# Photonic Stopband Filters Based on Graphene-Pair Arrays
## 3 Photonic Bandgaps
As mentioned above, the real part \(\text{Re}(\sigma)\) decides the optical loss in graphene. Since \(\text{Re}(\sigma)\) decreases with the increase in the light frequency for a given value of \(\tau\), the transmittance increases by turning up the frequency of light as shown in Figure 6a. Otherwise, it also shows that, by fixing the incident frequency, a lower value of \(\tau\) corresponds to a bigger \(\text{Re}(\sigma)\), which means that a larger optical loss may be induced. However, the imaginary part of surface conductivity \(\text{Im}(\sigma)\) is not affected by \(\tau\) as shown in Figure 6c. The value of \(\text{Im}(\sigma)\) decreases with the increase in the light frequency for different values of \(\tau\), but the three surface conductivity curves almost coincide with each other for \(\tau\) = 0.3, 0.4 and 0.5 ps. The parameter of \(\text{Im}(\sigma)\) governs the real equivalent permittivity of graphene, so one cannot affect the stopbands in the transmission spectrum by modulating the phenomenological relaxation time. Figure 7a demonstrates the transmittance in the parameter space. The parameter space is composed of the incident angle and the light frequency. For the left incident light, the fringes of brighter color in transmission space represent the photonic band gaps of light, i.e., the stopbands. For a sharp contrast, we have taken the logarithm of transmittance. The gap widths of these fringes decrease from the left to the right as the light frequency increases. For each stopband, the fringe profile in transmission moves toward higher frequency by promoting the incident angle. There is a full transmission band between the first stopband and the second stopband. We have denoted the full passband by passband I, which is sandwiched by two dotted lines. It means that, no mater how the incident angle changes, the incident light waves can transmit this structure as a light frequency locates in passband I. Furthermore, there is also another full transmission band labeled by passband II. Passband II is local between the second stopband and the third stopband. The width of passband II is narrower than that of passband I. There are two passbands in the transmission parameter space and as the frequency increases, there is no new full transmission band that arises. Figure 6: (**a**) Dependence of the transmission spectra on the phenomenological relaxation time \(\tau\). (**b,c**) Real part and imaginary part of graphene surface conductivity varying with \(\tau\), respectively. The graphene chemical potential is set as \(\mu\) = 0.4 eV for (**a–c**). For four given incident angles, Figure 7b provides the corresponding transmission spectra of light wave. It shows that the spectrum profile moves rightwards on the whole by increasing the incident angle. The bandgaps widths of the same ordinal stay unchanged. Therefore, the stopband position of light frequency in the transmission spectrum can be modulated by changing the incident angle as well. For a fixed wavelength, the Z-axis component of the corresponding wave vector is minished by turning up the incident angle, so the bandgap position moves towards a shorter wavelength. The system is a one-dimensional periodic structure along the Z-axis, and the XY-plane is infinite. In experiments, as long as the sizes of the device in the X and Y directions are two orders of magnitude larger than the incident wavelength, the design conditions can be satisfied. Large areas and high-quality monolayers of graphene and dielectric layers can be stacked layer by layer by chemical vapor deposition [32, 33]. This is a common and well-established technique that has been widely used in the manufacture of integrated circuits. Otherwise, graphene arrays have been utilized for the electron acceptors experimentally as well [34, 35]. Therefore, it would be desirable to realize the stopband filters based on graphene-pair arrays in laboratory.