## 3 Results and Discussion  
With the significant enhancement and localization of the electric field in the silicon nanostructures, enhanced optical bistability could be achieved. As per the hysteresis phenomenon between the input and output lights, optical bistability relies on the Kerr nonlinear medium inside a resonator. The relative permittivity of the Kerr nonlinear medium under strong irradiation has a nonlinear relationship with respect to the electric field, which is determined as follows:  
\[\varepsilon_{\mathrm{r}}=\varepsilon_{\mathrm{L}}+\chi^{(3)}\;\mid E\;\mid^{ 2}. \tag{2}\]  
where \(\varepsilon_{\mathrm{L}}\) is the linear relative permittivity, E is the electric field inside the nonlinear medium, and \(\chi^{(3)}\) is the third-order nonlinear coefficient of the Kerr medium. For the proposed mirror-induced MAS-based metasurfaces, the nonlinear medium is silicon, with the nonlinearity of silver being neglected due to its inferior intrinsic nonlinearity and weak distribution of local fields. Rather than traditional silicon, we choose hydrogenated amorphous silicon with a \(\chi^{(3)}\) of 2.84348 \(\times\) 10\({}^{-19}\) m\({}^{2}\)/V\({}^{2}\). From Equation (2), it can be seen that the \(\varepsilon_{\mathrm{r}}\) is estimated by \(E\), while \(E\) is determined by the \(\varepsilon_{\mathrm{r}}\) of the resonator. Equation (2) gives rise to a self-consistent problem, which requires the iterative numerical solution of \(E\) (local electric field) using COMSOL [23; 24]. The numerical implementation of bistability starts from the outside of the hysteresis loop. In each step of the self-consistent calculation, the electric field distribution from the previous step is used as the initial value. This guarantees convergence to the correct branch of the hysteresis loop. The upper hysteresis branch is obtained by gradually increasing the intensity of incident light. The lower hysteresis branch is obtained by reducing the intensity of incident light from a high level. When the input intensity enters the hysteresis loop region from either direction, the iteration process slows down, indicating a significant nonlinear process within the bistability region. For optical bistability generated from resonances, the incident operational wavelength \(\lambda_{\mathrm{in}}\) must be chosen to satisfy the critical condition as follows [40]:  
\[\left|(\lambda_{\mathrm{res}}-\lambda_{\mathrm{in}})/(w_{\mathrm{FW}}/2) \right|>\sqrt{3} \tag{3}\]  
Figure 4: Average EF of the electric field by asymmetrically optimizing the proposed mirror-induced MAS-based metasurface. (**a**) Dependence on \(L_{1}\) and height duty ratio \(H_{2}/(H_{1}+H_{2})\). \(W_{1}=210\) nm. (**b**) Dependence on \(W_{1}\) and \(H_{2}/(H_{1}+H_{2})\). \(L_{1}=\)190 nm. Except when specified, parameters are the same as those in Figure 1a. where \(\lambda_{\text{res}}\) is the resonant wavelength and \(w_{\text{FW}}\) is the full width half maximum (FWHM) of the resonance. Since the proposed metasurface has a strong resonance, and thus a narrow FWHM, the incident operational wavelength \(\lambda_{\text{in}}\) should be carefully selected to ensure a good compromise between the wide distance of bistability thresholds and the large ON/OFF ratios. We have already chosen \(\lambda_{\text{in}}=\lambda_{\text{res}}+1.5w_{\text{FW}}\). Figure 5a demonstrates the hysteresis loops of the mirror-induced MAS-based metasurfaces as a function of the incident intensity. For this, generated from the proposed metasurface after asymmetrical optimization, it can be seen that, when the incident intensity increases, the reflectance maintains high values steadily, which is considered as the ON state. As the incident intensity rises to reach the threshold of \(I_{\text{ON-OFF}}=8.5\) MW/cm\({}^{2}\), the bistable switch transforms from ON state to OFF state, and the reflectance drops to a small value of R = 0.14 from R = 0.79. The ON/OFF ratio is calculated to be 5.6.