## 3 Application to Reflectarray for Compact Antenna Test Range
### Wideband Multi-Frequency Optimization of the Near-Field Figures of Merit  
The next step is to carry out a multi-frequency optimization of the reflectarray layout to improve performance in the frequency range of interest. To this end, the multi-frequency optimization algorithm of Figure 2 is employed. The synthesis is carried out at three frequencies within the 5G FR n257 band, 27 GHz, 28 GHz and 29 GHz. In addition, two linear polarizations are optimized, considering requirements in both magnitude and phase ripples. Thus, a total of 12 FoM are optimized. In order to ease the optimization and facilitate convergence towards a solution, a multi-stage procedure is adopted [46]. A total of six stages are considered: first, only the dipoles oriented in the \(\hat{x}_{a}\) direction are optimized by means of the \(T_{x}\) variable. In the second stage, only the dipoles oriented in the \(\hat{y}_{a}\) are optimized by means of the \(T_{y}\) variable. Then, in the third stage, both variables \(T_{x}\) and \(T_{y}\) are optimized at the same time for all reflectarray unit cells. For the last three stages, a similar procedure is followed, but increasing the number of DoF per reflectarray element. Now, the DoFs are the lengths of all dipoles but maintaining the cell symmetry with \(L_{a_{1}}=L_{a_{3}}\) and \(L_{b_{1}}=L_{b_{3}}\) (see Figure 4a), i.e., \(T_{x_{1}}=L_{b_{1}}=L_{b_{3}}\), \(T_{x_{2}}=L_{b_{2}}\), \(T_{x_{3}}=L_{a_{4}}\), \(T_{y_{1}}=L_{a_{1}}=L_{a_{3}}\), \(T_{y_{2}}=L_{a_{2}}\), \(T_{y_{3}}=L_{b_{4}}\). The new variables are \(T_{x_{i}}\) and \(T_{y_{i}}\), \(i=1,2,3\), for each reflectarray element. Thus, in the fourth stage, only variables \(T_{x_{1}}\), \(T_{x_{2}}\) and \(T_{x_{3}}\) are considered, while, in the fifth stage, the optimizing variables are \(T_{y_{1}}\), \(T_{y_{2}}\) and \(T_{y_{3}}\). Finally, in the sixth and last stage, all six variables per element are optimized. By following this procedure with the algorithm of Figure 2, and imposing requirements of a maximum magnitude ripple of 1 dB and a maximum phase ripple of 10\({}^{\circ}\) for both linear polarizations and three frequencies, the results of Figure 7 are obtained. The main cut (\(y=0\) mm) is shown for both linear polarizations at the three considered frequencies for magnitude and phase. Now, the magnitude and phase fully comply with the requirements of a plane wave inside region \(\Omega_{1}\) where the FoM are defined (see Figure 3). The improvement in compliance of the QZ is better seen in Figure 8. This represents, for linear polarization \(X\) at the three frequencies of interest, the magnitude and phase in region \(\Omega_{1}\) that complies with the requirements of a plane wave before and after the multi-frequency optimization. The starting layout does not fully comply with specifications at any of the frequencies, although at the central frequency the phase almost does, as also shown in Table 1. After the optimization, the requirements are met for both magnitude and phase at the three frequencies for which the FoM were optimized. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Compliance (\%)**} & \multicolumn{4}{c}{**Ripple**} \\ \cline{3-10}
**Reflectarray** & **Frequency (GHz)** & \multicolumn{2}{c}{**Pol. X**} & \multicolumn{2}{c}{**Pol. Y**} & \multicolumn{2}{c}{**Pol. X**} & \multicolumn{2}{c}{**Pol. Y**} \\ \cline{3-10}  & & **Mag.** & **Phase** & **Mag.** & **Phase** & **Mag. (dB)** & **Phase (\({}^{\circ}\))** & **Mag. (dB)** & **Phase (\({}^{\circ}\))** \\ \hline Before POS & 28 GHz & 25.51 & 49.79 & 26.75 & 57.00 & 7.57 & 26.35 & 6.61 & 20.95 \\ \hline \multirow{3}{*}{After POS} & 27 GHz & 81.69 & 76.34 & 66.77 & 63.89 & 1.56 & 20.91 & 1.86 & 24.12 \\  & 28 GHz & 60.60 & 97.84 & 69.24 & 96.40 & 1.80 & 11.51 & 2.16 & 13.59 \\  & 29 GHz & 39.77 & 89.51 & 36.93 & 50.31 & 3.17 & 15.43 & 5.02 & 22.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance of the reflectarray before and after the POS with the GIA at a plane 0.5 m away from the antenna.