# Photonic Stopband Filters Based on Graphene-Pair Arrays
## 3 Photonic Bandgaps  
For a transverse magnetic (TM) wave, as it normally impinges on the periodic structures composed of graphene pairs from the left, Figure 2a gives the transmittance versus the frequency of light. The relaxation time is set as \(\tau=0.5\) ps, and the graphene chemical potential is given by \(\mu=0.4\) eV. The Bragg period number is \(N=40\). There are four photonic bandgaps in the spectrum in our given interval of frequency, and they are respectively defined as the first, the second, the third and the fourth bandgap in turn from left to right. The bandgap width declines as the frequency increases. The widths of the first \(\sim\) fourth bandgaps are 1.44, 0.74, 0.48 and 0.36 THz in order, and their corresponding center frequencies are 11.05, 21.04, 31.24 and 41.51 THz, respectively. On both sides of each bandgap, there are two passbands as well. Otherwise, the average transmittance of light in each transmission band can be elevated by increasing the incident frequency. The photonic bandgap is analogous to the energy band structure of electrons in semiconductors. The bandgaps in transmission spectra can naturally be utilized for band-stop filters [11], while the edge states at the bandgaps can greatly localize the electric field, which has extensively been applied to realize optical solitons [16; 17], topological boundary states [27; 28] and low threshold optical bistabilities [29; 30; 31]. The corresponding working wave of the filters is in the range of [6 um, 37.5 um], which locates at the middle-infrared band. As the TM polarized light wave is incident on the graphene-pair arrays from the left, keeping the parameters above unchanged, the homologous reflection spectrum is provided  
Figure 2: (**a,b**) Transmission and reflection spectra for the normally incident light waves from the left, respectively. The spatial intervals are set as \(d_{1}=10\) nm and \(d_{2}=10\) μm, respectively. in Figure 2b. There are four reflection pulses, viz. high reflectance sections, in the reflection curve as the frequency of light wave changes. The four frequency bands of high reflection exactly agree with the stopbands in the transmission spectrum, respectively. Otherwise, in the high reflection frequency bands, the reflectance is close to 1 at the rising edge of each reflection pulse and slightly lowers as the frequency increases. However, the value of reflection drops abruptly at the falling edge of each reflectance pulse. Since the chemical potential is one of the parameters which can affect the surface conductivity of graphene, the equivalent permittivity of graphene is a function of the chemical potential \(\mu\). Subsequently, one can modulate the transmission performance of the periodic structure by the graphene chemical potential flexibly. For different values of chemical potential, Figure 3a provides the transmittance varying with the frequency of light wave around the first stopband. As a TM-polarized light is normally left incident on the arrays, the falling edge position of the first band always stays unchanged as the chemical potential increases. The transmittance is equal to zero at the first bandgap, which means that there is no light that can transmit this periodic structure as an incident light wavelength locates at the stopband. By increasing the frequency of light, there is an inflection point at the right of each transmission stopband, and then the transmittance increases as the frequency passes the inflection point. We define the interval between the falling edge and the inflection point as the stopband width. The inflection point at each stopband is moving to the right by modulating the chemical potential to a larger value. In other words, the bandgap width enlarges as the chemical potential increases. On the right of each bandgap, the transmittance is not zero and it increases with the increase in light frequency.