# A Ka-Band Cylindrical Paneled Reflectarray Antenna
## 2 Design Principle  
The proposed configuration, shown in Figure 1, is composed by a linearly-polarized feed situated along the \(x\) axis at a height \(f\) from the origin. The feed illuminates a reflecting surface having an aperture \(D\) and \(L\) along the transversal (\(y\) axis) and longitudinal (\(x\) axis) plane, respectively. If the feed structure is symmetric, so is the antenna behavior and each half, shown in Figure 1b, can be independently characterized. The reflecting surface is divided into \(2N\) reflectarray panels distributed along a curved surface having cylindrical symmetry. Different strategies can be applied to define the size of the panel in the transversal direction and to choose the surface profile. For the case at hand, the reflecting surface has been obtained, discretizing along the \(y\) axis a parabola having focus \(f\), so that each half of the antenna is composed by \(N\) panels, namely \(S_{1}\),..., \(S_{N}\). The \(i\)th panel (\(S_{i}\)) has a width \(W_{i}\) and it is composed by a dielectric slab having thickness \(h\) and relative dielectric permittivity \(\varepsilon_{\rm r}\). The phase of the reflected field has been locally controlled by adjusting the width of \(M_{i}\) strips, which resonate along the antenna E-plane and are extended throughout the entire length of the panels. With respect to a conventional rectangular patch antenna, this approach is preferable because it fully exploits the cylindrical symmetry of the structure, concurring to increase the overall radiation efficiency while preserving the antenna linear polarization. The strips have a periodicity \(W\) while the width of the \(j\)th unit cell of the \(i\)th panel is \(W_{ij}\). As shown in Figure 1a, the electric field radiated by the feed impinges in the \(ij\) cell with an angle \(\theta_{ij}\) after traveling a \(r_{ij}\) long path. To obtain a beam in the broad side direction, all signals reflected by each cell \(ij\) must reach the feed with the same phase. This condition is fulfilled if the strip width \(W_{ij}\) of the involved cell is chosen to satisfy the following phase condition:  
\[-kr_{ij}+\phi_{ij}=0 \tag{1}\]  
where \(kr_{ij}\) is the phase shifting related to the distance between the center of the cell and the feed as shown in Figure 1a, \(k\) is the free space propagation constant and \(\phi_{ij}\) is the phase of the re-irradiated signal from the unit cell evaluated with the Ansys HFSS (2019 R2, ANSYS, Inc, Canonsburg, PA, USA, 2019) [14] model, as shown in Figure 2a. The dimension of the ith segment \(W_{i}\) is derived from the angle \(\theta_{i}\) as  
\[W_{i}=\frac{D}{2 Ncos(\theta_{i})} \tag{2}\]  
The aperture efficiency of the proposed configuration is related to the dimension of the unit cell \(W\) and to the number of segments \(N\). These two parameters can be chosen to maximize the antenna gain, \(G\), defined as [12]  
\[G(\theta)=D_{F}(\theta)\frac{\sum_{i=1}^{2N}\sum_{j=1}^{M_{i}}\abs{b_{ij}}^{2} }{\sum_{i=1}^{2N}\sum_{j=1}^{M_{i}}\abs{a_{ij}}^{2}}\eta_{s} \tag{3}\]  
where \(D_{F}(\theta)\) is the directivity of the feed, \(a_{ij}\) and \(b_{ij}\) are the field incident to and reflected by the _ij_ cell, respectively, and \(\eta_{s}\) is the spillover efficiency. \(\eta_{s}\) is calculated as the power collected by the array divided by the power radiated by the source. The power collected by the array can be evaluated as described in [13] using the \(a_{ij}\) coefficients defined below. The incident and reflected field were evaluated using a finite element method (FEM) tool [14] to simulate the feed and the single strip radiation pattern in the transverse section and assuming a uniform longitudinal distribution. Each strip was characterized using periodic boundary conditions (shown in Figure 2) and using an AD350A  
Figure 1: Paneled cylindrical reflectarray geometry: (**a**) Section view of the entire geometry; (**b**) 3-D view of half structure.