## 4 All Dielectric Unit Cell
### Lens Design  
The design process of the lens is based on geometrical optics according to (2) and (3). Applying these equations to a transmitarray with the optics defined previously (\(r=-100\) mm) (central feed) and to focus on a spot at 150 mm (\(F_{TA-f_{0}}\)), the phase distribution of the elements (\(\varphi_{TA}(\rho)\)) along the transmitarray surface of Figure 7a is obtained. The equivalent \(\epsilon_{r,eff}\) of each cell is shown in Figure 7b. Then, to physically implement this phase shift, the cells have to be accurately designed. In the design process of the cells, the dimensions of the air gaps, \(W\) and \(L\), are adjusted to introduce a phase shift \(\varphi(\rho)\) similar to \(\varphi_{TA}(\rho)\). The adjustment is made cell-by-cell using the phase response of Figure 6b. The amplitude response is used to discard the cells with high transmission losses. The final layout is composed by the cell that minimizes the differences between its phase shift and the theoretical response of Figure 7.  
The entire antenna was analyzed in CST Microwave Studio in a full-wave simulation. The transmitarray is illuminated with a pyramidal horn antenna of 15 dBi gain and placed at several positions, from \(\theta=0^{\circ}\) to \(\theta=\pm 20^{\circ}\). The electric field for the \(xoz\) plane and each  
Figure 6: Cell response as a function of the air gap dimension at 28 GHz and normal incidence. (**a**) Phase (deg); (**b**) amplitude (dB). Figure 7: (**a**) Transmission phase shift (deg) of the transmitarray elements along the surface to focus the near-field. (**b**) Effective dielectric constant of each cell to produce the desired phase shift. position is shown in Figure 8, and the near-field spots are simultaneously compared in Figure 9.  
Both figures show that the transmitarray radiates a perfectly near-field spot for the central position. The transmitarray is designed to focus at \(z=150\) mm; even so, the maximum of the field is compressed to 125 mm. When using the conjugate phase approach, the phase of the antenna elements is in phase at the focal point, but the maximum is located at a position closer to the antenna [25]. Regarding the other feed positions, the transmitarray can generate as many spots as the feeds used, noting that there is a physical limitation due to the antenna aperture size used as the feed. Depending on the linear polarization used, this horn antenna can be placed with an angular distribution of \(10^{\circ}\) or \(15^{\circ}\). The locations of the spots are similar to the angular distribution of the feeds, and the maximum of the spots is barely compressed for angles closer than \(15^{\circ}\). In this approach, the horn antennas are supposed to work in close, but different carriers, having a high isolation between horns. However, the isolation was also computed assuming the same frequency carrier for all the feeds, which can be considered the worst case. Figure 10 shows the setup with CST Microwave Studio to analyze the S parameters  
Figure 8: Normalized simulated amplitude electric field on the plane \(xoz\) for several positions of the feed. (**a**) \(\theta=0^{\circ}\), (**b**) \(\theta=5^{\circ}\), (**c**) \(\theta=10^{\circ}\), (**d**) \(\theta=15^{\circ}\), and (**e**) \(\theta=20^{\circ}\)  
Figure 9: Normalized near-field focused spot comparison for several feed positions, \(\theta\in[0,20]^{\circ}\). of the horn antennas, considering the lens and three horns placed at \(\theta=0^{\circ}\) (Port 1), \(\theta=15^{\circ}\) (Port 2), and \(\theta=-15^{\circ}\) (Port 3). The magnitude of \(s_{11}\), \(s_{22}\), and \(s_{33}\) shows that the three horn antennas are matched since this parameter is below \(-15\) dB from 26 to 30 GHz (see Figure 11a). Additionally, the magnitudes of \(s_{21}\), \(s_{31}\) (\(s_{21}=s_{31}\)), and \(s_{32}\) are shown in Figure 11b, and the level is almost below \(-20\) dB in the entire band. These results show a high isolation between adjacent horn antennas working in the same frequency carrier.