## 3 Design of the Bifocal DRAA Configuration
Note also that a conventional monofocal DRAA would employ a circular main RA of 1.8 m diameter instead of an elliptical one. As stated before, the reason for oversizing the main RA is related to the application of the bifocal technique to reduce the beam spacing in the \(xz\)-plane by a large factor. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Size Main-RA & 3.55 m \(\times\) 1.81 m \\ Size Sub-RA & 0.79 m \(\times\) 0.79 m \\ Angle of tilting Sub-RA & 10\({}^{\circ}\) \\ Geometrical center Main-RA & [0, 0, 0] mm \\ Geometrical center Sub-RA & [\(-\)1288, 0, 3003] mm \\ Phase center of feed F\({}_{1}\) & [\(-\)2043, 0, 2222] mm \\ Phase center of feed F\({}_{3}\) & [\(-\)1937, 0, 2204] mm \\ Phase center of feed F\({}_{5}\) & [\(-\)1830, 0, 2185] mm \\ Virtual focus related to F\({}_{3}\) & [\(-\)1854, 0, 4847] mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric parameters of the DRAA configuration. Figure 3: Geometry of the proposed DRAA configuration, formed by an elliptical main RA and a circular sub-reflectarray: (**a**) 3D view, (**b**) lateral view and (**c**) front view. The bifocal technique has been applied to the proposed DRAA configuration in order to obtain 0.56\({}^{\circ}\) separation between adjacent beams in the \(xz\)-plane, which involves a large reduction of the beam deviation factor (by a factor close to 2) with respect to the equivalent monofocal DRAA. The two focal points of the bifocal antenna have been selected to match the phase centers of the feeds F\({}_{1}\) and F\({}_{5}\) depicted in Figure 3. The beam directions associated to each focus have been chosen as \(\theta_{b1}=20.12^{\circ}\), \(\varphi_{b1}=0^{\circ}\) and \(\theta_{b5}=17.88^{\circ}\), \(\varphi_{b5}=0^{\circ}\), in order to avoid blockage from the sub-RA and to provide 0.56\({}^{\circ}\) separation between beams produced by adjacent feed-horns. The computation of the bifocal phase distributions required on both reflectarrays has been performed by means of an iterative ray-tracing algorithm, which alternates transmitted rays from F\({}_{1}\) (the first focus of the DRAA) with received rays that are directed to F\({}_{5}\) (the second focus). A detailed description of the ray-tracing algorithm can be found in [20]. The rays coming from F\({}_{1}\) are reflected, first, by the sub-RA, and then, by the main RA, leaving the structure with an angle (\(\theta_{b1}\), \(\varphi_{b1}\)). On the other hand, the received rays coming at an angle (\(\theta_{b5}\), \(\varphi_{b5}\)) are reflected, first, by the main RA, and then, by the sub-RA, being finally directed to F\({}_{5}\). When the rays impinge on the surface of the sub-RA or the main RA, the following equations are applied, which relate the angles of incidence (\(\theta_{inc}\), \(\varphi_{inc}\)) and reflection (\(\theta_{ref}\), \(\varphi_{ref}\)) of the rays with the partial derivatives of the bifocal phase distribution (\(\Phi_{bifocal}\)) with respect to the longitudinal and transversal length variables:  
\[\frac{\partial\Phi_{bifocal}(x,y)}{\partial x} =\frac{2\pi}{\lambda}\left(\sin\theta_{inc}\cos\varphi_{inc}- \sin\theta_{ref}\cos\varphi_{ref}\right) \tag{1}\] \[\frac{\partial\Phi_{bifocal}(x,y)}{\partial y} =\frac{2\pi}{\lambda}\left(\sin\theta_{inc}\sin\varphi_{inc}- \sin\theta_{ref}\sin\varphi_{ref}\right) \tag{2}\]  
The phases from the previous monofocal DRAA design have been used to obtain the initial conditions for the phase derivatives along the cross-section of both reflectarrays (parallel to the \(y\)-axis), as explained in [20]. This will allow keeping the monofocal characteristic of the DRAA in the azimuth plane, which will be combined with a bifocal design in the \(xz\)-plane to provide closer beams (separated by 0.56\({}^{\circ}\)). After several iterations of the ray-tracing procedure, a set of phase derivative samples is obtained on the surface of both reflectarrays.