## 3 Analysis of Design Steps  
The steps of the design procedure outlined in Section 2 are here described and discussed. First, we choose the substrates, assigning the relative dielectric constant and the loss  
Figure 2: Flow chart of optimization procedure.  
angle. As a possible choice, we suppose to use Rogers RT6002 (\(e_{r}=2.94\), \(tan\delta=0.0012\)) for the dielectric 1 and the RO4003 (\(\epsilon_{r}=3.55\), \(tan\delta=0.0021\)) for the dielectrics 2 and 3, since these materials are frequently used in space applications. The permittivity of honeycomb layers has been fixed to 1.08 and the loss angle to \(8.3\times 10^{-4}\). The fixed design parameters are reported in Table 1 and have been chosen appropriately, following the considerations discussed in the previous section. The optimized parameters are shown in Table 2 for each optimization step.  
Figure 3 shows the results in terms of H and V return loss and H/V isolation after the first step of optimization. As one can see, the results are not good for any of the frequencies considered, although we chose the lengths of the resonant elements (patch and slot) equal to a half-wavelength at the center of the band.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value [mm]** & **Parameter** & **Value [mm]** \\ \hline \(h_{1}\) & 0.17 & \(T_{S}^{H}\) & 0.96 \\ \hline \(h_{4}\) & 0.02 & \(TL_{S}^{H,V}\) & 0.3 \\ \hline \(h_{5}\) & 0.02 & \(V_{OFFSET}\) & 0.67 \\ \hline \(D1\) & \(11\) & \(H_{OFFSET}\) & 1.01 \\ \hline \(D2\) & \(11\) & \(T_{S}^{V}\) & 1.3 \\ \hline \(W_{S}^{H,V}\) & 0.47 & \(W_{1}^{H,V}\) & 0.44 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The fixed design parameters.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value at Step 1** & **Value at Step 2** & **Value at Step 3** \\  & **[mm]** & **[mm]** & **[mm]** \\ \hline \(h_{2}\) & 0.6 & 0.79 & 0.79 \\ \hline \(h_{3}\) & 0.6 & 0.74 & 0.74 \\ \hline \(L_{S}^{H}\) & 1.6 & 2.11 & 2.11 \\ \hline \(L_{S}^{V}\) & 1.3 & 1.3 & 1.47 \\ \hline \(WA_{1}\) & 2.59 & 2.59 & 3.2 \\ \hline \(WA_{Figure 4 shows the results in terms of H and V return loss and H/V isolation after the second step of optimization. H polarization has an impedance bandwidth of about 50% with a return loss of \(\geq\)10 dB.  
Finally, Figure 5 shows the final results of optimization. The impedance has a bandwidth about 50% (2.8-4.71 GHz) for both polarizations, with a return loss \(\geq\)10 dB. The H/V isolation is greater than 30 dB over the entire matching band. It can be seen in the last step of optimization that the impedance of the H polarization has remained practically unchanged. This is due to the isolation that between the two polarizations, obtained thanks to the mutual position between the two slots.  
The number of iterations is variable; it depends both on the initial choice of parameters and on the width of the variability interval. On average, 150 iterations are required for the first optimization step and 100 iterations for the second step.  
Figure 4: H and V polarization return loss and H/V isolation after second step of optimization.  
Figure 3: H and V polarization return loss and H/V isolation after first step of optimization.