## 3 Near Field Models of Reflectarray and Transmitarray Antennas
### Comparison of the Near Field Models
In the case of the near field model based on the superposition of the far fields radiated by each array element, the relative error is slightly larger, and it also decreases as the near field is computed further away from the array. However, even for a plane at \(z=0.1\,\mathrm{m}\), which is \(13\lambda\) at the working frequency, the error is very low. Figure 3: Comparison of the near field radiated by a center-fed transmitarray in (**a**) magnitude and (**b**) phase at the different \(z\) planes as calculated by the near field model based on the superposition of far field (FF) contributions of each array element, and the near field (NF) model based on the radiation equations for different values of \(N_{x}\) and \(N_{y}\) to assess the convergence in the numerical evaluation of the double integrals. Near field is plotted in the NFCS. In light of these results, it is clear that both models offer practically the same results in the computation of the near field for spatially fed planar arrays. However, from a computational point of view, the model based on the superposition of far fields is more interesting. Indeed, as shown in Table 2, this model is at least one order of magnitude faster than the model based on the radiation equations. This is particularly important when employing the model in optimization procedures to shape the near field, since the analysis routine where the near field is computed is invoked many times. Then, the model based on the radiation equations could be used, if desired, to assess the validity of the results that were obtained employing the approximate model based on superposition of the far fields radiated by the array elements. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{\(\boldsymbol{z=0.1\,\mathrm{m}}\)} & \multicolumn{3}{c}{\(\boldsymbol{z=1\,\mathrm{m}}\)} \\ \hline  & \(\boldsymbol{E_{x}}\) & \(\boldsymbol{E_{y}}\) & \(\boldsymbol{E_{z}}\) & \(\boldsymbol{E_{x}}\) & \(\boldsymbol{E_{y}}\) & \(\boldsymbol{E_{z}}\) \\ \hline FF contributions & 2.22\% & 2.21\% & 3.20\% & 0.22\% & 0.22\% & 0.32\% \\ NF (\(N_{x}=N_{y}=1\)) & 0.09\% & 0.33\% & 3.10\% & 0.06\% & 0.14\% & 0.15\% \\ NF (\(N_{x}=N_{y}=3\)) & 0.01\% & 0.02\% & 0.18\% & 0.00\% & 0.01\% & 0.01\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relative error of the field components calculated with the different near field models when compared with the model based on the radiation equations with \(N_{x}=N_{y}=5\), which is taken as reference. Figure 4: Comparison of the near field radiated by a center-fed transmitarray in magnitude and phase for the plane \(z=0.1\,\mathrm{m}\) generated by the model based on the superposition of far field contributions of each array element (**top** row), and the near field model based on the radiation equations for \(N_{x}=N_{y}=5\) (**bottom** row). The relative error between both simulations taking the bottom row as reference is 2.2% for both components of the field. Near field is plotted in the NFCS. The considered reflectarray is square and comprised of \(N=1936\) elements in a regular grid of \(44\times 44\). The periodicity is \(5.35\,\mathrm{mm}\) in both dimensions, which is approximately half a wavelength at the central frequency of operation, \(28\,\mathrm{GHz}\). The phase center of the feed is placed at \(\bar{r}_{f}=(-0.16,0,0.24)\,\mathrm{m}\) with regard to the reflectarray center and generates an average edge illumination taper between \(-14.7\,\mathrm{dB}\) and \(-16.5\,\mathrm{dB}\) in the range \(26.5-29.5\,\mathrm{GHz}\).