# Compact Beam-Scanning Reflectarray Antenna with SLL Reduction Using In-Plane Panel Translations
## 3 Multi-Stage Design
### SLL Reduction  
The design process above introduces unwanted phase aberrations on the reflected aperture field. These result in beam distortions in the form of high SLLs, which are particularly prominent at specific panel positions. As presented in Figure 3b, the maximum SLL after the scanning range optimization is \(-9.3\) dB at \(d_{-5}\). Ideally, these aberrations should be evenly spread across the entire scanning range so that no particular panel position is too affected. In order to mitigate the highest side lobes and to spread these aberrations more evenly across the scanning range, a second design stage is defined. The objective of this stage is to mitigate the SLLs of the most affected panel positions. The response of the RA panel will be refined to avoid partial collimation effects in the direction of the side lobes. To do so, an iterative approach similar to the first design stage will be used, except in this case the objective is to reduce collimation in the specific direction of the side lobes, as opposed to improving collimation in the direction of the main beam. Each iteration begins by identifying the panel position with the highest side lobe, \(d_{k}\). The side lobe direction, \(\theta_{k}^{SL}\), is selected as an additional intended beam direction for \(d_{k}\). As a result, \(d_{k}\) has two radiation objectives as follows: one for collimating towards \(\theta_{k}\) (fixed in the previous design stage) and the other for \(\theta_{k}^{SL}\). This secondary radiation direction is assigned a negative weight \(w_{k}^{SL}\) in the optimization process. Effectively, this means that when the optimization process will be biased to favor a phase law that collimates towards \(\theta_{k}\) when the panel is at \(d_{k}\) (through a positive optimization weight \(w_{k}\)), while reducing  
Figure 3: Calculated scanning performance throughout the design process: main beam direction and gain (**a**) and SLL (**b**) for each panel position. partial collimation effects towards the identified SLL direction \(\theta_{k}^{SL}\) (through a negative weight \(w_{k}^{SL}\)). This technique works by reducing collimation effects towards specific directions. For this reason, it can only be applied once the RA response is nearly finalized and the direction and level of the side lobes are well established. The response of the RA must not be altered significantly during this refinement stage; otherwise, it is possible for the side lobes to shift to a different direction rather than be suppressed. For this reason, this design stage is a refinement stage, where large changes to the phase law should be avoided. To that end, the set of panel positions and their radiation directions, as well as the optimization weight, are fixed after the ninth iteration (end of the first design stage). Furthermore, only two side-lobe suppression iterations are performed, and high values for \(-w_{k}^{SL}\) are avoided. The two refined panel positions are \(d_{-5}\) and \(d_{0}\), and the suppressed side-lobe angle is listed in Table 1. The effects of this refinement are shown in Figure 3, as follows: the main beam level and radiation directions are mostly unchanged between stages 1 and 2 (Figure 3a), but the maximum SLL is reduced from \(-9.3\) dB to \(-11.7\) dB (Figure 3b). The initial and final phase responses and the offset between the two are shown in Figure 4. The \(-9\) dB illumination taper is overlaid for the positions \(d_{\pm 11}\) to highlight the region of the aperture that is most relevant for each panel position. The main differences between both responses occur on the edges of the panel. That is because the illumination is lower there for the central feed positions, and thus, more effective optimizations are possible with little effect on the other beams. Figure 4: Calculated unifocal (\(\phi_{n}^{(0)}\)) (**a**) and optimized (\(\phi_{n}^{(11)}\)) (**b**) phase laws for the presented RA antenna. The difference between the two is shown in (**c**) in terms of \(\phi_{n}^{(11)}-\phi_{n}^{(0)}\).