# Tunable Mid-Infrared Graphene Plasmonic Cross-Shaped Resonator for Demultiplexing Application
## 2 Resonator and Analysis
In the air-graphene-air system, the dispersion relation of SPPs travelled on graphene can be represented by [53]  
\[\beta_{SPP}\ =\ k_{0}\sqrt{1-\left(\frac{2}{\eta_{0}\sigma_{G}}\right)^{2}} \tag{4}\]  
where \(\beta_{SPP}\), \(k_{0}\), and \(\eta_{0}\) are respectively the propagation constant of SPPs, the wave vector of incident light, and the impedance of air. The transfer function _tf_, of the proposed graphene plasmonic resonator structure by use of coupled mode theory (CMT) method is [54]  
\[tf\ =\ \frac{\frac{1}{Q_{w}}}{j2\delta+\frac{1}{Q_{i}}+\frac{1}{Q_{w}}} \tag{5}\]  
where \(Q_{w}\) and \(Q_{i}\) are the quality factors of the resonator which are respectively describing the waveguide coupling loss and the intrinsic loss. \(\delta\) is used for normalization of the frequency \(\omega\) and is defined as \(\delta\ =\ \frac{\omega-\omega_{0}}{\omega_{0}}\). Transmission \(T\) is obtained by \(T\ =\ \left|tf\right|^{2}\)[54]. Transmission spectrum of the MIR graphene plasmonic resonator of Figure 1 obtained by numerical FDTD and theoretical CMT methods are given in Figure 2a. For the transmission spectrum which is obtained by CMT method, \(Q_{w}\) and \(Q_{i}\) are respectively considered as 40 and 100. \(\left|E\right|\) Field profiles of the resonator for the resonance wavelength of \(\lambda\) = 7.5 \(\upmu\)m and the non-resonance wavelength of \(\lambda\) = 6.5 \(\upmu\)m are respectively given in Figure 2b,c. The SPPs are excited by use of a dipole point source with electric polarization which is considered 2 nm above the horizontal input graphene [55]. In the resonance wavelength, the incident lightwave couples effectively to the resonator and then, it decouples to the output graphene waveguide. In the non-resonance wavelength, the incident lightwave cannot couple to the resonator and it is blocked. The real part of the \(\beta_{SPP}\) is related to the \(\sigma_{G}\) (Equation (4)) and \(\sigma_{G}\) is related to the \(\mathrm{E_{f}}\) (Equation (1)). As the \(\mathrm{E_{f}}\) increases, the real part of the \(\beta_{SPP}\) decreases (depicted in Figure 2d). The wavelength of the transmission peak shown in Figure 2a should thus exhibit a blue shift when the Fermi energy increases. That is how the transmission spectrum can be tuned by the change of the Fermi energy of graphene. Figure 2: (**a**) Transmission spectrum of the resonator structure of Figure 1 obtained by numerical FDTD and theoretical CMT methods. \(|E|\) Field profiles of the resonator structure of Figure 1 for (**b**) the resonance wavelength of \(\lambda\) = 7.5 \(\upmu\)m, (**c**) the non-resonance wavelength of \(\lambda\) = 6.5 \(\upmu\)m, and (**d**) the dependence of the real part of the \(\beta_{SPP}\) (Equation (4)) of the graphene layer for different Fermi energies of graphene. The resonance condition of the proposed resonator of Figure 1 could be expressed by [56]  
\[2\beta_{SPP}L+2\delta\ =\ 2p\pi \tag{6}\]  
where \(\beta_{SPP}\ =\ \frac{2\pi n_{SPP}}{\lambda}\), \(n_{SPP}\) and \(\lambda\) are respectively the effective refractive index of SPPs and the resonance wavelength. The phase change at the ends of the ribbon is denoted by \(\delta\) and \(p\) which are integer numbers. While the incident wavelength is much larger than the ribbon length, we can assume \(p=1\). Therefore, the resonance wavelength could be given by [56]  
\[\lambda\ =\ \frac{2\pi n_{SPP}L}{\pi-\delta} \tag{7}\]  
Transmission spectra of the proposed graphene plasmonic resonator of Figure 1 for three different values of the resonator length L, the Fermi energy E\({}_{\text{f}}\), the resonator width W, and the angle between the ribbon and the horizontal line of the cross-shaped resonator \(\theta\) are simulated and given in Figure 3a-d, respectively. As depicted in Figure 3a, by increasing the resonator length, L, the resonance wavelength increases which shows a red shift. By Equation (7), it is clear there is a direct relationship between the resonance wavelength and the resonator length.