# Reflection Enhancement and Giant Lateral Shift in Defective Photonic Crystals with Graphene
## 3 Reflectance Enhanced with Graphene  
Now we calculate the transmission and reflection spectra as the TM wave obliquely impinges upon the defective PCs incorporated without graphene. The incident angle is \(\theta=20^{\circ}\). Figure 2a shows the transmittance spectra for three different Bragg periodic numbers \(N=3\), \(4\), and \(5\), respectively. The transmittance varies with the normalized frequency \((\omega-\omega_{0})/\omega_{0}\), where the photonic bandgap of PCs is \(\omega_{0}=4\omega\mathrm{a}\mathrm{o}\mathrm{a}\mathrm{c}\mathrm{s}\mathrm{e }\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{ n}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{n}\mathrm{l}\mathrmThere was a peak in each spectrum near the zero point of normalized frequency. The maximum value \(T_{m}=1\) at the spectrum center \((\omega-\omega_{0})/\omega_{\text{perp}}=0.065\) corresponds to the transmittance of the DM, denoted by a star symbol. The transmission spectrum width was smaller for a larger Bragg periodic number \(N\), demonstrating its good frequency selectivity. Figure 2b shows the reflectance spectrum. The reflectances of the DM were all as low as zero for the three Bragg periodic numbers. This shows that there was no light reflected from the PCs, as the incident frequency identified with the DM. The PCs became transparent for this specific frequency wave, so the DM is also known as a transmission mode. Figure 2c shows the phase of the reflection coefficient. The phase varied with the normalized frequency. There was a \(\pi\) phase hopping at the DM in each phase curve for different Bragg periodic numbers. The reflectance was zero at the DM, which resulted in uncertainty in the phase of the reflection coefficient. Figure 2d shows the distribution of the DM. It illustrates that the electric field was mainly localized at the defect layer, and the electric field exponentially decayed as it departed from the center along the \(z\)-axis. We embedded single-layer graphene in the middle of the defect and investigated the influence on transmittance, reflectance, and phase of the reflection coefficient with the interaction between the graphene and the electric field of the DM. Figure 3a shows the transmittance spectra. One can find that the transmittance of the DM was no longer constant, and decreased with the increase of the Bragg periodic number \(N\). On the contrary, the reflectance of the DM increased with the Bragg periodic number, as shown in Figure 3b. In other words, the reflection intensity of light could be greatly enhanced by increasing \(N\). Figure 3: The calculated spectra for the defective photonic crystals (PCs) incorporated with graphene. (**a**) The transmission spectrum \(T\). (**b**) The reflection spectrum \(R\). (**c**) The phase of the reflection coefficient. The chemical potential was \(\mu_{c}=0.15\) eV. Figure 2: The calculated spectra for the defective photonic crystals (PCs) without graphene. (**a**) The transmission spectrum \(T\). (**b**) The reflection spectrum \(R\). (**c**) The phase of the reflection coefficient. (**d**) Electric field intensity (\(1E_{x}1\)) distribution of the defect mode. The reflection coefficient is represented by \(r=|r|\exp(i\varphi)\), where \(\varphi\) is the phase change of the reflected beam. The graphene inset at the defect therefore apparently influenced the reflection behavior of light\(-\)especially the reflection phase. Figure 3c shows that the reflection phase varied continuously with the normalized frequency, but did not change abruptly at the DM. As the frequency increased, the phase changed dramatically around the DM. This behavior indicates that a large lateral shift of the reflected beam may have been induced, since the lateral shift was proportional to the curvature of the phase curve. The surface conductivity is a function of the chemical potential of graphene, so the optical properties can be readily modulated by the chemical potential.