Fig. 1. Enhanced second- and third-harmonic generation from an asymmetric nonlinear metasurface. Schematics of a nonlinear metasurface consisting of a periodic array of pairs of amorphous silicon bars placed on top of a glass substrate. The geometric asymmetry parameter, α, of the meta-atom is defined as α = ∆w/w.

Fig. 2. Simulated linear transmission spectra of the metasurface. (a) Dependence of transmission on the asymmetry parameter α under normal incidence and for x-polarized plane-wave excitation. The green dashed line corresponds to α = 0.075. (b) Dependence of transmission on the angle of incidence θ, determined for α = 0.075 and for x-polarized plane-wave excitation. (c, d) Same as in (a) and (b), respectively, but determined for y-polarized plane-wave excitation.

Fig. 3. Eigenmode analysis in case of α = 0.075. Band diagrams with respect to (a) k_y and (b) k_x. Only modes with Q-factors larger than 50 are included. Red solid (blue dotted) lines are modes excited by x-polarized (y-polarized) plane-waves. (c–h) In-plane electric field profiles of the eigenmodes at the Γ point, presented in order of increasing wavelength. The yellow dashed lines indicate the interfaces between amorphous silicon and air.

Fig. 4. Simulated dependence of Q-factor of the metasurface eigenmodes on the structural asymmetry parameter α for modes excited by (a) x-polarized and (b) y-polarized incident plane-waves. The calculations are performed at the Г point.

Fig. 5. (a) Top-view SEM image of a fabricated metasurface corresponding to α = 0.075. (b) Comparison of experimentally measured linear transmittance and the corresponding numerically calculated spectrum at α = 0.075. The purple arrow denotes the resonance that is used to illustrate the measured power-dependent second-harmonic and third-harmonic signals presented in (c). (c) Measured power-dependent output SH (red dots) and TH (blue dots) signals from the fabricated sample with α = 0.075. The dependence of nonlinear emissions is measured at the resonance with (pump) wavelength of 1620 nm under field polarization oriented along the x-axis. The dots represent the measured data while the black dashed lines denote the fitting lines of SHG and THG power.

Fig. 6. Enhancement factor η_E of the averaged electric field amplitude within the amorphous silicon resonators under plane- wave illumination with the asymmetry of the metasurface being α = 0.075.

Fig. 7. (a) Measured enhancement factor η_SHG of SHG from the all-dielectric metasurface with α = 0.075. An amorphous silicon slab with the same thickness as that of the metasurface is used as a reference. The inset shows a zoom-in of the enhancement factor. (b) Simulated linear transmittance for asymmetry parameter α = 0.075. (c, d) Numerically calculated surface and bulk contributions to SHG, respectively, when α = 0.075. Red (blue) lines correspond x-polarized (y-polarized)

Fig. 8. (a) Measured enhancement factor of THG from the resonant metasurface described by α = 0.075. (b) Numerically calculated TH intensity spectra corresponding to the same metasurface. Red (blue) lines correspond to x-polarized (y- polarized) incident plane-waves. In both figures, the upper ticks of the x-axis denote the TH wavelength.

Fig. 9. Fourier analysis of numerically calculated (a) transmitted SHG at 1590 nm (under x-polarized incident plane-wave), which corresponds to the resonant peak of highest measured SHG enhancement, and (b) transmitted THG at 1747 nm (under y- polarized incident plane-wave), which corresponds to the resonant peak of largest THG enhancement in experiments. Fourier analysis is performed for an asymmetry parameter of α = 0.075.

Fig. 10. Simulated dependence of SHG from the amorphous silicon metasurface with respect to the structural asymmetry parameters α in cases of (a) surface contribution under x-polaried excitation, (b) surface contribution under y-polarized excitation, (c) bulk contribution under x-polarized excitation, and (d) bulk contribution under y-polarized excitation.

Fig. 11. Numerical simulation of asymmetry-dependent TH intensity under plane wave excitation with polarization along (a) x and (b) y directions, respectively. In both figures, the upper ticks of the x-axis denote the TH wavelength.