## 3 Results and Discussions
### Plasmonic Lens Design
A clear off-axis focus spot appears at the predicted location, which agrees with our design's objective. Let us also notice a transmission efficiency of about 25%. Now let us apply the stitching operation as described previously. The result of this post-processing is displayed in Figure 8b where we plot the field's intensity distribution of the completed lens excited by an incident TM plane wave at normal incident. It is worth noting that this two metacells stitched process leads to an increase, in both the size and the number of the constituent elements of the metalens. In this first example, a \((L,n_{p})=(12.55\)\(\upmu\)m,\(125)\) metacell yields \((d,N_{p})=(24.99\)\(\upmu\)m,\(248)\) metalens with more than two times transmission efficiency enhancement: the efficiency of the stitched device increases from 25% to 60%. Figure 8c,d show the values of the (nanoridges, nanoslits)-widths sequence \([e_{k}]_{k}\) with respect to their locations labelled by \(k\). Now let us increase the metalens feature, i.e., both the size and the number of nsnr elements. In this second example, the structure is a \(d=41\)\(\upmu\)m-wide metalens constituted of \(N_{p}=408=205+205-2\) nsnr elements. The primary metacell, i.e., the \([A,D]\) subsection, is specifically designed as a 20.55 \(\upmu\)m-wide off-axis metalens with 205 nsnr elements. Figure 9a presents the normalized transmitted field's distribution. The deflection of focal point to the extreme limit of the design domain, with an efficiency of 30% is clearly visible. As shows Figure 9b, by stitching together both aperiodic metacells, namely the subsection \([A,D]\) and \([B^{\prime},A^{\prime}]\), the on-axis intensity is highly improved from 30% to 70%. Figure 9c,d show the values of the (nanoridges, nanoslits)-widths sequence \([e_{k}]_{k}\) with respect to their locations labelled by \(k\). Figure 7: Design of a one dimensional flat metalens: illustration of the two-metacells-stitched approach. Figure 8: Design of a plasmonic silver-flat-metalens of \(d=24.99\) μm-wide consisting of \(N_{p}=248\) silver-nanoridges and nanoslits focusing a normal incident TM-polarized plane wave at a wavelength \(\lambda=0.637\) μm at a distance of \(15\lambda\). (**a,b**) presents the cartography of the normalized magnetic field intensity in the plane \((X,O,Z)\). (**c,d**) shows the values of the (nanoridges, nanoslits)-widths sequence \([\varepsilon_{k}]_{k}\) with respect to their locations labelled by \(k\). Numerical parameters: \(\lambda=0.637\) μm, \(N_{p}=125\), \(e_{z}=400\) nm, TM polarization. (**a**) \(d=12.55\) μm; (**b**) \(d=24.99\) μm; (**c**) \(n_{p}=125\); (**d**) \(N_{p}=248=125+(125-2)\). Figure 9: _Cont._  
Finally we present a case of an optical near-field subwavelength spatial focusing originated from the interference of higher spatial harmonic field distribution. The focus length is set \(z_{s}=3\lambda\) in the free space. Since the foci is closed to the structure, it is possible to obtain a high-focusing of an incident light by enhancing the near field (evanescent field) contribution. We show in Figure 10 the results of a configuration of a gold metalens. As can be seen in Figure 10a,b, a subwavelength LFWHM can be reached: the LFWHM is equal to \(304.5\) nm \(\leq\lambda/2\) in the case of the gold plasmonics metalens designed here. Figure 11a presents the cartography of the normalized magnetic field intensity in the plane \((X,O,Z)\). Figure 11b shows the values of the (nanorridges, nanoslits)-widths sequence \([e_{k}]_{k}\) with respect to their locations labelled by \(k\). Some other results not presented in this paper, including information of required cputimes, some.gif animations, are available on [[https://photonicsnum.com](https://photonicsnum.com)]([https://photonicsnum.com](https://photonicsnum.com)) (accessed on 15 May 2021). Figure 10: Design of \(d\)-wide gold plasmonic metalens consisting of \(N_{p}\) gold nanorides and nanoslits focusing a normal incident TM-polarized plane wave at a wavelength \(\lambda=0.637\) μm at a distance of \(3\lambda\).