## 3 Results and Discussions
### Plasmonic Lens Design  
We design metalenses consisting of a perforated metal film, capable of focusing a normally incident TM plane wave (with \(\lambda=0.637~{}\upmu\)m), at a focal distance \(z_{f}=15\lambda\) in the vacuum (\(\
u^{(1)}=1\)). The perforated film with a height \(h=400\) nm is deposed on a SiO\({}_{2}\) substrat. To ensure the convergence of the polynomial expansion, \(n=4\) polynomials are used on each subinterval \(l_{x}^{(k)},k=1:N_{p}\). As initial conditions of the algorithm, we use 10 various random initial sequences of nsnr (nanoslits and nanorides) with elements widths \([(\epsilon_{k}^{old},\epsilon_{k}^{new})]_{k}\), generated according to equation Equation (1). The maximal number of iterations \(t_{max}\) is set to 20. In this first example, each initial layout is made of \(N_{p}=105\) nsnr elements distributed on a d = 10.75 \(\upmu\)m-wide interval. The minimum size constraint is set to 50 nm. The other numerical used-defined parameters are \(\alpha_{0}=0.1\), \(s_{1}=25\%\) and \(s_{2}=1\%\). Results are presented in Figure 4. These figures show, in the focal planes, the line scans of the normalized magnetic field intensity along the \(Ox\)-axis (\(z=0\)) (Figure 4a) and along the \(Ox\)-direction \(x=0\) (Figure 4b). Although any gradient-based method is known to be very sensitive to the initial conditions, as show these figures, for all the ten random initial conditions, that a very large part of the ten designed metalenses maintain satisfactory efficiencies close to 30% for the current lossy plasmonic lens. Both the lens size and the number \(N_{p}\) of nsnr elements can be exploited to manage the focus spot size as is clearly shown in Figure 5. In these figures, we plot, the line scans of the normalized magnetic field intensity in the transverse (\(z_{f}=0\)) (Figure 5a) and in the longitudinal \(x=x_{0}\) (Figure 5) focal plane respectively, for four values of the couple \((N_{p},d)\)\((N_{p},d)\in\{(105,10.75~{}\upmu\text{m}),(125,12.75~{}\upmu\text{m}),(165,16.75~{} \upmu\text{m}),(185,18.75~{}\upmu\text{m})\}\). Regarding these results, the ratio of the focal intensity to the intensity of incident wave increases with an increasing number of \((N_{p},d)\) couples. In the case of \((N_{p},d)=(105,10.75~{}\upmu\text{m})\), this efficiency is close to 29%. While it reaches 38% for a 18.75 \(\upmu\text{m}\)-width lens composed of \(N_{p}=185\) metallic-nanorods/nanoslits. Figure 4: Design of a 10.75 \(\upmu\text{m}\)-wide plasmonic metalens consisting of \(N_{p}=105\) silver-nanorides and nanoslits focusing a normal incident TM-polarized plane wave at a wavelength \(\lambda=0.637~{}\upmu\text{m}\) at a distance of 15\(\lambda\). Line scans of the normalized magnetic field intensity in the transverse (\(z_{f}=0\)) (**a**) and in the longitudinal \(x=0\) (**b**) focal plane respectively, for ten random realizations of initial conditions. Numerical parameters: \(\lambda=0.637~{}\upmu\text{m}\), \(N_{p}=105\), \(h=400~{}\upmu\text{m}\), TM polarization. (**a**) \(|H_{y}(x,0)|^{2}/\int_{\mathcal{I}}|H_{y}^{inc}(x,z_{s})|^{2}dx\); (**b**) \(|H_{y}(x_{0},z)|^{2}/\int_{\mathcal{I}}|H_{y}^{inc}(x,z_{s})|^{2}dx\). Now we analyse the impact of the \((N_{p},d)\) couple on both the LFWHM (lateral full width half maximum) and on the AFWHM (axial full width half maximum). As shown in Figure 5a,b, the LFWHM hardly depends on the lateral feature of the metalens. In Figure 5a the LFWHM is approximately equal to 565.2 nm. Contrary to the LFWHM, the AFWHM strongly depends on the width of the metalens. As shown in Figure 5, the AFWHM decreases when the lens size \(d\) and number of patterns \(N_{p}\) increase. For example, in the case of the silver-plasmonic metalens considered in our investigation, the AFHW is equal to 6916.9 nm for \((N_{p},d)=(105,10.75\) um) and it decreases to 2704.2 nm for \((N_{p},d)=(185,18.75\) um), yielding a better high-quality cylindrical focal spot.