# Dielectric Slabs-Based Lens for Millimeter-Wave Beamforming
## 3 Discussion and Simulation Results  
Figure 1a shows the horn antenna with the designed lens. The antenna port is excited by a Gaussian pulse, polarized along the y-axis and propagating along the +z-axis. The numerical solution is performed for open boundaries of the electromagnetic problem (far-field problem) using the finite integration technique (FIT) solver in CST. The performance of the proposed lens is examined in four different design cases. Lenses 1 and 2 represent the proposed dielectric slabs-based lenses with different slab thicknesses. The effect of the air gaps or epoxy materials between the slabs is also investigated in lenses 3 and 4, respectively. Figure 4 shows the horn antennas' radiation pattern without and with the designed lens in Figure 3. The first curve represents the radiation pattern of a horn antenna without any lens. The second one (lens 1) is the radiation pattern of the reconfigurable lens with \(t_{\Delta}=0.2\) mm, and 55 slabs. The third one represents the radiation pattern of the reconfigurable lens (lens 2) with \(t_{\Delta}=1\) mm, and 11 slabs. The air gaps are investigated in lens 3, in which the slab thickness is 0.9 mm, air gap thickness is 0.1 mm, and 11 slabs. Lens 4 is similar to lens 3 except replacing air gaps with epoxy materials (\(\epsilon=4,\sigma=0.2\)\(S\)/m). Figure 4 shows that the co-polarized radiation patterns of all designed lenses are identical. The gain for each one is around 30 dBi at 28 GHz, and \(10^{\circ}\) HPBW. Air gab's effect is minimal and can be neglected, as shown in the dotted blue line. The effect of the epoxy materials appears only in increasing the side lobes; however, this effect is still small. The results prove that the small gap of air or epoxy does not degrade the performance of the designed lens. Increasing the number of slabs from 11 slabs with \(t_{\Delta}=1\) mm \(=\lambda/10\) to 55 slabs with \(t_{\Delta}=0.2\) mm \(=\lambda/100\) do not give any extra enhancements. Therefore, the recommended slab thickness is \(\lambda/10<t_{\Delta}<\lambda/4\) according to the available commercial dielectric thickness and the effective medium theory. In addition, the gain, HPBW, and sidelobe level can also contribute to the selection of the number of the used slabs. Figure 3: The graded effective refractive index of eleven dielectric slabs, with aperture area diameter, \(a=64\) mm, and slab thickness, \(t_{\Delta}=1\) mm. The reflection coefficient of the antenna port with and without the designed lens 2 is shown in Figure 5a. For both results, the \(S_{11}\) is below \(-10\) dB over all the investigated bands. Figure 5b shows the cross-polarization gain with and without the designed lens 2. There is evidence of enhancement in the cross-polarization level when lens 2 is used. Figure 6 shows the co-polarized radiation pattern of the horn antenna with the designed lens (lens 2), validated using two different full-wave solvers, which are the finite element method (FEM) and the finite integration technique (FIT) solvers in CST. The finite element method (FEM) uses swiping frequency excitation waves with a tetrahedral mesh of 234,024 cells. The FIT solver uses the gaussian excitation pulse with a hexahedral mesh of 851,250 cells. Figure 4: (**a**) The designed lens 2 with the horn antenna. (**b**) The co-polarized radiation pattern of the horn antenna with and without a dielectric slabs-based lens using the finite integration technique (FIT) solvers in CST. The aperture area of the designed lens is \(a=64.52\) mm, and the horn antenna is placed at the optimized focal point (\(f=50.62\) mm). Until now, we have only tested the radiation pattern parameters for one horn antenna. Therefore, 16 switchable rectangular horn antennas are introduced to examine the proposed lens for millimeter-wave beamforming for different positions of a single excited antenna. In each time, only one antenna element is on, and the other elements are off.