## 3 Application to Reflectarray for Compact Antenna Test Range
### Initial Layout  
The multi-frequency DLO requires an initial reflectarray layout that must be generated prior to the use of the technique. Here, the layout is obtained following a multi-step procedure [44] in which, first, a phase-only synthesis (POS) at 28 GHz is carried out to obtain a phase-shift distribution such that it radiates the desired NF at the frequency of synthesis. To this end, the generalized intersection approach (GIA) [45] particularized for a reflectarray NF model [24] is employed. For the POS, the initial phase-shift distribution shown in Figure 5a was used. It corresponds to a far field-focused reflectarray pointing at \(\theta=20^{\circ}\) (see Figure 3). After the POS with the GIA, the phase-shift distribution of Figure 5b was obtained.  
Figure 4: **(a)** Sketch of the employed unit cell, consisting of two layers of parallel and coplanar dipoles, arranged in two sets of parallel dipoles to control both linear polarizations independently. **(b)** Phase response of the unit cell for normal and oblique incidence at 28 GHz, 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 \((\theta=40^{\circ},\varphi=30^{\circ})\).  
Then, the lengths of the dipoles are adjusted element-by-element so that they provide the required phase-shift obtained after the POS. For this process, variables \(T_{x}\) and \(T_{y}\) defined in (12) are employed, such that \(T_{x}\) controls the phase-shift for linear polarization \(X\) and \(T_{y}\) for polarization \(Y\). Figure 6 shows a comparison of the NF before and after the POS at 28 GHz for polarization \(X\). As can be seen, even though the field does not fully comply with the imposed specifications, it greatly improves when compared with the starting point before the POS. Indeed, the magnitude ripple is reduced from a maximum of 7.6 dB before the POS to a maximum of 2.2 dB after the POS. For the phase, the maximum ripple was reduced from 26.4\({}^{\circ}\) to 13.6\({}^{\circ}\).  
The POS enables obtaining a layout that works relatively well at the frequency of design, 28 GHz in this case. However, performance rapidly deteriorates as the frequency is shifted. This can be seen in Table 1, which gathers information concerning the compliance according to the imposed specifications (maximum ripple in magnitude of 1 dB and in phase of 10\({}^{\circ}\)), as well as the maximum ripple in magnitude and phase. As can be seen, even though the magnitude ripple decreases slightly at 27 GHz compared to the central frequency, the phase ripple increases considerably. In addition, both magnitude and phase ripples  
Figure 5: (a) Initial phase-shift distribution of a far-field-focused reflectarray at 28 GHz. (b) Optimized phase-shift distribution after the POS with the GIA at 28 GHz.  
Figure 6: Near-field comparison before and after the POS with the GIA for the (a) magnitude and (b) phase for the cut \(y=0\) mm at 28 GHz.  
deteriorate considerably at the upper frequency of 29 GHz, indicating that a wideband multi-frequency optimization is needed.