# A Novel L-Shaped Metalens for Ultra-Wide Band (UWB) Antenna Gain Improvement
## 3 Design and Results
### Metalens Design
Figure 3: Full-wave simulation of the gain \(G\) as a function of frequency for AVA (solid curve) and AVA with EHL having a thickness of \(d_{EHL}=0.832\) mm for the optimized effective permittivity \(\varepsilon_{effMG}\) and effective permeability \(\mu_{effMG}\) (dashed curve). Figure 2: Schematic of AVA with EHL (green region) in place of the metalens. For the SPRM approach, the S-parameters simulations of the stand-alone unit cell, i.e., before its integration with AVA geometry, are performed by considering the same substrate of the antenna. The top and the 3D views of the L-shaped unit cell with the relative geometric parameters are shown in Figure 5a,b, respectively. In the second design step, in order to retrieve the effective parameters of the metalens, the unit cell illustrated in Figure 6 has been considered. In the simulation, the \(\vec{E}\,k\)-plane of the unit cell corresponds to the Vivaldi \(xy\)-plane; see Figures 1 and 6. The plane EM wave excitation is obtained (see Figure 6) via the Waveguide Port 1, with open boundary conditions applied in the propagation direction. The perfect electric conductor (PEC) and the perfect magnetic conductor (PMC) boundary conditions are suitably applied. Figure 4: (**a**) Schematic of the designed L-shaped unit cell and (**b**) its equivalent circuit. Figure 5: Schematic of L-shaped unit cell: (**a**) top view, and (**b**) 3D view. The L-shaped unit cell geometry has been optimized in order to obtain a broadband gain enhancement to parity of the bandwidth [15; 16]. The geometry of the proposed unit cell offers a high degree of freedom for metalens optimization. The inductive effects can be tuned in different ways, depending on the EM polarization, by varying the length of the L-element independently along the \(x\)-direction, \(L_{x}\), and \(y\)-direction, \(L_{y}\). Similarly, the capacitive effects can be tuned by varying the gap widths of the splits \(g_{x}\) and \(g_{y}\). Moreover, changing the metal strip width \(w\) affects both the capacitive and inductive effects. Therefore, the L-shaped unit cell resonant behavior has been evaluated by changing all the above-mentioned geometrical parameters and the lattice lengths \(d_{x}\) and \(d_{y}\). For shortness, Table 2 directly reports the optimized values obtained in the third design step of the unit cell of the metalens integrated into the antenna. To identify these parameters, the exact number and distribution of cells, described in Section 3.3, have been considered. To give an insight into the metalens behavior, the simulated amplitudes Figure 7a and phases Figure 7b of the \(S_{11}\) and \(S_{21}\) parameters of the L-shaped unit cell versus the frequency are shown in Figure 7 for the optimized parameters of Table 2.  
The average values of the real part of the effective electric permittivity \(Re(\varepsilon_{eff,w})=4.98\) and of the effective magnetic permeability \(Re(\mu_{eff,w})=0.75\), retrieved with SPRM from the aforesaid \(S_{11}\) and \(S_{21}\) curves, are close enough to the theoretical optimized values \(\varepsilon_{eff/MG}=5.75\) and \(\mu_{eff/MG}=1.5\) identified in the first design step and provide an almost constant value of the effective refractive index over the whole frequency range. It is reported in Figure 7c. A congruent number of unit cells allows better results, taking into account the couplings between adjacent unit cells [34]. Therefore, a single cell, \(3\times 3\), and \(6\times 6\) clusters of unit cells have been also investigated, obtaining practically the same results. As a consequence, the frequency range \(f=3\div 15\) GHz can be considered a potential operation region for the antenna. This means that the L-shaped metalens can improve the antenna radiation performance in the whole band [15; 16]. Indeed, the designed lens has a higher average refractive index than that of the antenna substrate, so a waveguiding effect is possible. The EM field is better confined via the lens, enhancing the radiation beam in the end-fire direction.