Fig. 1. (a) Golden-angle spiral with b = 1.70 µm and a total of 3000 points; (b) nonlinear photonic quasi-periodic spiral structure based on rearrangement of α, with the two construction parameters α₁ being the golden angle and α₂ = 1.57, b = 1.70 µm; (c) nonlinear photonic quasi-periodic spiral structure based on rearrangement of b, with the two construction parameters b₁ = 1.50 µm and b₂ = 2.00 µm; (d)–(f) Fourier spatial spectra of (a)–(c).

Fig. 2. (a) Golden-angle spiral with b = 1.70 µm [an element of quasi-periodic spiral shown in Fig. 1(b)]; (b) Vogel’s spiral with α = 1.57 and b = 1.70 µm [the other element of quasi-periodic spiral shown in Fig. 1(b)]; (c) golden-angle spiral with b = 1.50 µm [an element of quasi-periodic spiral shown in Fig. 1(c)]; (d) golden-angle spiral with b = 2.00 µm [the other element of quasi-periodic spiral shown in Fig. 1(c)]; (e)–(h) Fourier spatial spectra of (a)–(d).

Fig. 3. (a) Typical second-harmonic pattern generated by the fabricated nonlinear photonic quasi-periodic spiral. Three ring-like patterns are modulated by the designed structure via nonlinear Raman–Nath diffraction. The crosswise line that lies in the middle is produced by spontaneous domains. (b) Geometrical relation of the phase-matching condition of nonlinear Raman–Nath diffraction; (c) variation tendency of external angle relying on fundamental wavelength.

Fig. 4. (a) Experimental and theoretical curves of second-harmonic intensity depending on external diffraction angle; (b) phase-matching diagram of third-harmonic generation in desired nonlinear photonic quasi-periodic spiral; (c) calculated pattern of cascaded third-harmonic and second-harmonic waves.