## 3 Results and Discussions
### Plasmonic Lens Design
It is worth noting that some optimized devices yield original unexpected asymmetric on-axis metalenses, even though they are under a normal incidence. Figure 6a,b presents the cartography of the normalized magnetic field's intensity in the plane \((X,O,Z)\) associated with the optimized geometry of Figure 6a. In Figure 6a, we report the values of the (nanorodes, nanoslits)-widths sequence \([e_{k}]_{k}\) with respect to their locations labelled by \(k\). No symmetry property can be extracted from these results, indicating that sophisticated energy exchange between the modes in the design area takes place and these interactions cannot be explained or/and modeled thanks to some trivial rules. The modes contained in the design area of the structure are not symmetric but their combination yields a symmetric spatial distribution of the electromagnetic far field. Without enforcing some symmetry properties, the optimization process freely evolves towards unexpected exotic structures. From the above study, we anticipate that a high-quality focusing spot can be reached by increasing the size and the nsnr-number \(N_{p}\) of the structure under optimization. However, numerically speaking, increasing the lens's geometric features is computationally expensive, since both the number of the design variables \(N_{p}\) and the number of basis functions required for the electromagnetic field numerical approximation increase also. To circumvent this drawback, we introduce a computationally less expensive approach termed the two-meta-cells-stitched approach. In this approach, a metalens constituted of \(N_{p}\)-nsnr elements is viewed as an arrangement of two interlocked subsections, \([A,D]\cup[B^{\prime},A^{\prime}]\) denoted meta-cells, as is shown in Figure 7. Only the meta-cell \([A,D]\) constituted of \(n_{p}\) nsnr-elements is simulated as a off-axis aperiodic metalens. The complementary \(n_{p}-2\) meta-cell \([B^{\prime},A^{\prime}]\) is built as the symmetric of the a \(n_{p}-2\) nsnr elements of a sub-meta-cell \([A,B]\) of the meta-cell \([A,D]\) (\([A,B]\cup[A,D]\)). Figure 6: Design of a plasmonic silver-flat-metalens of \(d=18.75\) μm-wide consisting of \(N_{p}=185\) silver-nanorides and air-gap focusing a normal incident TM-polarized plane wave at a wavelength \(\lambda=0.637\) μm at a distance of \(15\lambda\). (**a**) shows the values of the (nanorodes, nanoslits)-widths sequence \([e_{k}]_{k}\) with respect to their locations labelled by \(k\). (**b**) presents the cartography of the normalized magnetic field’s intensity in the plane \((X,O,Z)\). Numerical parameters: \(\lambda=0.637\) μm, \(N_{p}=52,e_{z}=400\) nm, TM polarization. (**a**) \(|H_{y}(x_{0},z)|^{2}/\int_{Z}|H_{y}^{inc}(x,z_{s})|^{2}dx\); (**b**) \(|H_{y}(x,0)|^{2}/\int_{Z}|H_{y}^{inc}(x,z_{s})|^{2}dx\). To illustrate the efficiency of the above idea of two-meta-cells stitched principle, we design two types of metalenses of different widths, \(d=24.99\)\(\upmu\)m and \(d=41\)\(\upmu\)m constituted of \(N_{p}=248=125+125-2\) and \(N_{p}=408=205+205-2\) nsnr elements respectively. The other parameters are \(h=400\) nm, and the focal length is set to \(15\lambda\). For each example, first we design a \(L\)-wide \([A,D]\) metacell to achieve efficient far field focusing of the transmitted field at \((x_{0},z_{s})\) so that the focus spot is located at the extreme limit of the computation domain. Typically, \(x_{0}\) is chosen very close to \(L\). Second, the second sub-metacell namely \([B^{\prime},A^{\prime}]\) is created from the previous optimized off-axis metalens \([A,D]=[A,B]\bigcup[B,D]\) as the symmetric of the \([A,B]\) subsection. The completed final on-axis device is then obtained by stitching together the \([A,D]\) and \([B^{\prime},A^{\prime}]\) subsections. First, Let us consider the case of 24.99 \(\upmu\)m-wide metalens made of 248 nsnr elements which \([A,D]\) subsection is a 12.55 \(\upmu\)m-wide metacell made of 125 nsnr elements. Figure 8a presents the normalized transmitted field's intensity distribution on \((X,O,Z)\) plane.