# Three-Dimensional Broadband and Isotropic Double-Mesh Twin-Wire Media for Meta-Lenses
## 3 Design of Luneburg Lens  
Now, we design a 3D Luneburg lens operating up to \(30\,\mathrm{GHz}\) based on the proposed unit cell with tailored sinusoidal amplitudes. According to the parametric studies in Figure 3a, the period is chosen to be \(p=$1\,\mathrm{mm}$\) for a good homogenization below \(30\,\mathrm{GHz}\). The lens radius \(R\) is set to be \(R=$20.5\,\mathrm{mm}$\), containing \(41\) unit cells in its diameter. This configuration produces a fine Cartesian discretization of the ideal index map, as depicted in Figure 4a. To implement this discretized map, the sinusoidal amplitude of the wires in the unit cell is gradually reduced with an increasing distance \(r\) from the lens center, and it becomes zero, i.e., the wires are straight, in the lens outer surface. Since the curves of refractive indices in Figure 3c are almost non-dispersive, the dimensional synthesis is not subject to a strong frequency dependency. Here, it is performed at \(24\,\mathrm{GHz}\) based on the effective refractive index (normalized to the minimum value of \(n=$1.4646$\)) as a function of the amplitude \(A\), as provided in Figure 4b. In Figure 4c, we synthesize a dimensional map of the sinusoidal amplitude required at each spatial location in the lens. The full structure of the designed lens is visualized in Figure 5.  
Figure 3: Effective refractive index of the double-mesh twin-wire unit cell as a function of: (**a**) period \(p\); (**b**) diameter of wire \(d\); (**c**) amplitude of undulation \(A\). The solid lines represent the data for \(\overline{\Gamma}\overline{X}\), and the dashed lines for \(\overline{\Gamma}\overline{R}\). Unless otherwise specified, the dimensions are \(A=$0.2\,\mathrm{mm}$\), \(N=$1\,\mathrm{mm}$\), and \(d=$0.06\,\mathrm{mm}$\). Figure 4: **(a) Discretized refractive index of Lens. (b) Normalized effective refractive index as a function of \(A\) at 24 GHz. (c) Synthesized dimensional map of the meta-lens. **  
Figure 5: **(a) 3D synthesized meta-Luneburg lens (with zoom-in view of some unit cells). (b) Dispersion diagram of unit cell with different boundary conditions; the dimensions as denoted in Figure 1a are \(p=1\,\)mm, \(A=0.2p\), and \(d=0.06p\), while the solid lines represent the data for \(\overline{\Gamma\!X}\) and the dashed lines for \(\overline{\Gamma\!M}\). The unit cell is defined to be the one in Figure 1c off-shifted by half the period in the \(x/y/z\)-directions. (c) Equivalent 2D synthesized meta-Luneburg lens (with zoom-in view of port and its field distribution). **  
The design of the meta-lens is validated by full-wave simulations in CST Microwave Studio. We note that the propagation characteristics in the \(\Gamma\)XM-plane is very similar if the periodic boundary conditions along \(\hat{z}\) are replaced with perfect magnetic conductor (PMC) boundaries, as demonstrated in Figure 5b. Therefore, it suffices to analyze one slab of the lens with PMC boundaries on the top and bottom, as illustrated in Figure 5c. Additionally, the modal field in the twin-wire structure at the boundary of the lattice is TEM-like, similar to the field in a coaxial transmission line. This allows us to excite the lens with a simple feeding structure such as the coaxial port, as illustrated by the inset of Figure 5c. Note that, since the mode supported by the lens is confined between the metallic wires, it is inherently mismatched from the one in free space. For ease of demonstration, we embed the lens in an effectively homogeneous background composed by the straight wires with \(A=0\), i.e., the same as those in the lens exterior. In this way, we obtain the absolute value of electric field \(|E|\) contours in the cutting plane \(z=0\) as in $20.5\,\mathrm{mm}$0 Here, the lens produces a uniform wavefront and aperture illumination across a wide operating bandwidth from DC to 30 GHz, although only 12-24 GHz are explicitly exemplified for better visualization. This confirms the good isotropy and low dispersion of the proposed double-mesh wire structure studied in Section 2.