## 3 Application to Reflectarray for Compact Antenna Test Range
### Antenna Definition, Near-Field Specifications and Unit Cell Characterization  
A square reflectarray comprised of \(N=1936\) elements distributed in a regular grid of \(44\times 44\) unit cells is chosen. The periodicity is \(p_{x}=p_{y}=4.29\,\mathrm{mm}\) in both directions, which is approximately \(0.4\lambda\) at \(28\,\mathrm{GHz}\). The phase center of the feed is placed at \(\vec{r}_{f}=(-79.3,0,200)\,\mathrm{mm}\) with regard to the reflectarray center (see Figure 1). The feed generates an illumination taper that varies between \(-15.4\,\mathrm{dB}\) at \(27\,\mathrm{GHz}\) and \(-16.3\,\mathrm{dB}\) at \(29\,\mathrm{GHz}\). The goal is to design a reflectarray that acts as a probe in a CATR system, where an antenna or device under test is placed in the region where a plane wave is generated in front of the antenna. The reflectarray feed would emit a wave, which is transformed into a plane wave by the reflectarray and is then collected by the device being measured. Thus, regarding specifications, the goal is to shape the near field in a plane in front of the reflectarray in such a way that it complies with a maximum magnitude ripple of \(1\,\mathrm{dB}\) and a maximum phase ripple of \(10^{\circ}\). To this end, these FoM are computed in a tilted plane (see Figure 3) at \(z=0.5\,\mathrm{m}\) (in the ACS), whose rotation is defined by \(\theta=20^{\circ}\), \(\varphi=0^{\circ}\), \(\psi=0^{\circ}\) (see Appendix in [24]). The region is subsequently divided into three subregions: \(\Omega_{1}\) where the FoM are calculated, \(\Omega_{2}\) which defines a transition, and \(\Omega_{3}\) which comprises the rest of the plane. Although regions \(\Omega_{2}\) and \(\Omega_{3}\) are not used for this particular optimization, they will become relevant when comparing the new proposed technique with others described in the literature in Section 3.4. In addition, regions \(\Omega_{1}\) and \(\Omega_{2}\) are circles with diameter \(d_{1}=100\,\mathrm{mm}\) and \(d_{2}=200\,\mathrm{mm}\), respectively. The chosen unit cell is depicted in Figure 4a. It consists of two layers of parallel and coplanar dipoles, arranged such that each set of four dipoles control the phase-shift for one linear polarization. The substrate has a thickness of \(h_{A}=h_{B}=0.787\,\mathrm{mm}\), a relative permittivity of \(\varepsilon_{A}=\varepsilon_{B}=2.33\) and a loss tangent of \(\tan\delta_{A}=\tan\delta_{B}=0.0013\). The separation center-to-center between parallel dipoles is \(0.9\,\mathrm{mm}\) and the dipole width is \(0.4\,\mathrm{mm}\). This unit cell provides up to eight DoF for optimization purposes (the dipole lengths). However, to analyse the electromagnetic response of the unit cell and to obtain  
Figure 3: Schematic representation of the antenna optics. The NF plane is tilted \(\theta=20^{\circ}\) with regard to the reflectarray and is divided into three regions. The FoM are defined in region \(\Omega_{1}\), while region \(\Omega_{2}\) defines a transition and \(\Omega_{3}\) comprises the rest of the plane. the initial layout (see Section 3.2), two variables are defined, \(T_{x}\) and \(T_{y}\), that are related to the dipole lengths as follows:  
\[ \begin{split} L_{a_{1}}&=0.5T_{y}\ ;\ L_{a_{2}}=T_{y}\ ;\ L_{a_{3}}=0.5T_{y}\ ;\ L_{a_{4}}=T_{x}\\ L_{b_{1}}&=0.5T_{x}\ ;\ L_{b_{2}}=0.9T_{x}\ ;\ L_{b_{3}} =0.5T_{x}\ ;\ L_{b_{4}}=0.93T_{y}.\end{split}  \tag{12}\]  
These scaling factors were determined after carrying out a parametric study to obtain a smooth variation of the phase-shift for \(T_{x}=T_{y}\) at several frequencies and angles of incidence. Figure 4b shows the phase-shift at 28 GHz for several angles of incidence. As can be seen, the electromagnetic response of the unit cell presents good angular stability while providing a phase-shift larger than 360\({}^{\circ}\), making it suitable for reflectarray design and optimization [35].