## 3 CPA-LPs and EPs Splitting  
As an example, we consider a one-dimensional anti-PT-symmetric PC consisting of 20 periodic unit cells, _viz._\(N=20\), embedded in a homogeneous background, such as in air, shown in Figure 1. The parameters are \(d=125\)\(\upmu\)m, \(n=1\), \(\Delta n=0.5\), \(\theta=5^{\circ}\). In our numerical model, we have neglected these material nonlinearity and dispersion as a simplification. Now, we calculate the transmission and reflection spectra as the TM wave obliquely impinges upon the anti-PT-symmetric PCs from the left and right sides. Figure 2a plots the transmittance spectrum, where we observe two resonant states associated with sharp transmission peaks \(T=3.01\times 10^{3}\) and \(1.27\times 10^{2}\) in the curve marked by black stars, respectively locating at \(\omega=2.3\times 10^{13}\), and \(3.8\times 10^{13}\) rad/s. Figure 2b gives the reflectance spectrum for light incident from the left. It shows that there are also two maxima in the curve of \(R_{1}\). The positions of the peaks are identical in the transmittance and reflectance spectra. This phenomenon demonstrates that two CPA-laser states may exist around the two maxima. The values of the nadirs in \(R_{1}\) equal nearly to zero, considering the limit of calculation accuracy. The reflection coefficient is represented by \(r=|r|\text{exp}(iq)\), where \(\varphi\) is the phase change of reflection beam. The phase of the reflection coefficient varies with the angular frequency as shown in Figure 2c. There are \(\pi\) jumps in phase at the nadirs of \(R_{1}\). When the reflectivity is zero, the phase uncertainty of the reflection coefficient is caused. It is notable that, around the peaks of the transmittance and reflectance, the reflection coefficient phase changes dramatically with the change of angular frequency. The dramatic change in phase may induce novel physical effects. For light incident from the right, Figure 2d gives the reflectance spectrum \(T_{2}\), which is not identical with the spectrum \(R_{1}\). Therefore, the reflection is anisotropic for light incident from the left and right sides. However, the corresponding frequencies of the two maxima in \(R_{2}\) are identical with the values in \(R_{1}\). The phase of the reflection coefficient is shown in Figure 2e. The two reflection coefficient phase curves do not seem to coincide superficially. However, these two curves are really the same if we subtract these \(\pi\) jumps in phase at the zero points in the curves of \(R_{1}\) and \(R_{2}\). Figure 1: Schematic of the one-dimensional anti-parity–time (PT)-symmetric photonic crystals (PCs), where the complex refractive indices are distributed in \(n(z)=n(-z)\). For the primitive unit cell layers A and B, the thickness is \(d=125\)\(\upmu\)m and the complex refractive indices are \(n+\Delta n+0.01 qi\), and \(n-\Delta n+0.01 qi\), respectively. The imaginary part is positive for gain material and negative for loss material. In the parameter space composed of the gain-loss factor and angular frequency, Figure 3a gives the transmittance of light as light incident from the left or right. We can see that there are some poles in the parameter space such as at CPA-LP\({}_{1}\) (\(q_{\text{LP1}}=1.975\), \(\omega_{\text{LP1}}=2.3\times 10^{13}\) rad/s), and so on. The extreme values of the poles approve them to be CPA-laser states, so we call the poles in the parameter space CPA-LPs. The CPA-laser state is a defective mode or resonant state. The output energy of the laser state is provided by the gain in the dielectrics. Figure 3b plots the reflectance of light incident from the left. The maxima of reflectance coincide with the maxima of transmittance in the parameter space, which identifies that the poles of reflection and transmission correspond to the CPA-LPs as well. Otherwise, we find some discrete EPs denoted by EPs\({}_{1}\) in parameter space. The zero point of reflection is confirmed to be EP based on the scattering matrix \(S=[t,\,r_{1};\,r_{2},\,t]\)[63].