## 3 Design of the Bifocal DRAA Configuration  
The proposed DRAA system has been designed by adopting a Cassegrain configuration, as can be seen in Figure 3. The main geometrical parameters of the DRAA are given in Table 1. The antenna consists of an elliptical flat main-RA of size \(3.55\) m \(\times\) 1.81 m, and a circular flat sub-RA of 79 cm diameter, with a relative tilting of \(10^{\circ}\) between them. An array of five adjacent feed-horns placed in the \(xz\)-plane (the antenna symmetry plane) was initially considered for the antenna illumination (feeds F\({}_{1}\) to F\({}_{5}\) in Figure 3), which corresponds to the central row of feeds in Figure 2b. A realistic model of a Ka-band feed-horn [3] has been used: the horns have a diameter of \(54\) mm and provide around \(-12\) dB illumination on the sub-RA edges at 20 GHz for a subtended angle of \(36^{\circ}\). The proposed DRAA geometry has been derived from a Cassegrain dual-reflector configuration, where both reflectors have been replaced by flat reflectarrays. The equivalent parabolic reflector for the elliptical flat main-RA presents the following parameters: dimensions (projected) \(3.36\) m \(\times\) 1.81 m, focal distance F = 4.72 m, and clearance C = 1.57 m. The focal location for the equivalent parabolic reflector coincides with the position of the virtual focus related to the central feed (F\({}_{3}\)) of the proposed antenna. The phase center of F\({}_{3}\) has been placed at the real focus of the Cassegrain system, and then, the adjacent feeds (F\({}_{1}\), F\({}_{2}\), F\({}_{4}\) and F\({}_{5}\)) have been situated with a separation of 55 mm between their phase centers (allowing 1 mm margin to properly accommodate the horns). The sub-RA dimensions and tilting angle have been finely adjusted after carrying out a trade-off with the bifocal algorithm,  
Figure 2: (a) Proposed bifocal DRAA configuration with an elliptical main reflectarray (RA) to produce multiple adjacent beams in dual polarization, (b) feed cluster and (c) beam configuration associated to the proposed antenna to produce a complete multibeam coverage in the Ka-band. in order to ensure an appropriate illumination on the main RA when the bifocal technique is applied to reduce the beam deviation factor and obtain closer beams (separated by 0.56\({}^{\circ}\)), at the same time as avoiding the blockage from the sub-RA and the feed-horns. Note that the Cassegrain system allows working with a smaller sub-RA size than in the case of the compact-range system used in [20, 27]. A cell period of 10 mm has been considered for both reflectarrays to avoid working with an excessive number of reflectarray elements; however, in a realistic design the period should be lower (e.g., around \(\lambda\)/2 at 20 GHz) to avoid grating lobes and to provide larger bandwidth. A reference monofocal design of the proposed DRAA system has been carried out at 20 GHz, considering that the antenna focus is placed at the phase center of the central feed (F\({}_{3}\)), whose associated beam will radiate in the direction \(\theta_{b3}=19^{\circ}\), \(\varphi_{b3}=0^{\circ}\) with respect to the normal vector to the main-RA surface. The monofocal phase distribution required on the sub-RA has been computed from the phase delay associated to the difference between two paths: the path between the phase center of F\({}_{3}\) and the sub-RA plane, and the path between the virtual focus related to F\({}_{3}\) and the sub-RA plane. Then, the monofocal phase distribution on the main RA has been obtained from the progressive phase distribution that generates the required focused beam, considering that the feed was placed at the virtual focus. The resultant monofocal phase distributions for both reflectarrays are depicted in Figure 4. The monofocal DRAA will provide around \(1.1^{\circ}\) of beam spacing (as will be shown later). Since the feed-horns are placed adjacent to each other, a smaller beam spacing would require overlapping feeds.