# Millimeter-Wave High-Gain Dual-Polarized Flat Luneburg Lens Antenna with Reflection Cancellation
## 2 Performance Comparison between Conventional and Flat Luneburg Lens  
The design of an FLL can follow the transformation optics as described in [20], where the permittivity and permeability tensors, i.e., \(\overset{=}{\varepsilon}^{\prime}\) and \(\overset{=}{\mu}^{\prime}\), are firstly computed following the transformation of structures, i.e., transforming from a spherical Luneburg lens into a flat one, as shown in Figure 2. Note that the original spherical LL shown in Figure 2a has a radius as \(R\), and its corresponding cylindrical FLL shown in Figure 2b has the same radius as \(R\) and a thickness as \(2\times b\). Hence, the constitution parameters for the FLL can be calculated as  
\[\overset{=}{\varepsilon}^{\prime}=\varepsilon\frac{\overset{=}{\Lambda}^{ \Lambda}\overset{=}{\Lambda}^{T}}{\left|\overset{=}{\Lambda}\right|} \tag{1}\]  
\[\overset{=}{\mu}^{\prime}=\mu\frac{\overset{=}{\Lambda}^{=}\overset{=}{ \Lambda}^{T}}{\left|\overset{=}{\Lambda}\right|} \tag{2}\]  
where \(\Lambda\) is a Jacobian matrix defined by the transformation in a cylindrical coordinate (\(\rho\), \(\varphi\), \(z\)) to match up the radially rotational symmetry of either a conventional LL or an FLL with its axis along \(z\)-direction, as  
\[\overset{=}{\Lambda}= \begin{bmatrix}1&0&0\\ 0&1&0\\ \frac{2\rho}{R^{2}-\rho^{2}}&0&\frac{b}{\sqrt{R^{2}-\rho^{2}}}\end{bmatrix}  \tag{3}\]  
Hence, both \(\overset{=}{\mu}^{\prime}\) and \(\overset{=}{\mu}^{\prime}\) are nondiagonal matrixes, which indicates that anisotropic materials should be adopted for the transformed FLL. This is relatively complex for engineering implementation. Hence, an approximation is applied to convert the obtained anisotropic FLL into an isotropic one to simplify the implementation, by both substituting the permittivity tensor for its component \(\varepsilon_{\rho\rho}\) and replacing the permeability tensor with \(\mu_{r}=1\). In this case, the FLL can be implemented by using isotropic materials with gradient index distribution. As pointed out in [20; 22], the approximation would increase the focal length and deteriorate lens scanning performance. To further understand the degeneration of gain and bandwidth performance, a comparison was made between the conventional LL and its transformed FLL at the _Ka_-band using a simulation. The radius of the Luneburg sphere was set as \(R=15\) mm, and it was constructed with 12 equal-thickness-layer concentric spherical shells, as shown in Figure 2a. The corresponding flat lens has the same radius and a thickness as \(2\times b=5\) mm, and it was decomposed into 12 equal-thickness layers along either the axis (longitudinal direction) or radius of the cylinder, as shown in Figure 2b. A compression rate of \(R/b=6\) was adopted for the FLL design, and the dominate range of the transformed permittivity was enlarged from the original [1,2] to [6,12], multiplied by the compression ratio. Each layer of both two lenses were modeled with perfect dielectrics whose permittivities are specified with the desired value, and the dielectric loss was omitted during simulation. Both the two LLs were excited with a WR28 open-ended waveguide over 25-35 GHz, and Figure 3 shows the simulated \(S_{11}\) and realized gain of two lenses. It was observed that the FLL exhibits a narrower bandwidth in terms of both \(S_{11}<-10\) dB and gain flatness, although at the two resonance frequencies of 26.2 GHz and 31.8 GHz, a compatible performance was achieved for the FLL, comparing with that of the conventional LL. Besides the two frequencies, the gain curve of FLL was lower than that of the conventional LL, and the largest discrepancy was around 2.5 dB. Figure 4 shows the simulated radiation patterns of both lenses at 30 GHz, where the gain of the FLL is 18.1 dBi and that of the conventional LL is 18.6 dBi.