## 3 Design of the Bifocal DRAA Configuration
The phase derivative samples are interpolated by polynomials, and then integrated to obtain the bifocal phase distributions that must be implemented on both reflectarrays. The resultant phase distributions obtained by this technique can be seen in Figure 5. Note that the main RA presents a larger number of 360\({}^{\circ}\) cycles along its vertical axis than in the monofocal design, while the sub-RA phases have changed from a convex to a concave characteristic (both effects are produced by the large degree of reduction in the beam spacing). Figure 4: Monofocal phase distributions (in degrees) on (**a**) the sub-RA and (**b**) the main RA. The bifocal DRAA has been simulated using an in-house analysis software, assuming ideal reflectarray cells in both surfaces that provide the required phase-shift. The electromagnetic field radiated by the horns has been modeled using a cosq(\(\theta\)) function with q = 28, which provides -12 dB illumination on the sub-reflectarray edges for a 36\({}^{\circ}\) subtended angle. The accuracy of the analysis and design software has been validated in previous works with satisfactory results [20,27], including the design, fabrication and testing of several DRAA prototypes in the Ku and Ka bands. The comparison of the radiation patterns in the \(xz\)-plane at 20 GHz with the beams generated by the bifocal DRAA (in solid colored lines) and those produced by the equivalent reference monofocal DRAA (in dashed lines) is shown in Figure 6.  
The separation between adjacent beams in the \(xz\)-plane is 0.56\({}^{\circ}\) for the bifocal DRAA, while it is around 1\({}^{\circ}\)-1.1\({}^{\circ}\) for the monofocal DRAA. The gain of the bifocal beams varies from 50.44 dBi to 49.85 dBi, the scan loss is 0.59 dB, and the side-lobe levels are lower than -21 dB with respect to the maximum. On the other hand, the scan loss in the monofocal antenna is 2.5 dB (the gain varies from 48.5 dBi to 46 dBi) and the extreme beams are quite broadened. Note that the gain of the bifocal beams is a couple of dB higher than the gain of the monofocal beams, since the use of an oversized main RA produces narrower beams. According to these results, the bifocal DRAA is able to produce much closer  
Figure 5: Bifocal phase distributions (in degrees) on (**a**) the sub-RA and (**b**) the main-RA. Figure 6: Simulated radiation patterns at 20 GHz in the \(xz\)-plane of the beams generated by the bifocal DRAA (solid colored lines) and the beams produced by the equivalent monofocal DRAA (dashed lines), indicating the maximum side-lobe levels (SLL, in black dotted lines) that produce the interference among reuse beams (see Section 5). beams in the \(xz\)-plane than the reference monofocal antenna (beam spacing is reduced by a factor close to 2) with lower scan loss. The radiation patterns of the bifocal DRAA in the azimuth plane (orthogonal to the \(xz\)-plane) are determined by the performance of the equivalent monofocal antenna, whose phase distributions have been used as initial conditions for the bifocal synthesis. Hence, the beam spacing provided by the bifocal DRAA in the azimuth plane is similar to that of the monofocal DRAA. This can be checked in Figure 7, which shows the comparison of the radiation patterns at 20 GHz in the azimuth plane for the bifocal DRAA (in solid lines) and the monofocal DRAA (in dashed lines), considering two additional beams produced by two horns placed adjacent to F\({}_{3}\) (F\({}_{3\text{L}}\) and F\({}_{3\text{R}}\)). The separation between adjacent beams is around 1.1\({}^{\circ}\) for both antennas, which is almost twice the angular spacing achieved in the \(xz\)-plane by the bifocal antenna. Thus, the bifocal antenna preserves the monofocal characteristic of the reference DRAA design in the azimuth plane.