# Compact Beam-Scanning Reflectarray Antenna with SLL Reduction Using In-Plane Panel Translations
## 2 Antenna Definition  
The geometry of the presented antenna is shown in Figure 1. One-dimensional steering is achieved through in-plane linear panel translations, which are used to control the portion of the aperture that is illuminated. The position of the center of the panel is defined by an offset \(d\) in the \(x\)-axis, which relates to the steering angle. The feed is a commercial 11.5 dBi linear-polarized rectangular horn, model LB-28-10-C-KF from A-INFOMW. It is selected to achieve a compact setup thanks to its low directivity. The feed is located at \(x_{f}=-66.1\) mm, and a tilt of \(\alpha=26^{\circ}\) is introduced to avoid the beam distortions caused by feed blockage for the beam steering angles \(\theta\geq 0^{\circ}\) in the \(XZ\)-plane. The feed-to-panel distance \(F\) is set to 88 mm, or 8.66 \(\lambda_{0}\), at the design frequency of 29.5 GHz, and the size of the panel orthogonal to the translation is set to \(D_{y}=187.2\) mm. These two values result in an illumination taper of \(-11.6\) dB on the longer panel sides and a very low \(F/D_{y}\) ratio of 0.47. The E-plane of the feed and the reflected beam coincides with the \(XZ\)-plane for all beam directions.  
Designing an RA relies on the accurate modeling of the electromagnetic field on the panel. For any given panel position, the incident field is obtained from a full-wave HFSS simulation using Ansys HFSS (Release 2023R1) [28]. The tangential reflected and incident field components can then be related at each cell of the lattice as follows [29]:  
\[\begin{pmatrix}E^{r}_{x}(\vec{r}_{n})\\ E^{r}_{y}(\vec{r}_{n})\end{pmatrix}=\begin{pmatrix}\rho_{n,xx}&\rho_{n,xy}\\ \rho_{n,yx}&\rho_{n,yyy}\end{pmatrix}\cdot\begin{pmatrix}E^{i}_{x}(\vec{r}_{n} )\\ E^{i}_{y}(\vec{r}_{n})\end{pmatrix}, \tag{1}\]where \(\vec{r}_{n}=(x_{n},y_{n},z_{n})\) is the position of the \(n\)-th element. \(\rho_{n,xx}\) and \(\rho_{n,yy}\) are its direct linear reflection coefficients, which can typically be approximated as the ideal phase shifters, \(e^{i\phi_{n,x|y}}\). Furthermore, the cross-polarization coefficients \(\rho_{xy}\) and \(\rho_{yx}\) can be neglected if a cell with good polarization purity is selected. This simplifies the design process into selecting the shifts \(\phi_{n,x|y}\) that each element should introduce. For the presented antenna geometry and polarization, the focus is on the \(\phi_{n,x}\) phase response, and the \(x\)-notation will be dropped for clarity. To avoid grating lobes, square cells of side \(p=0.4\lambda_{0}\) (4.07 mm) are considered in the following analysis.  
It is known from array theory that beam collimation requires a progressive linear phase on the reflected field [30]. This can be achieved by selecting the phase shift for each cell to compensate for the delay of the impinging field, \(\angle E_{x}^{i}(\vec{r}_{n})\). When the panel is at an arbitrary position \(d\), and particularizing for \(\varphi=0^{\circ}\) for the intended scanned plane, the maximum collimation towards a direction \(\theta_{0}\) requires a phase law  
\[\phi_{n}=-k_{0}x_{n}\sin\theta_{0}-\angle E_{x}^{i}(\vec{r}_{n}). \tag{2}\]  
As the panel shifts out of this position, the beam is de-focused, changing its gain and pointing angle. To estimate the radiation performance of a given phase law and panel position, the incident field is first obtained for that panel position. Then, the reflected field on the panel is calculated using (1), assuming the phase shifting elements are ideal. Then, the radiation pattern is calculated using the second principle of equivalence, as described in [31]. To analyze the continuous translation path, it is discretized into finely spaced positions. For convenience, the discretization step is selected to match the element spacing \(p\). The notation \(d_{k}\) will be used to identify the position of the panel \(d_{k}=k\cdot p\).