## 3 Transmittance and Optical Bistability
The aperiodic quasicrystals of Fibonacci multilayer dielectrics can enhance the band-edge resonances [55], which can be further enhanced by increasing the gain in our PT-symmetric structure. The mode distribution is designed to show the localization of electric field for different resonance states, that is, the electric fields are restricted in different defect layers. It can be seen from the mode distribution diagram that the resonant states of P\({}_{1}\) and P\({}_{7}\) originates from the central defect layer, as shown in Figure 2c,i, and the electric field distribution of these two states is more concentrated than that of other resonant states. However, when compared with P\({}_{1}\), the resonant state of P\({}_{7}\) is more localized to the electric field. Therefore, the nonlinear of graphene is the strongest around P\({}_{7}\), and the threshold for realizing optical bistability is relatively low. The bistable state will be simulated near P\({}_{7}\) in the following study. This can not only enhance the third-order equivalent nonlinear refractive index of graphene, thereby reducing the bistable thresholds, but also obtain a large transmitted light intensity. Figure 2: (**a,b**) Transmission spectra for PT-symmetric lattices. For (**a**) the lattices incorporated without graphene and for (**b**) the lattices with graphene. The chemical potential of graphene is \(\mu\) = 0.05 eV and the momentum relaxation time is \(\tau\) = 0.8 ps. (**c**–**i**) The electric field intensity (\(|E_{x}|^{2}\)) distributions of the resonant states at the peaks P\({}_{1}\)–P\({}_{7}\) of \(T\) [labeled successively by blue stars in (**a**)]. We first give the nonlinear surface conductivity coefficient of graphene when it comes to calculating the bistable effect. We will continuously modify the nonlinear transmittance and transmission intensity through inverse TMM, and finally obtain the stable transmittance and output light intensity [9]. In this case, the dielectric constant of graphene is \(\varepsilon_{1}=1+\text{i}\sigma^{(1)}\eta_{0}/(kd_{g})\) is the linear dielectric constant and \(\chi_{g}{}^{(3)}=\text{i}\sigma_{3}\eta_{0}/(kd_{g})\) is the cubic volume susceptibility. The coefficient is governed by \(\sigma^{(3)}=-3\text{i}e^{4}V^{2}/8\pi\hbar\omega^{3}\mu\)[66; 67], where \(V\) is the Fermi velocity and, here, we set it as \(V\approx\text{c}/300\)[28]. The equivalent dielectric constant for graphene can be denoted by two parts, _viz._\(\varepsilon_{g}=\varepsilon_{1}+\chi_{g}{}^{(3)}|E_{2}|^{2}\). The calculation of nonlinear transmission is based on a nonlinear correction for the linear permittivity of dielectrics. The output electric field and magnetic field are \(E_{\text{o}}\) and \(H_{\text{o}}=E_{\text{o}}/\eta_{0}\), respectively. The total transfer matrix of dielectrics from the \(l+1\)th layer to the last layer is denoted by \(M^{\prime}=[m^{\prime}{}_{11},\,m^{\prime}{}_{12},\,m^{\prime}{}_{21},\,m^{ \prime}{}_{22}]\). The input electric field intensity of the \(l+1\)th layer \(|E_{z,l+1}|^{2}=|E_{\text{o}}|^{2}/T^{\prime}\), where \(T^{\prime}\) is the transmittance of light that impinges upon the dielectrics from the \(l+1\)th layer. Figure 3a plots the transmittance surface of light in the parameter space of the incident intensity of light and gain-loss factor \(q\). The slope of transmittance curve can be negative and positive, as \(q\) and incident intensity change under considering the third-order nonlinear correction for the equivalent index of graphene. We can see that the curve of transmittance is positive for a small \(q\leq 0.0275\) and the slope of transmittance curve can be negative for a great \(q>0.0275\). Meanwhile, around the maximum transmittance in each curve with a fixed value of \(q\), the transmittance abruptly changes with the incident light intensity. Otherwise, it shows there is a corresponding transmission curve on the transmittance surface for each gain-loss factor. The ridge of the curves family is composed by the maximum in each transmission curve.