Fig. 1. Three-dimensional structure diagram of Bessel-Gaussian beam emitter (dashed square area is inverse design region, which is composed of silicon and silica)

Fig. 2. Flow chart of inverse design

Fig. 3. Final results of inverse design. (a) FOM as a function of iterations, with regions Ⅰ, Ⅱ and Ⅲ denoting the graying, binarization and DFM stages, respectively, and insets illustrate the distributions of beam’s cross-sectional electric field intensity at iterations of 75, 220 and 330, respectively; (b) left panel is final material distribution of design area, and dot-dashed curve (dashed curve) in right panel represents the intersection of silicon and silica for over etching 9 nm (under etching 7 nm); (c) from left to right, the first and second figures are normalized field intensity distributions of the theoretical and optimized second-order Bessel-Gaussian beams, respectively (the insets illustrate interference patterns of Bessel-Gaussian beams with spherical wave),and the third and forth figures are the phase distributions of the theoretical and optimized second-order Bessel-Gaussian beams, respectively

Fig. 4. Analysis of etching error and wavelength disturbance. (a) Normalized electric field intensity distributions (upper panel) and phase distributions (lower panel) of optimized beam, and figures from left to right correspond to cases of over etching, standard etching and under etching, respectively. Insets illustrate the interference patterns of emitted beams with spherical wave. The lower right corner is marked as correlation of the beam; (b) correlation of optimized beam under different etching conditions in wavelength range from 1.5 µm to 1.6 µm. The inset shows the normalized transmission spectra of the emitter under three different etching conditions at 1.5 μm to 1.6 μm