## 3 Transmittance and Optical Bistability
The peak value in the transmission curve is the transmittance of the defect (resonant) state. Figure 3b shows the influence of \(q\) on the maximum transmittance. It demonstrates that the maximum increases with \(q\) and the incident light intensity that calls for achieving the resonant state can be decreased as \(q\) increases. Figure 3c is the profile of input-output intensity relation that changes with \(q\). The bistability can occur as the gain-loss factor \(q>0.0275\). We call the lowest \(q\) to realize OB for the critical factor. The slope of the transmittance curve can be negative when the local electric field is strong enough  
Figure 3: (**a,c**) Transmittance and transmitted intensity of light, respectively. The parameter space is composed of the gain-loss factor \(q\) and the incident intensity. (**b**) The peak transmittance and relevant incident intensity versus \(q\). (**d**) Threshold of bistability for different gain-loss factors. The chemical potential of graphene is \(\mu=0.05\) eV and the momentum relaxation time of graphene is \(\tau=0.8\) ps. The incident wavelength is \(\lambda=119.25\)\(\mu\)m.  
and the nonlinear correction of dielectric refractive index reaches the same magnitude of the linear refractive index. At this time, the input-output curve shows bistable profiles. The white dotted line divides the focusing area in the parameter space is divided into two parts. Part I represents the non-bistable region and part II represents the bistable region. Figure 3d demonstrates the lower threshold and upper threshold for OB as the bistability occurs. It shows that OB can be obtained as \(q>0.0275\). It is worth noting that, as the gain-loss factor increases, the interval between the upper and lower thresholds increases. At the same time, the upper threshold bistability increases with the increase of \(q\), while the lower threshold always decreases with the increase of \(q\). One can see that, when compared with the case for \(q=0.028\), the lower threshold of bistability could be lowered by 30% for \(q\) = 0.1 with a fixed incident wavelength. Modulating the gain-loss factor \(q=0.1\), the lowest incident light intensity in our structure is far below the threshold in projects by means of Fano-like resonance and Fano resonance [68; 69]. The lower threshold could be further decreased by enlarging \(q\). Figure 4a provides the peak value of transmittance at P\({}_{7}\) in the transmission spectra for different gain-loss factor \(q\). The curve observably demonstrates that the maximum transmittance increases with the increase of \(q\). The blue circles denote the transmittance of light for the lattices incorporated with graphene, and the red circles presents the transmittance, as the lattices are without graphene. For the same gain-loss \(q\), the transmittance of light that is injected in the structure with graphene is lower than that without graphene. The conductivity of graphene, the embedding graphene can lead to an increase in the reflectivity as a light impinges on the lattices, and graphene is a lossy material to light. In contrast to the absence of graphene, these two factors cause the decrease in transmittance, as the lattices contain graphene. Figure 4b gives the Q-factor of the microcavity in the PT-symmetric photonic lattices. We define the Q-factor as the center frequency over the FWHM of the transmission peak [39]. It shows that, as the gain-loss factor increases, the Q-factor also increases. When compared with the structure without graphene, the Q-factor is larger for the lattices that were incorporated with graphene. For \(q=0\), the values of Q-factor for the lattices without or with graphene are \(Q_{q=0}=519.2\) and \(361.9\), respectively. The Q-factor increases by 35% by modulating \(q=0.1\) in contrast to passive systems (\(q=0\)), as the graphene is embedded in the structure. For a resonant microcavity, the higher its quality factor, the stronger the confinement of the resonator to the light field.