## 3 Optical Bistability, Tristability and Multistability
It shows that there is a waterfall in Re(\(\sigma_{g}\)) as the wavelength and the surface conductivity change. The incident wavelength is \(\lambda\) = 2.066 \(\upmu\)m at the saltus step point in the surface conductivity for \(\mu_{c}\) = 0.30 eV, while the wavelength at the saltus step point is \(\lambda\) = 2.066 \(\upmu\)m for \(\mu_{c}\) = 0.55 eV. Therefore, the incident wavelength at these points of dramatic change in the surface conductivity is blueshifted as \(\mu_{c}\) increases, indicated by the red arrow. The chemical potential can be regulated by an external voltage on graphene. On the other hand, the transmit phase of wave in the arrays of graphene sheets, which can affect transmittance of light, is a function of the surface conductivity. Therefore, the transmittance of light can be flexibly regulated by an external voltage. The imaginary part Im(\(\sigma_{g}\)) of the surface conductivity represents the real part of the refraction index of graphene. For three given different values of chemical potential, Figure 6c demonstrates Im(\(\sigma_{g}\)) varying with the incident wavelength. There is a sink in each curve and the sink has a blue shift as \(\mu_{c}\) increases. However, this obvious feature has a very limited effect on the transmittance due to graphene being ultrathin. It may significantly disturb the complex phase of reflection coefficient. Figure 6d describes the imaginary part of surface conductivity in the parametric space composed of the chemical potential and incident wavelength. There is a groove in the schematic of Im(\(\sigma_{g}\)) indicated by the arrow. That means the incident wavelength at the valley of Im(\(\sigma_{g}\)) is redshifted as the chemical potential decreases. Drastic changes in the real index of refraction of equivalent dielectric of graphene have only a slight effect on the transmittance. If we keep on increasing the incident light intensity, a settled input intensity of light can correspond to several transmittances with a positive slope as shown in Figure 7a. This phenomenon indicates that tristability or multistability could be induced in the input-output relationship as the incident intensity is strong enough. In addition, the nonlinear transmittance may overlap for a stronger incident intensity of light. For the incident wavelength \(\lambda\) = 1.308 \(\upmu\)m, we can see that the linear transmittance is larger at 0.5 eV than that at 0.40 or 0.45 eV in Figure 4a. Interband transition dominates for \(\hbar\omega_{1}\) < 2\(|E_{F}|\), so the chemical potential affects electron transition in graphene. The nonlinear part of the equivalent refractive index of graphene is proportional to the incident intensity of light. A change in refractive index induces phase shifts of the transmission coefficient in the arrays of graphene sheets. Therefore,  
Figure 6: (**a,c**) Re(\(\sigma_{g}\)) and Im(\(\sigma_{g}\)) versus the incident wavelength for some given chemical potentials of graphene, respectively. (**b,d**) Re(\(\sigma_{g}\)) and Im(\(\sigma_{g}\)) changing with the chemical potential and incident wavelength. The red arrow indicates the blueshift of incident wavelength at the saltus step point in surface conductivity as \(\mu_{c}\) increases. The green arrow notes the direction of valley shift of Im(\(\sigma_{g}\)) as \(\mu_{c}\) decreases. the transmittance varies with the incident intensity and the influence of incident intensity on the transmittance is more obvious for a stronger incident intensity of light. Figure 7b provides optical tristability and multistability for different values of chemical potential. For \(\mu_{c}=0.50\) eV, one can see that the vertical dotted line intersects the input-output curve at three points, i.e., \(S_{1}\), \(S_{2}\) and \(S_{3}\). This illustrates that for one input there can be three stable outputs, known as optical tristability. As light intensity keeps on increasing, multiple outputs appear for each input, which is known as optical multistability.