# Exceptional Points in Non-Hermitian Photonic Crystals Incorporated With a Defect
## 2 Non-Hermitian Photonic Crystals  
We constructed the non-Hermitian PCs, which were composed of two Bragg gratings and a defect, as shown in Figure 1. Two different dielectrics, A and B, arrayed alternatively to form two Bragg gratings. The dielectrics A and B were MgF\({}_{2}\) and ZnS, respectively. The defect containing dielectrics C and D was embedded in the Bragg gratings. The dielectrics C and D could be obtained by doping SiO\({}_{2}\) with high and low impurities of refractive indices, respectively. We denoted the refractive indices of dielectrics by \(n_{a}\), \(n_{b}\), \(n_{c}\), and \(n_{d}\), respectively. The thickness of each layer was set as a quarter of the optical wavelength, \(viz\), \(L_{a,b,c,d}=\lambda_{0}/4n_{a,b,c,d}\), where \(\lambda_{0}=100\)\(\upmu\)m is the midgap wavelength of the defect PCs. The refractive indices of materials were \(n_{a}=1.38\), \(n_{b}=2.35\), \(n_{c}=1.4+0.01\sigma i\), and \(n_{d}=1.6+0.01\sigma i\), respectively, where \(\sigma\) is the gain-loss coefficient. The positive gain-loss coefficient represented loss and the negative gain-loss coefficient represented gain. The gain in the defect could be realized by doping Ge/Cr or nonlinear two-wave mixing [47, 48], and the loss of material could be achieved by acoustic modulators [49]. The designed PCs could also be abbreviated to (AB)\({}^{N}\)CD(BA)\({}^{N}\), where \(N\) is the periodic number of Bragg gratings, and here it shows \(N=5\). In simulations, the incident wavelength was in the range of terahertz band and the transmission matrix method (TMM) was used to derive the reflection coefficient \(r\) and transmission coefficient \(t\)[18]. Subsequently, the reflectance of light and transmittance can be denoted by \(R=rr^{\star}\) and \(T=tt^{\star}\), respectively. The reflection coefficient is generally complex and can be expressed as \(\gamma\)/\(t=|\eta/\)  
Figure 1: **(a)** Schematic of one-dimensional non-Hermitian photonic crystals. The primitive unit-cell layers, A and B, array alternatively to form two Bragg gratings. The defect composed of dielectrics C and D is asymmetric with respect to the center. \(D_{ry}\) and \(D_{ty}\) are the lateral Goos–Hänchen shifts of the reflected beam and transmitted beam, respectively. (**b,c**) Reflectance and transmittance spectra for three specific values of the gain-loss coefficient, respectively. The asterisk represents complex conjugate operations. where \(\varphi_{rt}\) is the complex phase of the reflection/transmission coefficient. As a transverse magnetic (TM) polarization wave obliquely impinged upon the PC with the incident angle \(\theta\), the lateral GH shift of reflected/transmitted light beam was proportional to the slope of the reflection/transmission coefficient complex phase, with respect to the incident angle, that is \(D_{rphy}=-\lambda\mathrm{d}\varphi_{rt}/2\pi\mathrm{d}\theta\)[18], where \(\lambda\) is the incident wavelength. We set the incident light as a TM wave, and the incident angle was \(\theta=20^{\circ}\). When light obliquely impinged upon the PCs from the left, Figure 2a provides the reflectance spectra for three different values of gain-loss coefficient \(\sigma\), where \((\omega-\omega_{0})/\omega_{\mathrm{gap}}\) is the normalized frequency and \(\omega_{\mathrm{gap}}=4\omega_{0}\mathrm{arcsin}|(n_{b}-n_{a})/(n_{b}+n_{a}) ^{2}/\pi\) is the photonic bandgap [50]. The center frequency of the bandgap was defined as \(\omega_{0}=2\pi\mathrm{c}/\lambda_{0}\). One can see that there was a bandgap structure in the reflectance spectrum for each fixed coefficient and a defect mode located at the middle of the bandgap. The reflectance of defect mode was zero, _viz._\(R_{1}=0\) was the coefficient \(\sigma=0\). For \(\sigma>0\), such as \(\sigma=1\), the reflectance \(1>R_{1}>0\) and a partial incident light was reflected by the PCs. The reflectance surpassed 1 as the coefficient \(\sigma<0\). The loss and gain in the defect can both enhance the reflectance of the defect mode.