# Unidirectional Invisibility Induced by Complex Anti-Parity-Time Symmetric Periodic Lattices
## 4 Lateral Shift of Reflected Beam  
As an incident beam impinged upon the structure, the lateral shift of the reflected beam relative to the position predicted by geometry optics could be derived by \(\Delta=-d\varphi/dk\)[45]. For several different permittivities \(\varepsilon_{i}<\varepsilon_{i1}=0.02\), Figure 6a plots the lateral shift versus the angular frequency. The shift was positive and there was a peak in each curve around \(\omega_{1}\). The maximum shift was larger as the permittivity approached to \(\varepsilon_{i1}\) more. On the other hand, the shift was negative and there was a valley in each curve around \(\omega_{1}\) for the permittivities \(\varepsilon_{i}\geq\varepsilon_{i1}\) as shown in Figure 6b. Analogously, the maximum negative shift was larger as the permittivity was more near \(\varepsilon_{i1}\). In the parameter space, Figure 6c demonstrates the distribution of lateral shift of reflected beam around the \(\mathrm{ULP}_{1}\). For better contrast, we took the logarithm of the absolute lateral shift (i.e., \(\log_{10}\lvert\Delta/\lvert\lambda\rvert\)). The focusing region was separated into two parts labeled by I and II, respectively. The lateral shift was positive in part I, while it was negative in part II. Except for the \(\mathrm{ULP}_{1}\), the lateral shift approximated to zero along the dotted line. The negative and positive polarities of lateral shift converted at the \(\mathrm{ULP}_{1}\), approving the singularity of \(\mathrm{ULP}_{1}\) for the lateral shift of reflected beam.  
As light was incident on the lattices from the left, Figure 6d gives all of the maxima and minima of lateral shift in each curve. It shows that the positive and negative lateral shift were divided by \(\varepsilon_{i1}\) as the angular frequency changed. In the vicinity of the \(\mathrm{ULP}_{1}\), maximum lateral shift dramatically  
Figure 5: **(a,c)** The reflectance and phase of reflection coefficient varying with the permittivity and angular frequency for light incident from the right, respectively. **(b,d)** The reflectance and phase of reflection coefficient varying with the angular frequency for three different permittivities \(\varepsilon_{i}=0.01\), \(0.015\) and \(0.02\), respectively.  
climbed when the permittivity increased. The shift tended to infinity as \(\varepsilon_{i}\) reached to \(\varepsilon_{i1}\). At the ULP\({}_{1}\), the maxima experienced a step change from positive to negative. And then, the negative shift began to increase sharply with the increase of the permittivity. In our calculating accuracy, the positive and negative lateral shifts reached highly to \(4.39\times 10^{2}\) and \(-6.3\times 10^{2}\) times of the incident wavelength near the ULP\({}_{1}\), respectively. Subsequently, the shifts returned to an ordinary magnitude from the negative maximum as the permittivity moved away from the ULP\({}_{1}\). On the whole, one can see that the ULP\({}_{1}\) was a singular point of lateral shift and the polarities of lateral shift transformed at this point. Otherwise, around the permittivity \(\varepsilon_{i2}\), the curve of maxima was not smooth enough, indicated by the dashed box. Such an abnormal phenomenon resulted from the parameters close to the ULP\({}_{2}\). Distribution of electric field for the boundary mode induced great variation in the phase of reflection coefficient.