Fig. 1. (a) Schematic diagram of the nonlinear process. (b) Dispersion curve of the SPP on the gold film surface (red line) and the dispersion curve of the photons in vacuum (black line). The pump photon (pink), SPP (blue), and nonlinear photon (green) are connected by dashed lines, which represent the partially degenerate four-wave mixing interaction. (c) Spectrum detected at 5 nm above the gold film in simulation when the FWM happens.

Fig. 2. (a) Schematic of the grating structure used to excite the SPP. The arrow (blue) represents the polarization direction of the excitation light, which is radially polarized. (b) The E_x component of the SPP field generated by the third-order vortex radially polarized light incident on the grating. (c)–(h) The nonlinear signal with the x-polarization pump light when the coefficient is set as (c) 1.2 × 10⁻⁴ (m/v)², (d) 10⁻⁴ (m/v)², (e) 10⁻⁶ (m/v)², (f) 10⁻⁷ (m/v)², (g) 10⁻⁸ (m/v)², and (h) 10⁻⁹ (m/v)², respectively.

Fig. 3. (a)–(d) Right circular polarized components of the SPP field generated by the radially polarized light of l (l = 1–4) order vortex incident on the grating. (e)–(h) The nonlinear signals when the pump light is set to right circular polarized.

Fig. 4. (a) The structure to generate the PWB is designed based on the principle of the Fourier angle spectrum synthesis. O is the origin of the coordinates. (b) PWB excited under x linearly polarized light. (c) Nonlinear signal [yellow dashed areas in (a) and (b)] obtained under x linearly polarized pump light.

Fig. 5. (a),(b) Quadrilateral and hexagonal gratings, respectively. The blue circular arrow represents the polarization direction of the excitation light, which is left circular polarized. (c),(d) Left circular polarized and right circular polarized components of the SPP field generated by the quadrilateral grating under the left circular polarized excitation light. (e),(f) The corresponding nonlinear signals generated under the right and left circular polarized pump lights. (g)–(j) The corresponding results for the hexagonal grating.