## 3 Near Field Models of Reflectarray and Transmitarray Antennas
### Phase-Only Synthesis at 28 GHz  
Figure 7 shows the proposed wideband near field optimization strategy for spatially fed array antennas. It is divided into several stages in order to facilitate convergence towards a suitable solution. After defining the antenna and near field specifications, a phase-only synthesis is carried out at central frequency. Afterwards, the geometrical features of the unit cell are tuned so each element produces the desired phase-shift obtained from the POS. After this point, the near field radiated by the antenna closely complies with the specifications at central frequency. However, due to the narrow bandwidth nature of planar spatially fed arrays, it does not comply at other frequencies. Thus, this initial layout is used in a multi-stage direct layout optimization (DLO) at several frequencies, finally obtaining an optimized wideband layout. To obtain an array working in dual-linear polarization, the POS is carried out independently for both polarizations, while the DLO is performed considering specifications in both polarizations simultaneously. Figure 6: (a) Sketch of the employed unit cell, consisting of two stacked rectangular patches backed by a ground plane. The periodicity is \(p_{x}=p_{y}=5.35\,\mathrm{mm}\), and both layers use the same substrate from Rogers Co., DiClad 870 with thickness of \(30\,\mathrm{mil}\); (b) phase response of the unit cell for normal and oblique incidence, showing that the unit cell response is stable with the angle of incidence; (c) phase response of the unit cell for three different frequencies for oblique incidence with \((\theta=40^{\circ},\varphi=30^{\circ})\). For the POS, the generalized intersection approach (GIA) [33] will be employed. The POS assumes an ideal phase-shifter simplification in matrix \(M_{i}\), in which there are no losses (\(|\rho_{xx,i}|=|\rho_{yy,i}|=1\) for reflectarrays, or \(|\tau_{xx,i}|=|\tau_{yy,i}|=1\) for transmitarrays) and no cross-polarization (\(\rho_{xy,i}=\rho_{yx,i}=0\) for reflectarrays, or \(\tau_{xy,i}=\tau_{yx,i}=0\) for transmitarrays). In this way, only the phases of the direct coefficients are considered as optimization variables. Once the POS has converged to a near field that complies (or it is close to compliance) with the imposed specifications at central frequency, the sizes of the rectangular patches of the unit cell are adjusted such that the unit cell provides the required phase-shift that was obtained in the POS. This procedure can be summarized as follows. First, and for each reflectarray element, two phase shift tables are generated independently for each polarization. Then, the dimensions of the rectangular patches are adjusted using a linear equation. Finally, a zero-finding routine is employed to adjust both dimensions at the same time. During this process, a fixed relative size between the stacked patches is kept, \(T_{x_{1}}=0.77T_{x_{2}}\) and \(T_{y_{1}}=0.77T_{y_{2}}\). Further details on the GIA applied to POS may be found in [21; 33]. Since the employed algorithm is a local optimizer, the starting point of the optimization is very important. Thus, the starting phase-shift distribution will be such that it provides a focused near field at the center of the coverage zone. To that end, the conjugate-phase approach [34] is employed, which establishes that the phase of the field at the aperture should be:  
\[\angle\vec{E}_{\text{ap},i}=k_{0}\,|\vec{r}-\vec{r}_{i}^{\,\prime}|, \tag{35}\]  
for \(i=1,\ldots,N\). On the other hand, the phase of the field at the aperture is also equal to the phase introduced by the unit cell plus the phase of the incident field from the feed [32]:  
\[\angle\vec{E}_{\text{ap},i}=-k_{0}d_{i}+\angle\rho, \tag{36}\]  
Figure 7: Proposed wideband optimization strategy for spatially fed arrays using the generalized intersection approach (GIA). First, a phase-only synthesis (POS) at central frequency is carried out in order to obtain a phase-shift distribution from which the initial layout is obtained.