# Photonic Stopband Filters Based on Graphene-Pair Arrays
## 2 Graphene-Pair Arrays  
The graphene pairs are inserted in a bulk dielectric to form the graphene-pair arrays as shown in Figure 1. Here, the bulk dielectric is silicon dioxide (SiO\({}_{2}\)) with a refractive index of \(n_{d}\). Silicon dioxide is the substrate to install graphene pairs. The substrate material of silicon dioxide is homogeneous and presents isotropy in the THz band. The interval of the two graphene sheets in a graphene pair is denoted by \(d_{1}\) and here, is set as \(d_{1}=10\) nm. The spatial distance between two adjacent graphene pairs is \(d_{2}=10\)\(\upmu\)m. The parameters \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Structures** & **Authors** & **Optical Effects** & **Refs.** \\ \hline Graphene arrays & [PERSON], [PERSON] et al. & Spatial solitons & [15] \\ Graphene arrays & [PERSON], [PERSON] et al. & Optical bistability & [19] \\ Graphene multilayers & [PERSON], [PERSON] et al. & Plasmonic supermodes & [18] \\ Graphene multilayers & [PERSON], [PERSON] et al. & Rabi oscillations & [20] \\ Semi-infinite graphene arrays & [PERSON], [PERSON] et al. & Surface solitons & [16] \\ Graphene-pair arrays & [PERSON], [PERSON] et al. & Rabi oscillations & [21] \\ Graphene-pair arrays & [PERSON], [PERSON] et al. & Vector solitons & [17] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Research progress on graphene arrays. satisfy the condition \(d_{2}\gg d_{1}\). On the whole, the structure is periodic, and the spatial period of graphene pairs can be viewed as \(d=d_{1}+d_{2}\). Graphene is an ultrathin two-dimensional nanomaterial and is generally treated as an equivalent dielectric with a considerable thickness of \(d_{g}\). This equivalent method for graphene is reasonable under the condition of \(d_{g}<1\) nm, which has been verified by numerous experimental and simulating results [16; 17]. The equivalent permittivity of graphene can be expressed as  
\[\varepsilon_{g}=1+i\frac{\sigma\eta_{0}}{k_{0}d_{g}}. \tag{1}\]  
Here, the equivalent thickness of graphene is set as \(d_{g}=0.33\) nm. The symbol \(i\) is the imaginary unit. The parameter \(k_{0}\) represents the wave number of incident wave in vacuum and the vacuum resistivity parameter is denoted by \(\eta_{0}\). Other parameters are given as the following: the period number of graphene-pair arrays is \(N=40\) and the permittivity of SiO\({}_{2}\) is provided with a fixed value of \(\varepsilon_{d}=(n_{d})^{2}=2.1\). The conductivity of graphene is governed by the Kubo formula [19; 20; 21] and the complete expression is given by  
\[\sigma(\omega,\mu,\tau,T_{g})=-\frac{ie(\omega+i\tau^{-1})^{2}}{\pi\hbar^{2}} \Bigg{[}\int_{-\infty}^{+\infty}\frac{|\xi|}{\left(\omega+i\tau^{-1}\right)^{ 2}}\frac{\partial f_{d}(\xi)}{\partial\xi}d\xi-\int_{0}^{+\infty}\frac{ \partial f_{d}(-\xi)-\partial f_{d}(\xi)}{\left(\omega+i\tau^{-1}\right)^{2}-4 \left(\xi/\hbar\right)^{2}}d\xi\Bigg{]}, \tag{2}\]  
where \(f_{d}=1/(1+\exp[(\xi-\mu)/(k_{B}T_{g})])\) is the Fermi-Dirac distribution function, \(\hbar\) is the reduced [PERSON]'s constant, \(k_{B}\) is the [PERSON]'s constant, \(\xi\) is the electron energy, \(\mu\) is the chemical potential of graphene, \(T_{g}\) is the environment temperature, \(-e\) is the charge of an electron, \(i\) is the imaginary unit, \(\omega\) is the angular frequency, and \(\tau\) is the phenomenological relaxation time of electrons. The environment temperature is set as \(T_{g}=300\) K. The integration for the first term of Equation (2) can be rewritten by  
\[\sigma_{\rm intra}=i\frac{e^{2}k_{B}T_{g}}{\pi\hbar^{2}}\left\{\frac{\mu}{k_{B }T_{g}}+2\ln[\exp(-\frac{\mu}{k_{B}T_{g}})+1]\right\}, \tag{3}\]  
which dominates the scattering process of intraband electron-photon in graphene. The excited motions of electrons in graphene not only include intraband transitions, but also the interband transitions. The second term of Equation (2) represents the interband scattering.