Fig. 1. (a) Schematic of the proposed metasurface consisting of a silicon dioxide grating on top of a lithium niobate thin film on a silicon dioxide substrate. The yellow arrow k defines the wave vector of the emitted photons. k_⊥ and k_∥ are the transverse and longitudinal components of the wave vector, respectively. (b) Cross section of one unit cell of the metasurface; (c) wide-angle emission of photons from the subwavelength-thick metasurface, satisfying (d) energy matching and (e) transverse phase matching.

Fig. 2. (a) Quantum-classical correspondence between the SPDC and SFG; (b) CMT-predicted resonance wavelengths of the metasurface as a function of the transverse wave vector k_y at k_z = 0; (c)–(f) SFG efficiency as a function of the signal wavelength calculated by CMT (black circles) and COMSOL simulation (red lines) for different input transverse wave vectors of the signal (k_y,s = 0, 0.025 rad/µm) and wavelengths of the pump. The double of the pump wavelength is (c)–(d) 1569.84 nm and (e)–(f) 1567.33 nm, which are marked by the black dashed lines in (b).

Fig. 3. (a) SPDC brightness at the degenerate wavelength and (b) Schmidt number of the emitted photons as a function of the pump laser wavelength and Gaussian beam radius. Point A in (b) marks the peak Schmidt number at pump wavelength of 784.15 nm and beam radius of 200 µm. Point B corresponds to the same pump wavelength with a beam radius of 5 µm. (c), (d) SPDC emission patterns corresponding to the points A and B in (b), as indicated by labels.

Fig. 4. (a) Schmidt coefficients of the first 50 Schmidt modes corresponding to the point A in Fig. 3(b); (b)–(e) normalized amplitude and phase distributions of the first four Schmidt modes.

Fig. 5. (a) Schmidt coefficients of the first 50 Schmidt modes corresponding to the point B in Fig. 3(b); (b)–(e) normalized amplitude and phase distributions of the first four Schmidt modes.