# Three-Dimensional Broadband and Isotropic Double-Mesh Twin-Wire Media for Meta-Lenses
## 2 Dispersion Analysis of the Double-Mesh Twin-Wire Structure  
To aid in the fundamental understanding of the structure behaviors, we demonstrate the evolution of the proposed double-mesh twin-wire unit cell in Figure 1 and study the corresponding dispersion characteristics of those evolutionary geometries by examining their isofrequency contours in Figure 2. As depicted in Figure 1a, the proposed structure is derived from a meandered version of the single-mesh connected wires [32] that can be seen as a 3D extension to the meandered crossed-microstrip-line-based 'fishnet' metasurface studied in [19]. To simplify the parameterization of the unit cell, the mean path of a meandered wire with diameter \(d\) is defined to be a sine shape \(S(\
u)=A\sin\left(N\
u\pi/p\right)\) (\(\
u\equiv x,y,z\)), with \(p\) and \(N\) being the period and the number of turns within one period. The same definition of geometric parameters also applies to the structures in Figure 1b,c. As revealed by the varied contours with different cutting planes in Figure 2a, the lack of spatial symmetry of this structure leads to a strong anisotropy that hinders its use for 3D lens designs. To enhance symmetry in all spatial directions, another set of wires is added to connect and pair with the original one in each axis, thus a twin-wire mesh is constructed, as illustrated in Figure 1b. Although an improvement of isotropy is evidenced in Figure 2b, the wave propagation inside this structure undergoes severe dispersion due to the presence of cut-off frequency that yields the hyperbolic contour lines. To overcome this problem, an additional copy of the twin-wire mesh is introduced and transposed by half the period in all three Cartesian directions, as shown in Figure 1c. Since the two meshes are disconnected, there is no cut-off frequency, leading to low dispersion and isotropy over a wide bandwidth, as indicated by the spherical contours and their equidistant progression in Figure 2c. Similar phenomena are also reported in [33; 34], where these disconnected wire meshes are framed into the so-called 'non-Maxwellian' media. As an alternative viewpoint, one may find an analogy to higher symmetries [38; 39; 40; 41], and in particular to glide [37] and 1D twist symmetries [42; 43]. When glide symmetries are adequately applied to metasurfaces, they increase the effective refractive index of the unit cells, reduce their dispersion, and increase their isotropy [35; 37]. These distinctive properties make 2D glide symmetric structures a suitable candidate for lens applications [20; 23; 44; 45], motivating us to further explore the proposed double-mesh wire medium for 3D meta-lenses. A graded-index lens tailors the wave propagation by spatially varying the (effective) refractive index inside the lens medium that provides a sufficiently large tuning range for the index. In a meta-lens, this is usually achieved by locally modifying the geometrical details of the unit cell comprising the medium. In Figure 3, we present a parametric study of the effective refractive index of the proposed unit cell. We investigate the impact of the three main geometrical dimensions \(A,d,p\). Figure 1: Evolution of double-mesh twin-wire unit cell: (**a**) single-wire unit cell; (**b**) twin-wire unit cell; (**c**) double-mesh twin-wire unit cell. The mean path of the meandered wire is described by \(S(v)=A\sin\left(Nv\pi/p\right)\left(v\equiv x,y,z\right)\). \(p\) is the period of the unit cell and \(N\) stands for the number of turns within one period (\(N=1\) in all present schematics). Figure 2: Dispersion analyses of: (**a**) single-wire unit cell; (**b**) twin-wire unit cell; (**c**) double-mesh twin-wire unit cell. The dimensions, as denoted in Figure 1, are \(A=0.1\,\mathrm{mm}\), \(N=1\), \(p=1\,\mathrm{mm}\), and \(d=0.1\,\mathrm{mm}\). In Figure 3a, the effective refractive index for different periods \(p\) of the double-wire mesh structure is presented.