## 2 Lens Antenna: Reference and Reduced Height Designs
### Reduction of Lens Height (Model 2)  
Intensive studies have been carried out to lower the height of the lens [28; 29]. In [28], the height was reduced without degrading the antenna performance through the air cavity in the radiator part while maintaining the thickness of the lens wall for structural stability. In [29], the lens height was reduced by setting the gap from the antenna source to the lens and the air slab by considering the phase velocity of the radiated waves passing through each area of the lens sections. In this work, we design the dielectric constant of the first cylinder (Section 1) in contact with the radiating patch, as shown in Figure 2c in light green, which is lower than 2.5, with  
\[v_{p}=\frac{1}{\sqrt{\mu\epsilon}}\, \tag{1}\]  
such that the phase velocity is increased. As the phase velocity increases, the height of the dielectric lens can be reduced because a wavefront can reach the same position even through a shorter path, or via a reduced height. Thus, the reduced height cylinder yields the same performance as the reference lens [36]. The skirt part, Section 2, marked in light green, is also farther from the excited patch than the gray part of Section 2, as shown in Figure 2c. Therefore, as shown in Figure 2b, when the dielectric constant of the skirt portion is lowered, the height of the lens can be reduced. The overall height of the lens can be reduced this way by assigning a lower dielectric constant of the lens sections, as they are farther away from the excitation [19; 28; 29]. We carry out parametric studies on the dielectric constant (\(\epsilon_{r1}\)) of the light green section of model 2. The dielectric constant of  
Figure 1: Reference lens antenna simulation. (**a**) Reference patch antenna. (**b**) Reference lens antenna (model 1). (**c**) Reflection coefficient. (**d**) \(\theta\)-sweep radiation pattern (\(\phi=0^{\circ}\)). The design parameters are \(W=30\), \(L=30\), \(D=18.6\), \(W_{1}=70\), \(H_{1}=25\), \(H_{2}=22\), and \(H_{3}=35\). All units are in millimeters.  
the other parts is fixed as \(\varepsilon_{r2}=2.5\). The results are shown in Figure 3. As can be seen in Figure 3a, the resonant frequency is naturally shifted downward with a higher value of \(\varepsilon_{r1}\). The radiation pattern barely changes at the resonant frequency as shown by Figure 3b. However, as can be seen from Figure 3c, the front-to-back ratio is higher by 1 to 3 dB at each resonance when \(\varepsilon_{r1}\) is 2. Therefore, the dielectric constant of the light green section in model 2 is chosen as 2, with an operating frequency of 5.8 GHz.  
Figure 2: Dielectric lens antenna (model 2) simulation. (**a**) Simulation model. (**b**) Perspective view of the lens (disintegrated). (**c**) Side view of the lens (disintegrated). (**d**) Reflection coefficient. (**e**) \(\theta\)-sweep radiation pattern (\(\phi=0^{\circ}\)). The design parameters are \(H_{4}\) = 20, and \(H_{5}\) = 15. All units are in millimeters.  
Using this design strategy, the final lens design of model 2 is \(H_{4}\) = 20 mm, \(H_{5}\) = 15 mm, and \(e_{r1}\) = 2, and \(e_{r2}\) = 2.5. The total height of the lens is reduced to 70 mm, 1.353 \(\lambda\) at the same resonance frequency of 5.8 GHz (see Figure 2d). The simulated radiation pattern is very similar to that of the reference design, as shown in Figure 2e. The directivity of model 2 is 14.5 dBi, which is about the same directivity of the reference lens antenna.