## 3 Transmittance and Optical Bistability  
Figure 2a gives the transmission spectrum, as a TM wave normally impinges upon the lattices that are incorporated without graphene. The transmittance is defined as \(T\) = \(tt^{\star}\), where \(t\) is the transmission coefficient. We define the photonic bandgap of multilayers as \(\omega_{\mathrm{gap}}\) = \(4\omega_{0}\mathrm{arcsin}|(n_{b}-n_{a})/(n_{b}+n_{a})|^{2}/\pi\)[65], where \(\omega_{0}\) = 2\(\pi\)c/\(\lambda_{0}\) (\(\lambda_{0}\) = 150 \(\upmu\)m) is the center frequency of the bandgap. For three given gain-loss factors, such as \(q\) = 0, 0.05, and 0.1, there are seven peaks in each curve in the display range of abscissa. The abscissa represents the normalized frequency. The maxima of transmittance are denoted by P\({}_{1}\)-P\({}_{7}\) (being labeled by blue stats) in sequence. The transmittance at P\({}_{1}\) and P\({}_{7}\) will be grater than 1, that is, \(T\) > 1, since the gain existing in dielectrics for the gain-loss factor \(q\) > 0. The corresponding transmittance for P\({}_{7}\) is the largest, followed by P\({}_{1}\). The gain in dielectrics and the PT symmetry of structure induce the transmittance to exceed 1. The maxima P\({}_{1}\)-P\({}_{7}\) correspond to different defect modes, and the resonance of defect modes at P\({}_{1}\) and P\({}_{7}\) is stronger than that at other points. The transmittance of peak P\({}_{7}\) is as high as \(T\) = 15 for \(q\) = 0.1, while the maximum transmittance of other points is \(T\) = 2. We cut the transmittance near P\({}_{7}\) in order to make the transmittance of most other points more clear. Meanwhile, the transmittance near P\({}_{7}\) is shown as a whole in the inset. The dotted box is used to mark the area near the P\({}_{7}\) shown in the inset. The momentum relaxation time of graphene is \(\tau\) = 0.8 ps. This order of magnitude ps for \(\tau\) is often found in references. Further, it is found that the bistable threshold and threshold interval are moderate through simulation with \(\tau\) = 0.8 ps, which is convenient for us to conduct image display as a typical parameter. Figure 1: **(a)** Fibonacci sequence created by the law B\(\rightarrow\)BA, A\(\rightarrow\)B with F\({}_{1}\) = B, F\({}_{2}\) = BA and F\({}_{j}\) = F\({}_{j-1}\)\(|\)F\({}_{j-2}\) for \(j\geq 3\). **(b)** Schematic of PT-symmetric lattices with Fibonacci sequence dielectrics and graphene, where the indices of refraction are respected for \(n(z)=n^{\star}(-z)\) about \(z=0\). For the primitive unit-cell layers A, B, C, and B, the complex refractive indices are \(n_{a}=1.38+qi\), \(n_{b}=2.35-qi\), \(n_{c}=2.35+qi\), and \(n_{d}=1.38-qi\), respectively. Figure 2b gives the corresponding transmittance for the lattices with graphene embedded in the middle. When compared with the structure without graphene, P\({}_{7}\) is also at the maximum peak, but overall, the transmittance has gone down. Figure 2a,b show the transmittance of light in the lattices without and with graphene, respectively. The transmission curves are similar, because the aperiodic PT-symmetric lattices that cause the resonance have not changed. The addition of graphene, due to the loss of graphene, will only cause the decrease of transmittance, but it will not affect the resonance of the lattices and the distribution of mode field. Similarly, in both structures, we can enhance the transmittance by increasing the gain-loss factor. Light is reflected back and forth in the resonator. The loss in optical field energy results in a decrease of transmittance because of the loss that existed in graphene. Figure 2c-i orderly give the corresponding transverse distributions of electric field intensity (\(|E_{2}|^{2}\)) for the defect states at peaks P\({}_{1}\)-P\({}_{7}\), such as Figure 2c corresponding to P\({}_{1}\), Figure 2d corresponding to P\({}_{2}\), etc. The peaks P\({}_{1}\)-P\({}_{7}\) are all resonant states and P\({}_{1}\) and P\({}_{7}\) correspond to band-edge resonances.