# Reflection Enhancement and Giant Lateral Shift in Defective Photonic Crystals with Graphene
## 3 Reflectance Enhanced with Graphene
Figure 4a shows the transmittance of light in the parameter space composed of the normalized frequency and the chemical potential of graphene. We took the logarithm log\({}_{0}\)R for increasing image contrast. One can see that the transmittance was a function of these two parameters. As the chemical potential increased, the transmittance varied with the normalized frequency. For a fixed chemical potential, there was a maximum in each transmission spectrum. The maximum corresponded to the transmittance of the DM. This demonstrates that the frequency of the DM remained approximately constant as the chemical potential changed, while the transmittance changed abruptly near \(\mu_{c}\) = 0.41 eV. The transmission mode (i.e., the DM) corresponded to the maximum of transmission spectrum in the middle of the bandgap. Figure 4b shows the transmittance of the DM for different chemical potential. It shows that the maximum was \(T_{\mathrm{m}}\) = 0.29 when the chemical potential \(\mu_{c}\) < 0.41 eV, and the maximum was \(T_{\mathrm{m}}\) = 0.99 when the chemical potential \(\mu_{c}\) > 0.41 eV. Figure 4c shows the reflectance of light. It demonstrates that the position of minimum reflectance goes along with the DM in the parameter space as the chemical potential changed. Figure 4d shows the minimum reflectance in the bandgap. It shows the maximum changed with the chemical potential of graphene. The value was \(R_{\mathrm{m}}\) = 0.21 when the chemical potential \(\mu_{c}\) < 0.41 eV, and the minimum was \(R_{\mathrm{m}}\) = 0 when the chemical potential \(\mu_{c}\) > 0.41 eV. The surface conductivity of graphene is contributed by electron movement. The intraband transition of electron movement governs mainly for a low chemical potential, while the interband transition plays a leading role for a high chemical potential [24,31]. Figure 5a shows the real part of the surface conductivity of graphene. It varied with the chemical potential of graphene. As \(\mu_{c}\) < 0.41 eV, the real part of the surface conductivity of graphene was close to 0.6 S. High surface conductivity corresponds to a strong surface current, which can induce reflection enhancement in the DM. The real part of the surface conductivity of graphene was close to zero as \(\mu_{c}\) > 0.41 eV, in which the graphene presented complete transparency. Figure 5b shows the imaginary part of the surface conductivity of graphene. There was a minimum in the imaginary part curve as the chemical potential changed. The imaginary part of the surface  
Figure 4: (**a,c**) The transmittance and reflectance varying with the chemical potential and normalized frequency, respectively. (**b,d**) The transmittance and reflectance of the defect mode. The Bragg periodic number \(N\) = 5.  
conductivity is equivalent to the real part of the refractive index if we regard the graphene as an equivalent dielectric. Therefore, the center frequency of DM may vary with the chemical potential of graphene. Furthermore, the periodic structure of the system can also influence the reflectance of the DM, as shown in Figure 5c. The chemical potential took a fixed value of \(\mu\)\(\epsilon\) = 0.15 eV. One can see that the reflectance of DM increased with the increase of the Bragg periodic number \(N\), while the transmittance of the DM decreased. The resonance of the DM was stronger for larger \(N\). The enhancement of the local field where the graphene was located contributed to the interband transition of electrons as the Bragg number \(N\) increased.