, (c) 1040 nm, and (d)1160 nm

Table 1. Common ferroelectric crystals and their basic ferroelectric properties^[31]