## 3 Optical Bistability, Tristability and Multistability  
The reflectivity and transmittance are given in Figure 3a as a light impinges upon the arrays of graphene. One can see that there is an upward saltus step on the transmission curve. As the input wavelength is lower than 1.25 \(\upmu\)m, the transmittance \(T\) fluctuates around 0.5. While the wavelength is greater than 1.25 \(\upmu\)m, the transmittance approximates to 1 and fluctuates around this value. There are many peaks in transmission spectrum and these maxima in the curve correspond to resonant states. In addition, the reflectivity \(R\) is low and fluctuates around zero, indicating that most energy of light has been transmitted. The peaks of transmittance face to the zero points of reflectivity exactly. Interband transition of electrons dominates as a short wavelength is incident in the graphene sheets, inducing large loss in intensity of light wave, while the incident wave with long wavelength mainly triggers intraband transition. As the intensity of incident light is strong enough, so that the third-order nonlinear modification to the surface conductivity of graphene reaches the order of linear counterpart, the influence of light intensity on the transmittance and output intensity of light should be considered. For a fixed incident wavelength, Figure 3b provides the relationship between transmittance and the intensity of  
Figure 3: (**a**) Linear transmission spectrum and reflection spectrum of low-intensity light impinging upon the arrays of graphene sheets. (**b**,**c**) Transmittance and output intensity of light wave changing with the input intensity of light wave, respectively. The wavelength of incident light is \(\lambda\) = 1.308 \(\upmu\)m. The chemical potential of graphene is \(\mu_{c}\) = 0.50 eV. incident light. It can be seen that the transmittance oscillates as the input light intensity increases. In addition, the amplitude of oscillation enlarges gradually. Furthermore, with the increase in intensity of incident light, the negative slope of transmittance curve appears, indicating that optical bistability may be achieved. Considering the nonlinear effect, the intensity of transmitted light increases with the incident intensity. Figure 3c provides the input-output relationship. Characteristics of this curve manifest so that it is similar to the hysteresis. By increasing the incident intensity of light, the output intensity makes an abruptly upward jump at \(I_{u}\), and the input-output curve varies along path I, while the output light intensity has a downward jump at \(I_{d}\) as the incident intensity declines, and the input-output curve goes along path II. So \(I_{u}\) is known as the upper threshold, and \(I_{d}\) is defined as the lower threshold. Therefore, for the fixed incident intensity between the values of \(I_{d}<I_{i}<I_{u}\), it corresponds to two stable resonant states \(S_{1}\) and \(S_{2}\). In general, the output-input intensity relation evolves along path I by increasing the light intensity, while it evolves along path II by decreasing the light intensity. The curve actually corresponds to a double valued function, similar to the hysteresis loop. The two paths represent different physical meanings and here we put them together in the same schematic to make a sharp contrast. Figure 4a shows the transmission spectra for three different external voltages on graphene. The value of chemical potential depends on the external voltages biased on graphene. The incident wavelength at the saltus step point of transmittance has a red shift as the chemical potential increases. The Fermi level, i.e., the chemical potential, is a critical value to characterize intraband and interband transitions in the energy band. Therefore, the critical incident wavelength triggering the intraband transition decreases as the external voltage on graphene rises. For different chemical potentials, Figure 4b displays their corresponding reflection spectra.