## 2 Antenna Design Procedure and Methods
### Design of the 1 to 16 Feeding Networks  
E-plane GGW corporate topologies provide compact feed networks, which were already used for a \(4\times 4\) slot array in [27]. However, E-plane GGW in 1-to-2 power distribution structure can cause output phase inversion, such as the input-output relationship of a magic tee in the E plane. Hence, an additional 180\({}^{\circ}\) phase shift is required to compensate for the phase difference and realize the broadside beam, which makes it difficult to further scale-up the array. On the other hand, implementing the feed network with RGWs can avoid such phase inversion [28]. However, RGW-based power dividers require additional impedance matching structures that occupy more space. These problems can be conveniently addressed when GGWs and RGWs are combined as part of the same distribution network [29]. Such an approach is optimal to feed arrays with symmetric and compact topologies and therefore was adopted to design the feeding network for the proposed antenna array.  
Figure 6 shows the respective top views of the K- and Ka-band layers of the antenna array along with their feeding networks. We highlighted GGWs in blue and RGWs in red. Sixteen antenna units are fed by the paralleled feeding network.  
Figure 7a shows the topology of the RGW-to-GGW power dividers used in the feed network. The protrusion at the end of the RGW can be used to optimize the input reflection coefficient of the power divider. Figure 7b shows the topology of the GGW-to-RGW dividers. The distance between the RGWs and the shorter end of the GGW is around \(\lambda\)g/4, where \(\lambda\)g is the wavelength of the guided wave in the GGW. This distance is chosen to generate the maximum E-field value at the crossing point between the GGW and RGWs to excite the parallel RGWs. It should be noticed that the RGWs have no protrusion in a GGW-to-RGW power divider structure. The dimensions of the RGW-GGW transitions are summarized in Tables 1 and 2.  
Figure 5: Electric field vectors inside the K-band cavity at different time steps within one period T. (a) t = 0. (b) t = T/4. (c) t = T/2. (d) t = 3T/4. The direction of wave propagation is coming out of the page.  
Figure 6: Top view of the horizontal cut planes going through two different layers of the proposed array. (a) Higher-frequency layer. (b) Lower-frequency layer. (GGWs: blue; RGWs: red.).  
Figure 8 shows the electrical field distribution of a section of the feed network that contains both GGW-to-RGW and RGW-to-GGW transitions. The out-of-phase field distribution in the two RGW sections to the right and to the left of the crossing point with the GGW can be clearly observed. To compensate for the 180\({}^{\circ}\) phase difference without having to use any additional phase shift lines, the output GGWs are placed back to back, which cancels the 180\({}^{\circ}\) phase difference. As a result, sixteen radiation units are uniformly excited with the same amplitude and phase.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(\mathbf{GGW_{1}}\) & \(\mathbf{GGW_{2}}\) & \(\mathbf{GGW_{3}}\) & \(\mathbf{GGW_{4}}\) & \(\mathbf{GGW_{5}}\) & \(\mathbf{GGW_{6}}\) \\ \hline width & 1.78 & 1.9 & 1.9 & 1.2 & 0.8 & 2.25 \\ height & 4.62 & 4.62 & 4.62 & 4.8 & 4.8 & 4.9 \\ length & 3.5 & 3 & NA. & 1.6 & 1.6 & NA. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dimensions of the GGW in the corporate-feed network (unit: mm).  
Figure 7: (**a**) RGW-to-GGW divider. (**b**) GGW-to-RGW divider. The fundamental parameters of each waveguide are indicated.