## 3 Results and Discussion  
### Beam Steering  
In general, a phase difference between adjacent elements in the array produces a sloped isophase plane, which deflects the beam and realizes beam steering. The beam deflection angle is proportional to the phase difference, which could be described by the generalized Snell's law [37; 38]:  
\[\sin(\theta_{t})n_{t}-\sin(\theta_{t})n_{i}=\frac{\lambda_{0}}{2\pi}\frac{d \varphi}{dx} \tag{1}\]  
where \(\theta_{i}\) and \(\theta_{t}\) are the incident angle and refractive angle, respectively. \(n_{i}\) and \(n_{t}\) are the refractive indexes of the incident and refractive region, and \(n_{i}=n_{t}=1\) in our case. \(\varphi\) is the abrupt phase introduced at the interface. Equation (1) implies that the refracted beam can have an arbitrary direction, provided that a suitable constant gradient of phase discontinuity along the interface is introduced. According to the simulations above and Equation (1), it is known that the designed 7 elements can be rearranged in a period, actively achieving three phase orders and beam steering angles. Table 2 shows the distribution of elements and their corresponding phases with different phase orders. The phase order N indicates a phase gradient of N \(\times\) 2\(\pi\)/7, which could be actively controlled by choosing the feeding lines. Considering both the size of the structure and the number of phase gradients, a patch array with \(7\times 7\) antenna elements is adopted and schematically shown in Figure 4. In this array, the desired phase gradient is oriented along the \(x\)-direction. Meanwhile, the phase remains constant along the \(y\)-direction. In addition, this patch array may obtain a beam steering order from \(-\)3 rd to 3 rd due to the symmetry of the proposed structure. Here, we only consider a one-dimensional case scanning in the \(x\)-direction. It can be easily predicted that the designed patch array may achieve a two-dimensional beam scanning by engineering the feeding network along both \(x\)- and \(y\)-directions (not shown here). Figure 3: Phase of radiation patch obtained from each arm of feeding line in (**a**) element 2 (**b**) element 3 (**c**) element 4 (**d**) element 5 (**e**) element 6 (**f**) element 7. Figure 5a shows the simulated radiation patterns of the electric field in the \(x\)-\(z\) plane from \(-3\)rd to \(-3\)rd phase gradient orders. It can be seen that the designed patch array produced directive beams with a wide scanning angle of \(110^{\circ}\), rotating from \(-55^{\circ}\), \(-34^{\circ}\), \(17^{\circ}\), \(0^{\circ}\), \(17^{\circ}\), \(34^{\circ}\), to \(55^{\circ}\), when the phase gradient was tuned as \(-6\pi\)/7, \(-4\pi\)/7, \(-2\pi\)/7, \(0\), \(2\pi\)/7, \(4\pi\)/7, and \(6\pi\)/7, respectively. These scanning angles agree well with the theoretical results obtained by Equation (1). In addition, the gains of all the arrays exceeded 20 dB, verifying the good performance of the designed tunable beam steering. To further demonstrate the accuracy of the beam steering, we plot in Figure 5b the near field distributions of the electric field in the \(x\)-\(z\) plane of the designed patch array. The simulation results of the engineered wavefront propagated towards oblique angles of \(-55^{\circ}\), \(-34^{\circ}\), \(-17^{\circ}\), \(0^{\circ}\), \(17^{\circ}\), \(34^{\circ}\), \(55^{\circ}\), when the phase gradient was tuned to be \(-6\pi\)/7, \(-4\pi\)/7, \(-2\pi\)/7, \(0\), \(2\pi\)/7, \(4\pi\)/7, and \(6\pi\)/7, respectively. Overall, from Figure 5, we see that the proposed actively tunable patch array can effectively control the radiation pattern. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Phase Order** & **Element** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline
3 & 0 & 3 & 6 & 2 & 5 & 1 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The phase distribution of array elements under different phase gradients (unit: \(2\pi\)/7 rad). Figure 4: Geometrical construction of the patch array.