# Three-Dimensional Broadband and Isotropic Double-Mesh Twin-Wire Media for Meta-Lenses
## 1 Introduction  
Many emerging technologies, e.g., high-resolution radars [1], and the next generation of mobile communications [2], are expected to operate at higher operational frequencies (above \(15\,\mathrm{GHz}\)). At these frequencies, the propagation loss is high. To mitigate this loss, high gain antennas must be used. In some of these applications, an angular coverage is also requested; therefore, these high-gain antennas require beam steering where low scan losses are an asset.  
A directive beam can be produced with a lens antenna. This type of antenna employs a lens to shape the radiation from a feeding element. Typically, the feeding element is placed at the focal point/line of the lens, and the lens transforms the radiation from the feed into a planar wave. A wide angular beam coverage with low scan losses can be obtained using rotationally symmetric lenses with multiple feeds placed along the focal line [3]. Due to the simple feed system, lens antennas can be cost-effective at high frequency [4, 5]. For example, in [1], a homogeneous spherical lens antenna with low scan losses in a \(180^{\circ}\) range is proposed. Similarly, in [6], a wide scanning homogeneous cylindrical lens antenna is designed. While a wide scan range is obtained in these homogeneous lenses, they suffer from spherical aberrations leading to low aperture efficiency and high side lobes. To overcome these limitations, a graded index lens can be used.  
In a graded index lens, the refractive index varies as a function of space. Traditionally, graded index lenses were difficult to manufacture. Three-dimensional (3D) graded index lenses were manufactured in concentric hemispherical layers, which is inconvenient and costly [7, 8]. Recently, it has been demonstrated that 3D graded index lenses can be manufactured using additive manufacturing (i.e., 3D printing) by spatially varying the densityof the deposited dielectric material [9; 10; 11; 12; 13; 14]. However, the use of dielectric introduces losses that are prohibitive at high frequency. A mostly metallic 3D-graded index lens is designed in [15]. However, the designed lens exhibits strong anisotropy, which degrades the scanning performance of the lens. Alternatively, two-dimensional (2D)-graded index lenses can be realized using concentric dielectric cylinders [16; 17] or mimicked with quasi-periodic structures [18; 19; 20; 21; 22; 23] or geodesic surfaces [5; 24; 25; 26]. However, the scanning is reduced to one dimension in the 2D lenses. Furthermore, the aperture of the 2D lenses is limited in one direction, which makes it difficult to reach gain values above 30 dBi.  
Although fully-metallic 3D printers are still expensive, researchers are actively working to make this technology cost effective [27; 28]. A cheaper solution is to produce fully metallic structures with metallic coating [29; 30; 31]. With these techniques, fully metallic complex 3D structures can be manufactured, which allows for the design of cost-effective metallic 3D-graded index lenses. The wire mesh is one such complex structure that has received attention by researchers due to its exotic electromagnetic behavior [32; 33; 34]. A single-wire mesh structure has all its metallic wires connected, which results in the structure having a cut-off frequency. The propagating modes in this structure are highly dispersive, and it is therefore not suitable for lens design. On the other hand, a double-wire mesh structure has two independent wire meshes and can support wave propagation without cut-off frequency with moderate dispersion [34].  
In this paper, we investigate the double-mesh twin-wire structure. We demonstrate that the dispersion can be reduced by off-shifting the two wire meshes with half the period in three Cartesian directions, similar to what has been observed in glide-symmetric structures [35; 36; 37]. We apply this double-mesh twin-wire structure in the design of a fully metallic graded index Luneburg lens.