# Reflection Enhancement and Giant Lateral Shift in Defective Photonic Crystals with Graphene
## 4 Large Lateral Shift around the DM  
Figure 6a shows the phase of the reflection coefficient. For each fixed chemical potential, there is a corresponding phase curve. The phase varied with the normalized frequency. The change in phase became particularly dramatic in the area around the DM. Especially, the phase approximated a step change at the DM as the chemical potential \(\mu\)\(\epsilon\) > 0.41 eV. To demonstrate the details of the phase change, Figure 6b depicts the phase curve for three different chemical potentials: \(\mu\)\(\epsilon\) = 0.25, 0.41, and 0.60 eV. One can see that the phase varied with the normalized frequency. Around the DM, the curvature was negative, but it became positive as the frequency extended to both sides beyond the DM. The interval of phase change at the DM enlarged as the chemical potential increased and the change in phase approached \(\pi\); even so, the phase curve still varied continuously and there was no hopping in phase. This means that a higher chemical potential may lead to a greater change in the phase of the reflection coefficient around the DM. The loss of graphene decreased greatly as the chemical potential exceeded 0.41 eV. Low loss in materials can induce a sharp change in the reflection coefficient phase [6]. Figure 5: (**a**,**b**) The real and imaginary parts of surface conductivity of graphene, respectively. (**c**) The transmittance and reflectance of the defective mode (DM) varying with the Bragg periodic number \(N\). Figure 6: (**a**) The phase of the reflection coefficient. The parameter space composed of the chemical potential and normalized frequency. (**b**) The reflection coefficient phase varying with the normalized frequency. The Bragg periodic number \(N\) = 5. As an incident beam impinges upon the structure, the lateral shift of the reflected beam can be derived by \(\Delta=-d\varphi/dk\)[33]. Figure 7a shows the lateral shift in the parameter space composed of the chemical potential and normalized frequency. For each fixed chemical potential, the lateral shift of the reflected beam can be positive or negative as the normalized frequency changes, and the extrema are very large. Therefore, for better contrast, we took the logarithm of the absolute lateral shift (i.e., \(\log_{10}\Delta 1\)). One can see that the lateral shift was large in the region around the DM in the parameter space. There was an obvious difference in the lateral shift near the chemical potential \(\mu_{c}=0.41\) eV. Figure 7b shows the lateral shift of the reflected beam for different chemical potential. The lateral shift is a function of the normalized frequency. There was a peak and two valleys in each curve of lateral shift. The extrema were \(\Delta=124\lambda\), \(-18.8\lambda\), and \(-18.9\lambda\) for \(\mu_{c}=0.25\) eV. The extrema were \(\Delta=2.74\times 10^{3}\lambda\), \(-118.6\lambda\), and \(-117.4\lambda\) for \(\mu_{c}=0.41\) eV. The extrema were \(\Delta=5.74\times 10^{4}\lambda\), \(-118.9\lambda\), and \(-119\lambda\) for \(\mu_{c}=0.60\) eV. Therefore, by increasing the frequency, the peak in the shift curve became sharper and the gap between the two valleys reduced. This super-sensitive lateral shift effect of reflected beam could be used for the detection of weak physical effects [34; 35]. The surface conductivity of graphene is a function of the chemical potential and normalized frequency. The loss is lower for larger chemical potential. Low loss can cause violent fluctuation in phase of reflection coefficient around the DM as the normalized frequency changes, and then the dramatic fluctuation in phase induces large lateral shift of reflected beam. We only used TM waves as the typical incident beam in the present study. In fact, when TE waves were used for calculation, similar results were obtained. When the structure is fixed, the reflection spectrum is a function of incident angle and incident wavelength.