## 2 Periodic Graphene Arrays  
Single sheets of graphene are embedded in silicon dioxide to form arrays of graphene with a fixed spatial period as shown in Figure 1. Silicon and silicon dioxide are usually used for waveguides or substrate [29; 30]. In the arrays, the thickness of dielectrics, viz., silicon dioxide, is noted by \(d\) and \(N\) is the period number of graphene. Graphene is an ultrathin two-dimensional material equipped with distinct electrical conductivity. Its surface conductivity is governed by the Kubo formula [1; 31]:  
\[\sigma_{1}(\omega,\mu_{c},\tau,T)=-\frac{ie(\omega+i\tau^{-1})^{2}}{\pi h^{2}} \Bigg{[}\int_{-\infty}^{+\infty}\frac{|\epsilon|}{(\omega+i\tau^{-1})^{2}} \frac{\partial f_{d}(\varepsilon)}{\partial\varepsilon}d\varepsilon-\int_{0}^ {+\infty}\frac{\partial f_{d}(-\varepsilon)-\partial f_{d}(\varepsilon)}{( \omega+i\tau^{-1})^{2}-4(\varepsilon/h)^{2}}d\varepsilon\Bigg{]}, \tag{1}\]  
where\(f_{d}=1/(1+\exp[(\varepsilon-\mu_{c})/(k_{B}T)])\) is the Fermi-Dirac statistical distribution, \(k_{B}\) denotes the Boltzmann constant, \(\tau\) is the momentum delay time, \(i\) is the unit of imaginary number, \(\varepsilon\) denotes the particle energy, \(\mu_{c}\) represents the chemical potential of graphene (also known as the Fermi level \(E_{F}\)), and \(T\) is the temperature, \(e\) is the unit charge of electron. The first term of Equation (1) represents the intraband electron-photon scattering process, which can be obtained after integration. \[\sigma_{\text{intra}}=i\frac{e^{2}k_{B}T}{\pi h^{2}}\bigg{\{}\frac{\mu_{c}}{k _{B}T}+2\ln[\exp(-\frac{\mu_{c}}{k_{B}T})+1]\bigg{\}}. \tag{2}\]  
Figure 1: Graphene arrays constructed from monolayer graphene. The substrate material is silicon dioxide. Graphene is arranged periodically, where the thickness of SiO\({}_{2}\) is \(d=10\) nm. The equivalent thickness of a graphene sheet is \(\Delta=0.3\) nm. The second term of Equation (1) represents the interband transition of electrons. For \(\hbar\omega\), \(|\mu_{c}|\gg k_{B}T\), it can be simplified as  
\[\sigma_{\text{inter}}=i\frac{e^{2}}{4\pi h^{2}}\ln\left[\frac{2\left|\mu_{c} \right|-h(\omega+i\tau^{-1})}{2\left|\mu_{c}\right|+h(\omega+i\tau^{-1})} \right]. \tag{3}\]  
The energy band of intrinsic graphene is provided in Figure 2a. Graphene exhibits strong broadband absorption which is 50 times higher than GaAs. For graphene, its surface conductivity is contributed by interband and intraband transitions, which are related to the frequency of photons and the chemical potential on graphene \(\mu_{c}=E_{F}\). The chemical potential of doped graphene can be controlled by chemical doping or external gate voltage [32]. For intrinsic graphene \(E_{F}=0\), there is only interband transition and no intraband contribution. Figure 2b gives the transitions of electrons in graphene for \(E_{F}>0\). The Fermi level is elevated as a negative gate voltage bias on graphene. The effect of this operation is similar to the case of \(n\)-doping. Interband transition only occurs as the incident photon satisfies the condition \(\hbar\omega_{1}>2|E_{F}|\), so the low-wavelength dynamic surface conductivity is dominated by interband contribution. In the THz band, the energy of the photon is relatively low. Interband transition dominates for \(\hbar\omega_{2}<2|E_{F}|\) and the conductivity of graphene is approximately in the form of a Drude model, similar to that of the metals. As the graphene is biased by a positive gate voltage, similar to the case of \(p\)-doped, the Fermi level degrades \(E_{F}<0\) as shown in Figure 2c. Interband transition occurs above the threshold of \(\hbar\omega_{1}>2|E_{F}|\) as well. Graphene could be viewed as an equivalent dielectric and numerous simulations have shown that the deviations caused by such emulating can be negligible as long as the equivalent thickness is less than 1 nm [33; 34]. As a transverse wave is incident in the arrays of graphene from the left and its intensity is low enough, the third-order nonlinear effect in graphene can be ignored.