# Compact Beam-Scanning Reflectarray Antenna with SLL Reduction Using In-Plane Panel Translations
## 3 Multi-Stage Design
### Scanning Range Enhancement  
A major design challenge of the proposed antenna geometry is that it is not clear how to relate the panel positions \(d_{k}\) and the beam directions \(\theta_{k}\) to achieve good scanning performance. As a result, a multi-focal optimization cannot be directly used to calculate the final phase law. Instead, the iterative process represented in Figure 2 is used, where the translation range is gradually expanded while reinforcing the natural beam scanning of the antenna geometry. The exact panel size required to accommodate the final translation range is initially unknown, so \(D_{x}\) is purposefully overestimated at this point. The design process starts with a unifocal analytical phase law for \(d_{0}\) and a beam angle of \(\theta_{0}=26^{\circ}\), calculated using (2). The scanning performance of this phase law is evaluated  
Figure 2: Multi-stage optimization process of the RA response. by calculating the radiation pattern for panel positions close to \(d_{0}\):\(d_{\pm 1}\), \(d_{\pm 2}\), and \(d_{\pm 3}\). These positions are selected as the first set of positions to optimize. For each of these positions \(d_{k}\), an intended radiation direction \(\theta_{k}\) is assigned. This intended direction matches the beam direction observed for the unifocal phase law. Having defined a set of positions and intended directions, the first multi-focal optimization can be performed to calculate a new phase law that equalizes the gain level for this small set of panel positions. The above process is repeated until a beam steering range from 0\({}^{\circ}\) to 45\({}^{\circ}\) is covered. For every \(l\)-th iteration, the set of panel positions is expanded. Then, the radiation performance of the previous phase law, \(\phi_{n}^{(l-1)}\), is evaluated at every panel position in the updated set. The intended beam direction \(\theta_{k}\) for every position is the calculated beam direction from the previous phase law. This reinforces the natural tilting of the beam, and it gradually updates the relation between the panel position and the beam pointing. Once the set of panel positions and its associated set of intended radiation directions are updated, another multi-focal optimization is performed to calculate a new phase law \(\phi_{n}^{(l)}\). In doing so, the weighting factors of every position \(w_{k}\) are tuned to ensure that scanning losses remain below 1.5 dB in the new optimized range. This process is repeated for a total of nine iterations, requiring 24 panel positions from \(d_{-12}\) to \(d_{11}\). The panel positions being considered and the intended radiation direction for each of them are listed in Table 1. Six positions are introduced in the first iteration (three on each side of \(d_{0}\)) and four in the second. Starting with iteration 3, only one position is introduced on each side of the previous translation range to ensure a slow and gradual increase in scanning range and to improve the convergence of the optimization algorithm. This iterative design technique progressively expands the usable beam steering range and adapts the radiation angles for each position. \begin{table}
\begin{tabular}{c|c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Iteration \(l\)**} & \multicolumn{10}{c}{**Stage 1**} & \multicolumn{10}{c}{**Stage 2**} \\  & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** \\ \hline \(d_{-12}\) & - & - & - & - & - & - & - & - & -  
The antenna scanning performance throughout the design process is shown in Figure 3a. The unifocal phase law achieves a high gain of 33.7 dBi at the focal panel position of \(d_{0}\), but it rapidly drops as the panel shifts out of this position. Within the final panel translation range, which spans between \(d_{-12}\) and \(d_{11}\), the unifocal phase law shows a minimum gain of 25.4 dBi, or approximately 8.3 dB of scan loss. After three iterations of the design process, the phase law has been optimized for the range between \(d_{-6}\) and \(d_{6}\).