Fig. 1. (a) Using a fork-shaped nonlinear grating, the fundamental Gaussian beam (ω₁) is converted into a second-harmonic (ω₂ = 2ω₁) vortex beam; (b) a three-dimensional fork-shaped nonlinear grating can convert the incident fundamental vortex beam into second-harmonic beams with different propagation directions and carrying different topological charge numbers.

Fig. 2. Diagram of experimental device for femtosecond laser direct writing of ferroelectric domain inversion structure.

Fig. 3. Ferroelectric domain structure of a three-layer fork-shaped nonlinear photonic crystal, obtained using a Čerenkov second-harmonic microscope^[33], is presented. (a) Three-dimensional structural imaging; (b)–(d) schematic representations of the first, second, and third fork-shaped structures.

Fig. 4. Distribution of the second-harmonic vortex beams generated by the three-layer fork-shaped structure in the far field is influenced by the topological charge carried by the fundamental vortex beam, which alters the position of the Gaussian fundamental beam. (a)–(c) display the far-field emission patterns of the second-harmonic vortex beams for a fundamental vortex beam with topological charges l_FB of −0.5, −1, and −1.5, respectively. (d)–(f) Present the corresponding theoretical simulations. (g)–(i) and (j)–(l) show the results when the sign of the topological charge of the fundamental vortex beam is reversed, with the position of the Gaussian beam shifting by 180° compared to the previous configuration.

Fig. 5. Power of the emitted second-harmonic (SH) vortex is presented as a function of the fundamental beam power, where the wavelength of the fundamental beam is 1560 nm.