# Unidirectional Invisibility Induced by Complex Anti-Parity-Time Symmetric Periodic Lattices
## 3 Unidirectional Invisibility and Laser Points  
We calculated the transmission and reflection spectra as the transverse magnetic (TM) wave obliquely impinged upon the complex anti-PT symmetric lattices. The incident angle was \(\theta=5^{\circ}\). Figure 2a shows the transmittance and reflectance spectra for light incident from the left. The transmittance \(T_{1}\) decayed rapidly and tended to a constant as the angular frequency \(\omega\) increased. However, there were two peaks in the curve of reflectance \(R_{1}\). The two maxima were \(R_{1}=10.5\) and \(7.97\times 10^{4}\) in the curve marked by black asterisks, respectively locating at \(\omega=1.83\times 10^{13}\), and \(4.29\times 10^{13}\) rad/s. When light was incident from the right side, Figure 2b shows the reflection spectrum and transmission spectrum. Initially, the transmittance \(T_{2}\) and reflectance \(R_{2}\) decreased rapidly with the increase of the angular frequency. Then, as the frequency increased, the reflectance and transmittance tended to be constant. For incident light at higher frequencies, the transmittance was close to zero. This phenomenon indicated that the media had anisotropy of the reflection as light was incident from the left and right. Therefore, for incident light with a certain frequency, the reflected light on the left could be enhanced greatly, but the reflected light on the right may be suppressed. These properties could be used to enhance the reflected light intensity from unidirection, as well as unidirectional invisibility. The transmission and reflection coefficients were represented by \(t=|t|\text{exp}(i\varphi_{t})\) and \(r=|r|\text{exp}(i\varphi_{r})\), where \(\varphi_{t}\) and \(\varphi_{r}\) were the phase shifts of the transmitted and reflected beams referred to the incident beam, respectively. Figure 2c shows the phases of the transmission coefficients of light from the left and right, respectively. One can see that the reflection coefficients in the left and right were isotropic. There were many step points in the curve, and the jump value was \(2\pi\), which was meaningless and  
Figure 2: (**a**,**b**) The transmission and reflection spectra for light incident from the left and right, respectively. (**c**,**d**) The phases of transmission coefficient and reflection coefficient for the complex anti-PT symmetric PCs, respectively. could be eliminated. However, near the poles of the transmittance, the phase of the transmission coefficient changed dramatically, which indicated that large lateral shift could have existed in the transmitted beam. Figure 2d shows the phase of the reflection coefficient. When light was incident from the left, the phase change of the reflection coefficient near the two maxima of reflectance was \(2\pi\) and \(\pi\), respectively, and the phase varied dramatically with the angular frequency around those asterisks. The \(2\pi\) phase jump point was stepwise and the step was meaningless. But the phase changed continuously around the \(\pi\) change point. When light was incident from the right, the phase curve had a maximum point. With the increase of angular frequency, the phase of reflection coefficient first increased, then decreased, and finally tended to a constant value. For the reflected beam, the change in phase was extremely sensitive to the angular frequency of incident light around the maxima of the reflectance, so this effect could be used for high-sensitivity sensors. To find the properties in the complex anti-PT symmetric lattices, we defined the permittivities of dielectrics A and B as \(\varepsilon_{a}=0.1+0.01\varepsilon_{i}i\) and \(\varepsilon_{b}=-0.1+0.01\varepsilon_{i}i\), respectively. The gain in the lattices could be obtained by nonlinear two-wave mixing or a Ge/Cr doped fiber [39; 40], while the loss could be achieved by an acoustic modulator [41]. Figure 3a gives the reflectance \(R_{1}\) as light was incident from the left. The parameter space was composed of the permittivity \(\varepsilon_{i}\) and angular frequency \(\omega\).