## 2 Design Method  
The proposed method is based on the generation of a sectorial beam reflectarray from the combination of several contiguous ones. An optimization algorithm is employed to find the best steering of the beams, whose combination generates the sectorial pattern with the desired beamwidth. SideLobe Level (SLL) reduction is also considered for optimization. A particle swarm optimization (PSO) algorithm [15] is chosen for this purpose. Figure 2 defines the coordinate system and the geometry parameters employed in the design method. The radiation pattern (RP) of a reflectarray can be estimated through array summation from the feed pattern, the geometry knowledge and an accurate characterization of the reflection coefficients of the elements (Equation (1)). The calculus of the RP is more useful in UV coordinates, because the optimization techniques employed to achieve the required radiation distribution are based on the comparison of the pattern with some masks, which are easily defined in UV space. \(E_{inc_{m,n}}\) is the incidence field from the feed at each element, \(|s_{11_{m,n}}|\) the modulus of reflection coefficient of each element and \(\phi_{m,n}\) the phase of the reflection coefficient at each element. \[E_{r}(u,v)=\sum_{m=1}^{M}\sum_{n=1}^{N}E_{inc_{m,n}}|s_{11_{m,n}}|e^{i\phi_{m,n} }e^{ik_{0}(ux_{m,n}+vy_{m,n})} \tag{1}\]  
The optimization vector or _particle_ is composed of 4 variables. For the \(k\)-th iteration: \(\mathbf{P_{k}}=[v_{b1,k};v_{b2,k};A_{b0,k};\phi_{cmt,k}]\). The first two variables of each particle are the \(v\) positions of the desired maximum radiation of two contiguous beams denoted as beams 1 and 2 (\(b1\) and \(b2\)). The \(u\) position for all the beams is fixed by the desired steering of the final  
Figure 2: Main geometrical parameters of a reflectarray antenna. sectorial beam \(\theta_{sa}\) (Equation (2)). It is expected that these two beams, together with the zero steering beam (\(v_{bb}\)) and the other two symmetrical beams at the zero left-side (\(v_{b-1}\) and \(v_{b-2}\)), achieve a sectorial beam of the required beamwidth. \[u_{b}=sin(\theta_{sa}) \tag{2}\]  
The required phases at each element \(\phi_{R}(x_{n},y_{n})\) to generate the sectorial beam are calculated as the weighted arithmetic mean of the phases required to generate each of the beams steering to \((\theta_{bi},\varphi_{bi})\) (Equation (3)), being \(tan\varphi_{bi}=\dfrac{u_{b}}{v_{bi}}\) and \(sin\theta_{bi}=\sqrt{u_{b}^{2}+v_{bi}^{2}}\). \[\phi_{R}(x_{n},y_{n})=\dfrac{\sum_{i=-2}^{2}A_{bi}\cdot k_{0}(d_{n}-(x_{n}cos \varphi_{bi}+y_{n}sin\varphi_{bi})sin\theta_{bi})}{\sum_{i=-2}^{2}A_{bi}} \tag{3}\]  
The phase distribution \(\phi_{R}(x_{n},y_{n})\) is taken for calculating the radiation pattern of a reflectarray of \(M\) row elements by \(N\) column elements through array theory formulation [16]. The weights employed are: \(A_{bb}\), obtained from the optimization process, for the phases that generate the zero steering beam, and those for the other 4 beam cases. \(k_{0}\) is the wavenumber at operation frequency, \(d_{n}\) the distance from feed to \(n\)-th element and \((x_{n},y_{n})\) the position of the \(n\)-\(th\) element. The real phases achievable with the elements (\(\phi\)), previously obtained by electromagnetic simulations, are employed, instead of ideal ones, to calculate the most reliable radiation pattern, as well as the losses (\(|s_{11}|\)) introduced by each element (Equation (1)). As it was probed in [14], the S-curve of the elements varies significantly from normal incidence \(\Theta_{inc\_norm}=0^{\circ}\) to maximum incidence in the border elements, so the real phases and losses for a set of incidence angles comprising all the possible range are employed \(\mathbf{\Theta_{inc}}=[0^{\circ},5^{\circ},10^{\circ},15^{\circ},20^{\circ},2 5^{\circ},30^{\circ},40^{\circ},50^{\circ}]\), assigning to each element the nearest angle of \(\mathbf{\Theta_{inc}}\).