## 2 Lens Antenna: Reference and Reduced Height Designs
### Reduced Height Lens Design with a Single 3D Printing Material (Model 3)  
In this section, we show that different effective dielectric constant values from the same printing material can be achieved by perforating holes in the printed body and applying the technique to the lens design of model 2. Although the dielectric lens of model 2 shows a radiation performance close to the reference design of model 1 with a 15% reduced height, it consists of two dielectric constant values. If two other materials are used during actual production with a 3D printer, the fabrication complexity should increase, and the expected performance would change due to the non-negligible dielectric effect of an adhesive attaching the other printed bodies. The overall fabrication time must also increase. We use a ring resonator method [37; 38; 39; 40; 41; 42] when adjusting the dielectric constant of bulk printed material, as shown in Figure 4. When compared with the frequency response of a reference value of 2.5 (the dielectric constant of printed resin) for the dielectric cylinder (Figure 4a), the effective dielectric constant of a material made of the same material but with holes inside should change [37; 43]. The dielectric constant of the material under test can be inferred in this way. From the black dash-dot line in Figure 4d, we can see that the ring resonator resonates at 5.6 GHz when the printed cylindrical resin with a dielectric constant of 2.5 is placed on top of the ring resonator. The cylinder is 30 mm in diameter and 15 mm in height, large enough to cover the ring resonator. We also see that the resonator resonates at 5.74 GHz (as shown in Figure 4, marked by a blue dashed line), when a cylinder with a dielectric constant of 2 in Figure 4b is placed. Based on this, a change in the resonant  
Figure 3: Parametric study of the dielectric constant (\(\varepsilon_{r1}\)) of the light green section. (**a**) Reflection coefficient. (**b**) \(\theta\)-sweep radiation pattern (\(\phi=0^{\circ}\)). (**c**) Front-to-back ratio. frequency can be checked when a perforated sample is placed on top of the ring resonator, as shown in Figure 4c. If its resonance frequency is found to be close to the sample dielectric constant of 2, it can be said that the structure is designed with an appropriate number of holes and size to have a dielectric constant close to 2. In Figure 4d,e, we conduct a parametric study on the transmission coefficient according to the size and number of the holes. The cubical hole dimension is depicted in Figure 4c. In Figure 4d, the transmission coefficients according to the hole size are given with the fixed number of holes of 208. It is clearly observed that the resonance of the transmission coefficient is shifted upward with a larger \(R\) of the holes, and it is closest to the reference when \(R\) is 2 mm. In Figure 4e, the transmission coefficient with the number of the holes is plotted, with \(R\) = 2 mm. We can see that the graph approaches the reference when the number of holes is 208. In other words, a larger hole will reduce the portion filled with resin and the effective dielectric constant will be reduced. By the same token, a smaller hole will bring the effective dielectric constant closer to 2.5. As shown by the red solid lines in Figure 4d,e, the resonant frequency is well matched with the case of bulk material with a dielectric constant of 2 (see the blue dashed line), when the cubical hole size is 2 \(\times\) 2 \(\times\) 2 [mm\({}^{3}\)] with a total number of 208 holes that are made down into the center of the cylinder. Before applying the configuration of Figure 4c to the lens design, we verify the computed expectations as shown in Figure 4 experimentally. The cylindrical sample is printed with the SLA 3D printer Formlabs Inc., Form 3, as shown in the photos in Figure 5a. The material properties are measured using the dielectric constant and loss tangent measurement system from KEYCOM [44].