# Exceptional Points in Non-Hermitian Photonic Crystals Incorporated With a Defect
## 3 Exceptional Points and Sharp Change in Phase  
Figure 2a demonstrates the reflectivity in the parameter space composed of the normalized frequency \((\omega-\omega_{0})\)/\(\omega_{\rm gap}\) and gain-loss coefficient. One can see there is a peak and a valley in the parameter space. The peak value was \(R_{1}=1.03\times 10^{5}\) and it can be viewed as \(+\infty\) in our calculating accuracy. The maximum point was the CPA-LP, which located at \([(\omega_{\rm LP}-\omega_{0})/\omega_{\rm gap}=0.0654\), \(\sigma_{\rm LP}=-0.79]\). The valley located at \([(\omega_{\rm EP1}-\omega_{0})/\omega_{\rm gap}=0.0654\), \(\sigma_{\rm EP1}=-0.1]\) and was an EP labeled by EP\({}_{1}\). The reflectance of light at EP\({}_{1}\) was \(R_{1}=1.33\times 10^{-6}\), which is approximately equal to 0 in our calculating accuracy. The non-Hermitian system can be viewed as a resonant cavity, of which the defect is the cavity body and the Bragg gratings are reflectors. The resonance cavity and gain-loss coefficient of the defect collectively induced the CPA-LP and EP\({}_{1}\). Next, we demonstrate the characteristics which approve the CPA-LP and EP. For the light incident from the left, the reflection coefficient can be written as \(r_{1}=|r_{1}|\exp(i\varphi_{r1})\), where \(\varphi_{r1}\) is the complex phase of reflection coefficient, as shown in Figure 2b. The EP\({}_{1}\) and CPA-LP were two singular points in the complex phase of the reflection coefficient. The complex phase \(\varphi_{r1}\) sharply changes around the EP\({}_{1}\) and CPA-LP. That is, a slight increase in the normalized frequency or in the gain-loss coefficient may lead to a great variation in the complex phase near the EP\({}_{1}\) and CPA-LP. For three specific values \(\sigma=-0.893\), \(-0.793\), and \(-0.693\), which were near the CPA-LP, Figure 2c provides the complex phase of reflection coefficient varying with the normalized frequency. For the coefficient \(\sigma=-0.793\), approaching to \(\sigma_{\rm LP}\), the complex phase change was \(\pm\pi\) around the CPA-LP as the frequency increased. For \(\sigma=-0.893\), lower than \(\sigma_{\rm LP}\), the slope of complex phase curve was negative, while for \(\sigma=-0.693\), larger than \(\sigma_{\rm LP}\), the slope of complex phase curve was positive and there was a meaningless \(2\pi\) hopping in the complex phase. Figure 2d demonstrates the complex phase cures for three given values, \(\sigma=-0.199\), \(-0.099\), and \(0.001\), around the EP\({}_{1}\). One can see that the complex phase change was also \(\pm\pi\) as the coefficient approached \(\sigma_{\rm EP1}\), such as \(\sigma=-0.099\). Different from the situation near the CPA-LP, the slope of complex phase curve was positive as the coefficient \(\sigma<\sigma_{\rm EP1}\), while the slope was negative as \(\sigma>\sigma_{\rm EP1}\) around the EP\({}_{1}\). The common characteristic was that the closer the parameters were to the CPA-LP and EP\({}_{1}\), the higher the slope of curve was, which may have resulted in a giant GH shift. Figure 2e shows the transmittance light in the parameter space. There was a peak in transmission in the parameter space as the normalized frequency and gain-loss coefficient changed. The maximum point was exactly at the CPA-LP, which approves the properties of the CPA laser state. The maximum transmittance was \(T_{\rm LP}=1.34\times 10^{5}\), which can be viewed as \(+\infty\) in our accuracy. The CPA-LP is a boundary state (the power of mode is mainly restricted at the interface of two different materials). The complex phase of transmission coefficient changed dramatically with the parameters around the CPA-LP, as shown in Figure 2f. There was a hopping in the complex phase of transmission coefficient at the CPA-LP and the variation was \(\pm\pi\). The hopping was meaningless \(\pm 2\pi\) at the other complex phase jump points, where the complex phase could therefore be viewed to be continuous. The resonance of defect cavity and the gain in dielectrics acted together to result in the CPA-LP.