# Unidirectional Invisibility Induced by Complex Anti-Parity-Time Symmetric Periodic Lattices
## 3 Unidirectional Invisibility and Laser Points
One could find that there were two poles in the parameter space, denoted by the unidirectional laser point (ULP)\({}_{1}\) (\(\varepsilon_{i1}=2\), \(\omega_{1}=1.8\times 10^{13}\) rad/s) and ULP\({}_{2}\) (\(\varepsilon_{i2}=1\), \(\omega_{2}=4.29\times 10^{13}\) rad/s), respectively. The corresponding maxima of the reflectance were \(R_{\text{UPL1}}=2.28\times 10^{4}\) and \(R_{\text{UPL2}}=4.44\times 10^{4}\). On the contrary, we plotted the reflectance as light impinged into the lattices from the right. The intensity of reflection beam was extremely faint as shown in Figure 3b. The reflectance sharply decreased with the increase of the angular frequency in area I and the value of reflection in area II approached to zero in our calculating accuracy. The properties demonstrated that there was strong dependence of reflection on the two incident directions from the left and right. Especially, the poles of reflectance arose in the parameter space for the light incident from the left, but the reflectionless phenomenon could have resulted for light incident from the right. The characteristics could be therefore utilized for the unidirectional laser. Figure 3c provides the distribution of the electric field intensity for the ULP\({}_{1}\). It shows that the power of the mode field was mainly localized at the first layer of the input port, so the ULP\({}_{1}\) could be viewed as a boundary mode. Figure 3: (**a,b**) The transmittance and reflectance varying with the permittivity and angular frequency for light incident from the left, respectively. (**c**) The electric field intensity (\(\left|E_{x}\right|^{2}\)) distribution at the unidirectional laser point (ULP\({}_{1}\)). We studied the phase properties of reflected and transmitted coefficients as the permittivity of the dielectrics and the angular frequency changed. Figure 4a plots the reflection phase in the parameter space. It shows the phase dislocated at the ULP\({}_{1}\) and ULP\({}_{2}\), which were the singular points in phase of reflected coefficient. To demonstrate the characteristics of the phase varying with the permittivity and angular frequency, Figure 4b plots the phase curves for three specific permittivities \(\varepsilon_{i}\) = 0.01, 0.015 and 0.02. One can see that the phase varied with the angular frequency. There was 2\(\pi\) phase jump at the frequency \(\omega_{1}\) for the curve corresponding to \(\varepsilon_{i}\) = 0.02 and \(\pi\) phase jump at the frequency \(\omega_{1}\). Actually, the curve was continuous at \(\omega_{1}\) since the 2\(\pi\) phase jump was meaningless. For \(\varepsilon_{i}\) = 0.015, the phase changed dramatically around the ULP\({}_{1}\) and ULP\({}_{2}\). This behavior indicated that large lateral shift of the reflected beam may have been induced, since the lateral shift was proportional to the curvature of the phase curve. For \(\varepsilon_{i}\) = 0.01, the phase changed abruptly with \(\pi\) at \(\omega_{2}\). The \(\pi\) change in phase justly approved the characteristics of the ULP\({}_{1}\) and ULP\({}_{2}\). The maximum property of transmittance resulted in the uncertainty in phase. We also gave the phase of transmitted coefficient in the parameter space as shown in Figure 4c. It manifested that the phase of transmitted beam also dislocated at the ULP\({}_{1}\) and ULP\({}_{2}\). Figure 4d demonstrates the phase changed with the angular frequency. For the above three specific permittivities, the phase abruptly stepped with \(\pi\) around the ULP\({}_{1}\) and ULP\({}_{2}\), and at the other step points, the phase change was the no meaning value of 2\(\pi\). For light incident from the right, Figure 5a shows the reflectance in the parameter space. One can see the focusing area in the parameter space was divided into two parts, viz part I and part II, which corresponded to the descent zone and saturation zone, respectively. The reflectance decreased with the increase of the angular frequency in the descent zone. However, the reflectance was saturated as the angular frequency further increased.