# Photonic Stopband Filters Based on Graphene-Pair Arrays
## 3 Photonic Bandgaps
(**b,d**) Real part and imaginary part of graphene surface conductivity by modulating simultaneously the light frequency and chemical potential, respectively. of light, changing the graphene chemical potential or modulating the light frequency are important measures in manipulating light propagation in the graphene-pair arrays. Around the parameters of \(\mu=0.2\) eV and \(f=100\) THz, there is an abrupt change in \(\mathrm{Re}(\sigma)\). It results from the transformation of electrons intraband transition to interband transition in graphene. Figure 4c gives the imaginary part of surface conductivity \(\mathrm{Im}(\sigma)\) changing with the light frequency. For different values of \(\mu=0.3\), \(0.4\) and \(0.5\) eV, it shows that \(\mathrm{Im}(\sigma)\) decreases with the increase in light frequency in each of the three corresponding curves. Similarly, for a given incident wavelength, a larger chemical potential induces a bigger value in \(\mathrm{Im}(\sigma)\). The imaginary part of surface conductivity determines the real part of \(\varepsilon_{g}\). Moreover, \(\mathrm{Im}(\sigma)\) is a function of the chemical potential \(\mu\). Therefore, we can modulate the stopband width by changing the structure parameter through an external gate voltage on graphene as shown in Figure 3c. Figure 4d describes \(\mathrm{Im}(\sigma)\) changing with the chemical potential and light frequency in the parameter space. It shows that \(\mathrm{Im}(\sigma)\) increases with the raise of the graphene chemical potential and decrease in the light frequency. The parameter of \(\mathrm{Im}(\sigma)\) determines the real part of graphene equivalent permittivity and \(\mathrm{Im}(\sigma)\) can be changed by \(\mu\), so we can regulate the transmittance by the graphene chemical potential flexibly. The interval between two adjacent graphene pairs is \(d_{2}\) and the spatial period of graphene-pair arrays is \(d_{1}+d_{2}\). The parameter \(d_{2}\) can also affect the transmission spectrum of light and the stopband width and the position in the transmission spectrum. For different values of \(d_{2}\), Figure 5 gives the transmittance by changing the frequency of the light wave. As a light impinges on the arrays from the left, it demonstrates that there are four stopbands in the frequency range of [10; 50] THz as the spatial distance is \(d_{2}=10\) \(\upmu\)m, and the stopband number respectively increases to \(6\) and \(8\) for \(d_{2}=15\) and \(20\) \(\upmu\)m. Therefore, one can regulate the stopband number by tuning the value of \(d_{2}\). As mentioned in Equation (2), the surface conductivity of graphene is a function of the parameter of \(\tau\), which is defined as the phenomenological relaxation time of electrons in graphene. Figure 6a gives the transmission spectra for different values of \(\tau\). One can see that all of the bandgap positions in the spectra hold still as the parameter \(\tau\) varies arbitrarily. However, on the whole, the photonic passbands in the transmission spectra are elevated with the value increases in \(\tau\). That is because the optical loss in graphene reduces as the phenomenological relaxation time rises. The phenomenological relaxation time only governs the surface conductivity of graphene but does not determine the geometric parameters of the structure composed of graphene-pair arrays. Therefore, the stopband gap positions in the transmission spectra are unaffected by \(\tau\). Figure 5: Transmittance varying with the frequency of light wave for different spatial intervals of two adjacent graphene pairs. The graphene chemical potential is \(\mu=0.4\) eV, and the relaxation time is set as \(\tau=0.5\) ps. The graphene surface conductivity \(\sigma\) is also a function of the phenomenological relaxation time \(\tau\). Figure 6b gives the real surface conductivity \(\text{Re}(\sigma)\) for different values of \(\tau\). For three given values of \(\tau\) = 0.3, 0.4 and 0.5 ps, the value of \(\text{Re}(\sigma)\) in each corresponding curve changes with the light frequency.