## 3 Non-Uniform Phase Distribution for Directional Radiation  
In order to control the reflection pattern, it is required to control the phase distribution on the interface, top for reflection and bottom for transmission, between the dielectric array and the space. Specifically in this study, it was expected that the phases on the surface of the two dielectric blocks were different, i.e., two guiding modes were exited simultaneously and they propagated along the two materials independently. We applied the simulation model in Figure 8a and the parameter settings were \(w=12\) mm, \(l_{1}=15\) mm, \(l_{2}=7.5\) mm, \(h=50\) mm, \(\epsilon_{\text{h}}=10.2\), \(\epsilon_{\text{l}}=4.3\), and \(g=15\) mm, where \(g\) is the distance in the \(y\) direction between the dielectric blocks and the upper \(y\) boundary. The simulation results are shown in Figure 8. As periodic boundaries were applied in both transverse directions in the models shown in Figures 4b and 8a, it does not affect the results whether the low-permittivity region is separated into two parts or not. The model in Figure 8a shows the phase distribution more clearly and, therefore, was adopted in this paper. The absolute values of the power density are plotted in Figure 8b-d at the frequencies of 7.7 GHz, 8.7 GHz, and 9.7 GHz respectively. The cut-plane is parallel to the \(xoz\) plane and cuts the dielectric blocks equally in the \(y\) direction. At 7.7 GHz, the wave propagates along the high-permittivity block (on the left), as shown in Figure 8b; at 8.7 GHz, it can be observed that the power density concentrates at both the middle of the two blocks and the low maximum intensity at 8.7 GHz indicates a high reflection at the top surface; then, at 9.7 GHz, the higher-order mode is excited in the high-permittivity block (the left one) and the power densities in the right block are close to those in the left block. The transverse size of the high-permittivity dielectric waveguide, denoted by \(l_{2}(\epsilon_{\text{h}})^{0.5}\), is larger than that of the low-permittivity one and so is the cut-off wavelength. Therefore, the guiding mode appears in the high-permittivity block at relatively lower frequencies. Figure 9 shows the phase distribution of the Ey component on the bottom interface between the dielectric material and the free space. Figure 9a-c are obtained under the frequencies of 7.7 GHz, 8.7 GHz, and 9.7 GHz, respectively. It can be observed that the difference between the phases corresponding to the two dielectric blocks, denoted as \(\Delta\varphi\), increases from 7.7 GHz to 9.7 GHz. The values of \(\Delta\varphi\) in Figure 9a-c vary because the two guiding modes propagate independently with different phase constants in the \(x\) direction. So, the expected non-uniform phase pattern can be obtained when the guiding  
Figure 8: Amplitude simulation model of (**a**) the array without metal wall and simulated power density distributions at (**b**) 7.7 GHz, (**c**) 8.7 GHz, and (**d**) 9.7 GHz. mode propagating along the low-permittivity block is also excited. In this situation, the microwave power is mainly constrained in the dielectric region; so, the phase distribution in the space is not plotted. The materials used are selected to be low-loss dielectric materials so that the conductivity can be neglected. The only electromagnetic property under concern is permittivity, which can influence the wave numbers of the excited modes. The permittivity will further determine the field pattern in the structure and the field distribution on the interface plane, which results in specific reflection characteristics. The non-uniform phase distributions result in specific radiation patterns, which are shown as a far-field direction diagram in Figure 10. It should be mentioned that the distance between the test material and the reflect array is \(\sim\)0.3 m in practice, which is long enough to consider the reflection pattern from the \(\sim\)0.2 m \(\times\sim\)0.2 m array as a far-field pattern at \(\sim\)9.7 GHz.