# Exceptional Points in Non-Hermitian Photonic Crystals Incorporated With a Defect
## 4 Applications and Giant GH Shift  
Figure 5a gives the distribution of the electric field for EP\({}_{1}\). It demonstrates that the power of electric field was mainly restrained in the defect. The electric field intensity decreased exponentially as the coordinate extended to the left and right from the 0 point. The distribution of electric field further manifested that EP\({}_{1}\) was the defect state (or transmission state). Figure 5b provides the electric field distribution of CPA-LP. The electric field mainly distributed the defect as well. This means that CPA-LP was also a defect state. Two Bragg gratings formed the resonance of cavity, together with some appropriate gain in defect, to result in the CPA laser state. Figure 4: (**a**,**b**) Real and imaginary parts of Hamiltonian eigenvalues varying with the gain-loss coefficient around the EPs, respectively. Figure 3: (**a**,**b**) Reflectance and complex phase of reflection coefficient varying with normalized frequency and gain-loss coefficient for a light incident from the right, respectively. In the parameter space, the complex phase of the reflection coefficient changed sharply as the normalized frequency and gain-loss coefficient increased near EPs and CPA-LP. Figure 6a gives the GH shift of reflected light around EP\({}_{1}\) and CPA-LP. The interested area is divided into three parts by dotted lines. EP\({}_{1}\) and CPA-LP are exactly on the dotted lines. At EP\({}_{1}\) and CPA-LP, one can see that the GH shift was infinite, so EP\({}_{1}\) and CPA-LP were the singularity of the GH shift. Except for EP\({}_{1}\) and CPA-LP, the GH shift on the dotted lines was 0. Otherwise, the GH shift in region I and III was positive, while the GH shift in region II was negative. The GH shift was greater as the parameters increasingly approached EP\({}_{1}\) and CPA-LP. In our accuracy, the positive GH shift could be as high as \(10^{6}\lambda\), and the negative GH shift could be as high as \(-10^{5}\lambda\). The sharp change in the complex phase of reflection coefficient near EPs and CPA-LP generated the giant GH shift. The singularity of the GH shift at EPs and CPA-LP was also an intriguing characteristic. Figure 6b illustrates the relationship between the GH shift and normalized frequency for five special gain-loss coefficients \(\sigma=-1.5\), \(\pm 1\) and \(\pm 0.5\). One can see that the GH shift changed with the normalized frequency. There was a peak in each curve as the coefficient \(\sigma>\sigma_{\text{EP1}}\) or \(\sigma<\sigma_{\text{LP}}\) and a valley existed in each curve as \(\sigma_{\text{LP}}<\sigma<\sigma_{\text{EP1}}\). The positive and negative maximum GH shifts were at the normalized frequency (\(\omega-\omega_{0}\))/\(\omega_{\text{gap}}=0.0654\). The incident beam was composed of many different frequency components of plane waves. We have showed that the light field penetrated more deeply  
Figure 5: (**a**) Electric field distribution of EP\({}_{1}\). (**b**) Electric field distribution of CPA-LP. Figure 6: (**a,c**) Lateral Goos–Hänchen (GH) shift of reflected light beam from the left and right, respectively. (**b**) GH shift for three special gain-loss coefficients versus normalized frequency. (**d**) Lateral GH shift of transmitted light beam. The parameter space is composed of the normalized frequency and gain-loss coefficient. It has rescaled the GH shift by taking logarithm log\({}_{10}\)\(|\)\(D_{y}\)\(|\) for clarity for (**a,c,d**). into the PCs as the parameters became closer to EP\({}_{1}\) and CPA-LP, so the dispersion of plane waves was more serious. For this reason, the reflected beam consisting of scattering waves had a giant lateral shift. Figure 6c gives the GH shift around EP\({}_{2}\) and CPA-LP as light was incident from the right. Similarly, EP\({}_{2}\) and CPA-LP were the singular points of the GH shift. Along the dotted lines, the GH shift was 0. The GH shift in region I and III was positive, while the shift in region II was negative.