# Millimeter-Wave High-Gain Dual-Polarized Flat Luneburg Lens Antenna with Reflection Cancellation
## 3 Flat Luneburg Design and Implementation
### Reflection Cancellation Method  
In this design, an air interval is introduced between adjacent layers in the FLL, which has three functions: (1) it can reduce the strong coupling between adjacent patches, which cannot be modeled during the retrieval of effective permittivity with only one layer unit cell, as shown in Figure 5a; (2) by tuning the air gap, the operating frequency and bandwidth can be adjusted to one that functions in the design of the frequency-selective surface (FSS) [23]; (3) the air gap can provide a reflection cancellation phenomenon, which can enhance the performance of the FLL in terms of gain and bandwidth.  
To illustrate the operating scheme of the reflection cancellation, an illustrated diagram is shown in Figure 7. Suppose that the two layers are separated with a distance equal to a quarter of the wavelength, i.e., \(\lambda/4\), the reflections occurring at the two interfaces on the same side of the two layers would be superposed with a phase difference of \(\pi\), resulting a reflection cancellation. Therefore, the air interval between adjacent layers provides additional design freedom to optimize the performance of the FLL. Note that the two-layer model shown in Figure 7 is frequency-dependent, but the real case in an FLL with multilayers is more complicated than in the two-layer model. In the FLL, there are multiple reflections occurring at each layer and mutual couplings among layers, all of which contribute to wideband performance. Hence, a parameter study was carried out by varying the thickness of the air interval, noted as the _airgap_, to investigate its influence on \(S_{11}\), gain, and FTBR simultaneously, as shown in Figure 8. Note that the air interval between any two adjacent layers was set identically to simplify both the optimization and implementation. The parameter study does not only result in a simple way of changing the air interval, as the FLL should also be re-designed with the revised air interval each time. This is because the variation in _airgap_ will change the total thickness of the FLL, resulting in a revised comparison \(R/b\) ratio. Meanwhile, the location of each layer varies with the changing in the air interval as well. Thus, the permittivity should be recomputed and discretized for the design of the unit cell at any specified _airgap_. Thanks to a script-based modelling method in Ansys HFSS, the parameter study can be executed in an automatic way.  
Figure 6: Dual-polarized microstrip patch antenna: (**a**) perspective view, (**b**) side view, (**c**) patch and feeding microstrip line. (\(l_{s}\) = 30, \(w_{1}\) = 0.7, \(l_{1}\) = 9, \(w_{2}\) = 0.48, \(l_{2}\) = 2, \(w_{3}\) = 0.3, \(l_{3}\) = 1, \(w_{\mathrm{p1}}\) = \(w_{\mathrm{p2}}\) = 2.7, \(h_{a}\) = 0.6, all in millimeters).  
As can be seen in Figure 8a, with the increasing in _airgap_, the resonances on the \(S_{11}\) curves shifted to low frequency as assumed. A good bandwidth of 26.1-31.6 GHz in terms of \(S_{11}<-10\) dB was achieved when the _airgap_ was 0.3 mm, and the gain curve exhibited good flatness as well. The FTBR exhibited similar trends in that it achieved its largest value when the _airgap_ is 0.3 mm, as shown in Figure 8b. Hence, _airgap_ = 0.3 mm was chosen for the design of the FLL, and it derived a final designed FLL with a total thickness of 5.75 mm. Though _airgap_ = 0.01 mm corresponds to a wide band in terms of \(S_{11}<-10\) dB, the gain has a large variation over the bandwidth, which mainly attributes to the strong reflection of the FLL.