## General Near-Field Synthesis Based on the Optimization of Relevant Figures of Merit  
### Calculation of the Near Field and Figures of Merit  
Let us consider the scenario shown in Figure 1. It consists of a reflectarray comprised of a total of \(N\) elements with regular periodicity \(p_{x}\times p_{y}\), illuminated by a feed whose phase center is placed at \(\vec{r}_{f}\) with regard to the array coordinate system (ACS) defined by \((x_{a},y_{a},z_{a})\). The reflectarray generates a near field in a volume in front of the antenna. Each point in space where the near field is computed is given by \(\vec{r}=(x,y,z)\). In addition, the near field is referenced, in general, to its own coordinate system (NFCS, or near-field coordinate system) given by \((x_{nf},y_{nf},z_{nf})\) which in general is translated and rotated with respect to the ACS. For a given point in space, the near field radiated by the reflectarray may be obtained as the contribution of the far field radiated by each element:  
\[\vec{E}_{\rm NF}(\vec{r})=E_{{\rm NF},x}\hat{x}_{nf}+E_{{\rm NF},y}\hat{y}_{nf }+E_{{\rm NF},z}\hat{z}_{nf}=T\sum_{n=1}^{N}\vec{E}_{{\rm FF},n}(\vec{r},\vec{ r}_{n}^{\prime}), \tag{1}\]  
where \(T\) is the matrix of change of coordinates from the ACS to the NFCS, which is defined by three angles \((\theta,\varphi,\psi)\); \(\vec{E}_{{\rm FF},n}(\vec{r},\vec{r}_{n}^{\prime})\) is the far field generated by the \(n\)-th element according to [PERSON]'s principle of equivalence [36]; and \(\vec{r}_{n}^{\prime}=(x_{n},y_{n})\) are the coordinates of the \(n\)-th element in the reflectarray coordinate system (see Figure 1). Further details on the near-field model, including the definition of matrix \(T\), may be consulted in [24]. From the near field \(\vec{E}_{\rm NF}(\vec{r})\) obtained in a volume \(\Omega\) in space, the next step is to obtain the relevant FoM. For this purpose, we consider a number of disjoint regions \(\Omega_{m}\)  
Figure 1: Sketch of the reflectarray configuration for general near-field synthesis based on the optimization of the near-field figures of merit. \(m=1,2,\ldots,M\) where those FoM may be calculated. These regions fulfil the following properties:  
\[\Omega_{i}\cap\Omega_{j}=\emptyset;\quad i,j=1,2,\ldots,M;\quad i\
eq j, \tag{2a}\] \[\Omega=\bigcup_{m=1}^{M}\Omega_{m}, \tag{2b}\]  
where \(\emptyset\) is the empty set. In short, these two properties mean that the regions \(\Omega_{m}\) fill all space \(\Omega\) without overlapping. Once the regions \(\Omega_{m}\) have been established, we can define the FoM on them. In general, the FoM will be a function \(f\) of the near field and, here, we employ the following notation:  
\[F_{k,\Omega_{m}}=f\Big{\{}\vec{E}_{\text{NF}}(\vec{r})\Big{\}}, \tag{3}\]  
where \(F_{k,\Omega_{m}}\), \(k=1,2,\ldots,K\) is the \(k\)-th FoM of a total of \(K\) FoM and it is defined in region \(\Omega_{m}\). Since we consider a general NF synthesis with magnitude and/or phase specifications, the FoM can be related to either or both. The definition of each particular FoM will depend on the goal of the optimization procedure. For instance, one may want to maximize the value of the near field in a particular region, for instance, of the \(\hat{x}_{nf}\) component for linear polarization \(X\). Since, for this case, the limiting FoM is the minimum value of the NF, the relevant FoM is:  
\[F_{k,\Omega_{m}}=\min_{\Omega_{m}}\{|E_{\text{NF},x}|\}, \tag{4}\]  
where \(\min_{\Omega_{m}}\) gives the minimum value in region \(\Omega_{m}\). In this way, one may maximize the minimum value of the field through the proper definition of the FoM. In the case of the present investigation, plane-wave generators usually define the QZ to fulfil specifications with a maximum ripple of phase of \(10^{\circ}\) and a maximum ripple of magnitude of \(1\) dB [27; 37].