# Affordable Thin Lens Using Single Polarized Disparate Filter Arrays for Beyond 5G toward 6G
## 2 Design of Single Polarized Thin Lens
### Lens Implementation Ustig Macro Design  
This section describes the procedure and results of designing the proposed lens by using the selected unit cells and the feed antenna, described in Sections 2.2 and 2.3, respectively. The design procedure is based on the general procedure described in [14; 15; 16; 21], but modified for the proposed single-polarized lens.  
1. The proposed lens is chosen to be located at a distance of 150 mm away from the \(4\times 4\) patch antenna array. The diameter of the selected aperture for the lens is chosen to be 150 mm. The value of \(f\)/\(D\) was chosen to be 1.0, where f is the distance between the feed antenna and the lens, and \(D\) is the diameter of the lens, to implement a compact and focused design. It should be noted that the target distance also determines the adjustable range of the phase shift.
2. The phase profile of the waves emitted from the \(4\times 4\) patch antenna array source is captured on the selected aperture at the distance of 150 mm. In this case, the phase profile is captured every 1.5 mm from the center of the lens considering the size of the unit cell, which is 1.5 mm.
3. The phase values needed for collimating radiated rays at all captured points are calculated and matched to the selected unit cells in Table 1. For the interval outside the range of tunable range from \(-280\) to \(0^{\circ}\), LP1 and BP5 are used as alternatives.
4. If the type of the unit cell at each point is the same as that at the next point, the area formed by these two points is grouped and classified as a zone.
5. It should be noted that variations in the ILs over the lens aperture might have the same effect as that of the non-uniform magnitude of feeds in a phased antenna array, leading to beam deformation related to beam width, side lobes, and so on [22; 23]. Table 2 lists the numbered zones, the number of selected unit cells in each zone, the required phase shift at the center of each zone, and the selected unit cells in each zone. In Table 2, it is assumed that the change in the phase shift due to the different incident angles of the unit cells constituting the proposed lens is not significant.  
Figure 10: Simulated radiation patterns of the \(4\times 4\) patch antenna array when input phase offsets are (a) \(0^{\circ}\), (b) \(45^{\circ}\), (c) \(90^{\circ}\), and (d) \(135^{\circ}\).  
Figure 12 shows that the designed lens and the \(4\times 4\) antenna array are separated by a selected distance \(f\). To achieve rapid full-wave simulations for the lens, a large-scale structure, unit cells are replaced by simple effective mediums. In Figure 13, the structure of the unit cell shown in Figure 12 is replaced by the effective mediums. 3D radiation pattern plots shown in Figure 14 are obtained by simulating the structure shown in Figure 13 using Ansys HFSS. Each beam pattern is plotted when the input phase offsets are set to be 0, 30, 60, and 90\({}^{\circ}\), before and after mounting the lens. A gain enhancement of approximately 16 dB is obtained using the proposed lens along the boresight. In the absence of the lens, each peak beam is observed at 0, \(-\)10, \(-\)20, and \(-\)30\({}^{\circ}\). The angles in which the peak beams are observed are changed to be 0, \(-\)1, \(-\)2, and \(-\)3\({}^{\circ}\) on adding a lens.  
Trade-offs between gain enhancement and beam steering that essentially occurs in convex lenses are well known [24; 25]. However, as a wireless backhaul system uses a beam steering coverage as small as a few degrees after antenna installation and maintenance, the presented coverage is allowable, as shown in Figure 15.