## 3 Near Field Models of Reflectarray and Transmitarray Antennas
### Multi-Frequency Optimization of the Near Field  
The multi-frequency optimization of the near field consists in a direct layout optimization using the GIA [21]. To accommodate specifications at a number of frequencies, the cost function (\(F\)) in the backward projector of the GIA is modified in the following fashion [36]:  
\[F_{\Omega}=\sum_{f=1}^{N_{f}}\sum_{l=1}^{L}w_{f}(\vec{r}_{l})\left(\left|\vec{E} _{\text{NF},f}(\vec{r}_{l})\right|^{2}-\left|\vec{E}_{\text{NF},f}(\vec{r}_{l} )\right|^{2}\right)^{2}\Delta\Omega, \tag{38}\]  
where \(\Omega\) is the region where the near field is computed; \(N_{f}\) is the total number of frequencies at which the direct layout optimization is performed; \(L\) is the number of points in which the region \(\Omega\) is discretized; \(w_{f}\) is a weighting function that depends on the frequency and observation point \(\vec{r}_{l}\); \(\vec{E}_{\text{NF},f}(\vec{r}_{l})\) is the computed near field at each iteration of the GIA and frequency; \(\vec{E}_{\text{NF},f}(\vec{r}_{l})\) is the reference near field in region \(\Omega\) resulting from applying the specification templates at a given frequency; and \(\Delta\Omega\) is the step of the discretization. Further details on the specifics of the optimization algorithm may be consulted elsewhere [21]. Unlike in POS, where the optimizing variables (sometimes also referred to as degrees of freedom or DoF) were the phases of the direct coefficients, the direct layout optimization employs the geometrical features of the unit cell as optimizing variables. In the present case, the dimensions of the rectangular patches are employed, \(T_{x_{1}}\), \(T_{x_{2}}\), \(T_{y_{1}}\), and \(T_{y_{2}}\) (see Figure 6). Thus, a total of 7744 DoF are available for the optimization. Moreover, the direct layout optimization will be carried out in several stages in order to facilitate convergence towards a suitable solution. First, only variables \(T_{x_{1}}\), and \(T_{x_{2}}\) will be considered at three frequencies, 27 GHz, 28 GHz and 29 GHz. In the second stage, the near field is optimized at the same three frequencies, but this time only considering as variables \(T_{y_{1}}\) and \(T_{y_{2}}\). Finally, in the third stage, all four variables per element are used in the optimization at the same seven frequencies of Table 3. In all stages, the specifications for both linear polarizations are imposed. In addition, the MoM-LP tool is directly employed in the optimization loop to obtain the real electromagnetic response of the unit cell at each considered frequency. In order to accelerate computations, the differential contributions technique [37] is used. The improvement in the coverage zone can be visually seen in Figure 9, where the normalized magnitude of the electric field for linear polarization \(X\) is represented at three different frequencies (26.5 GHz, 28 GHz, and 29.5 GHz) before and after the direct layout optimization. The field is only plotted at those points that comply with the requirement of 2 dB of ripple. Although the starting layout does not radiate a near field that fully complies with specifications at 28 GHz, Figure 9b shows that most of the coverage zone does comply, and only at one edge is the ripple higher than 2 dB, demonstrating the successful optimization carried out by the POS. However, due to the intrinsic narrow bandwidth behaviour of the reflectarray, at other frequencies, it does not comply.