## 3 Results  
Figure 2 shows the effective refractive index n\({}_{\text{eff}}\) of eigenmodes in the limited graphene sheet arrays for two different cases. The total number of graphene sheets is N = 40. For a single graphene sheet, there is only one TM polarized SPP mode. As a result, the system composed of 40 graphene sheets will support 40 supermodes. Figure 2a,b plot the real part and imaginary part of effective refractive index with \(\mu_{1}\) = \(\mu_{2}\) = 0.15 eV. In this case, the structure is with uniform graphene sheet and no interface is formed at the middle of the structure. Both the spectra of Re (n\({}_{\text{eff}}\)) and Im (n\({}_{\text{eff}}\)) are continuously varied and no gap is found in the spectra. As a result, no bound modes are supported in the system. The case is different if we tune chemical potential \(\mu_{2}\) of some graphene sheets. Figure 2c,d plot the real and imaginary parts of effective refractive index with \(\mu_{1}\) = 0.15 eV and \(\mu_{2}\) = 0.18 eV. In this case, the structure now can be regarded as two arrays with different sequences of chemical potential connected, and the center of the structure is the interface. The results show that there is evidently a SPP mode in the band gap of Re (n\({}_{\text{eff}}\)) with effective refractive index being n\({}_{\text{eff}}\) = 68 + 0.3i. The wavelength of SPPs is figured out as \(\lambda_{\text{SPP}}\) = \(\lambda\)/n\({}_{\text{eff}}\) = 1/68. On the other hand, there is also another trivial defect mode with largest real part of effective refractive index as shown in Figure 2c. Figure 3a,b plot the mode profile (lH) of topological and trivial defect modes, respectively. The dotted lines indicate the location of graphene sheet. Both modes are located at the center defect, but their field distributions are different. For the topological mode, the field at the center of two graphene sheets is almost vanished, while the field for the trivial defect mode is not vanished. As  
Figure 2: The effective refractive index of supermodes for two different structures. (**a,b**) are the real and imaginary parts of refractive index for \(\mu_{2}\) = 0.15 eV, respectively. (**c,d**) are Re (n\({}_{\text{eff}}\)) and Im (n\({}_{\text{eff}}\)) for \(\mu_{2}\) = 0.18 eV, respectively. In all cases, \(\mu_{1}\) = 0.15 eV. explained above, the topological bound mode emerges because the sequence of chemical potential of two graphene sheet arrays is different. The trivial defect mode is formed as the two graphene sheets around the defect constitute a region of different effective refractive index than the rest of the structure. Figure 4 shows the wave propagation of SPPs for two different structures with different chemical potentials. In both cases, the wave is launched from a single graphene sheet at the center waveguide. In Figure 4a, the graphene sheets are uniform with the chemical potential of graphene \(\mu_{1}=\mu_{2}=0.15\) eV. In this case, there is no bound mode supported in the system and the SPPs spread during the propagation, which is known as discrete diffraction. In Figure 4b, the chemical potential is \(\mu_{1}=0.15\) eV and \(\mu_{2}=0.18\) eV. As the SPPs are incident from the single graphene sheet at the center defect, most energy is confined at the center of the structure. Moreover, the intensity of light shows a beating pattern during the propagation. This is because the trivial and non-trivial defect modes are excited simultaneously. The surface conductivity of graphene relates largely to the chemical potential, which can be controlled by electrostatic and chemical doping [5]. Figure 5 plots the numerically calculated effective refractive index as chemical potential \(\mu_{2}\) changes form 0.13 eV to 0.17 eV. The chemical potential \(\mu_{1}\) is fixed at 0.15 eV. In Figure 5a, we plot the real part of refractive index. As \(\mu_{2}=\mu_{1}\), there are no bandgap or topological bound modes. The bandgap broadens when chemical potential is tuned away from 0.15 eV.