# Dielectric Slabs-Based Lens for Millimeter-Wave Beamforming
## 2 Dielectric Slabs-Based Lens Design  
The semi-spherical lens (see Figure 1a) is assumed as a slab with thickness \(d\) to investigate if it is an effective refractive index at each position on the lens. The lens slab consists of two materials, regardless the other material is air or not. Calculating the scattering parameters of the whole lens slab (Figure 1a) cannot give the required information about the effective refractive index at each position. Therefore, to investigate the value of the effective refractive index at any position on the slab lens, a small square point in the lens is taken and expanded to take slab shape (as shown in Figure 1b). For normal incident plane waves on the slab, the relationships between the complex refractive index (\(n_{eff}\)) with the S-parameters [23; 24] are given by:  
\[n_{eff}=\frac{-i}{\kappa d}\ln\left(\frac{S_{11}}{1-S_{21}\left(\frac{z_{eff}- 1}{z_{eff}+1}\right)}\right). \tag{1}\]  
where \(d\) is the slab thickness, \(\kappa\) is the free-space wavenumber, \(S_{11}\) is the reflected scattering parameter, \(S_{21}\) is the transmitted scattering parameter and \(z_{eff}\) is the complex effective wave impedance, which is given as:  
\[z_{eff}=\pm\sqrt{\frac{\left(1+S_{11}\right)^{2}-{S_{21}}^{2}}{\left(1-S_{11} \right)^{2}-{S_{21}}^{2}}} \tag{2}\]  
For passive materials, the real value of the imaginary value of the complex refractive index and the complex wave impedance must be greater or equal to zero. Therefore, the sign of \(z_{eff}\) must be determined according to those conditions. The investigated slabs consist of homogeneous polyetherimide (PEI) plastic material (\(\varepsilon=3\)) of thickness, \(t\), and air of thickness, \(d-t\).  
The scattering parameters (\(S_{11}\) and \(S_{21}\)) are determined numerically using a full-wave simulator for a vertically polarized plane wave normally incident on the investigated slabs with different \(t/d\) ratios (0, 0.2, 0.4, 0.6, 0.8, 1). A finite element method (FEM) solver in CST is used to obtain the approximate solution of the electromagnetic boundary problem. Floquet's ports are assigned using the unit cell boundaries with normal incidence.  
Figure 1: Physical model of a small position on the semi-spherical lens. (**a**) Semi-spherical lens, and (**b**) the effective slab of this position.  
The corresponding effective refractive index is computed from the numerical scattering parameters data (see Equation (1)) and presented in Figure 2.  
An analytical formula can be derived to describe the approximate effective refractive index (\(n_{eff}\)) of a slab consisting of two dielectric materials with permittivities \(\varepsilon_{1}\), and \(\varepsilon_{2}\) and thickness \(t\), and \((d-t)\), respectively, as given by:  
\[n_{eff}\approx\sqrt{(\varepsilon_{1}t+\varepsilon_{2}(d-t))/d} \tag{3}\]  
Equation (3) represents approximated formula to do an initial calculation of the target effective refractive index with different slab thicknesses. The numerical values of the effective refractive index of the investigated slab are similar to the analytical values computed from Equation (3), as shown in Figure 2.  
The effective refractive index behaves as a continuous refractive index of the semi-spherical lens when the change in the dielectric thickness, \(t_{\Delta}\) is fixed and very small. However, the behavior of the effective refractive index is stepped if the \(t_{\Delta}\) is considerable compared to the operating wavelength. Therefore, the dielectric slabs-based lens is designed by discretizing the semi-spherical dielectric lens to several slabs corresponding to the selected effective refractive index, as shown in Figure 3.  
The slab thickness is constant, whereas the radius of each slab is variable and depends on the total radius of the semi-spherical lens.  
Figure 2: The effective refractive index of the investigated slab with different \(t/d\) ratios, 0, 0.2, 0.4, 0.6, 0.8, 1, using a full-wave simulator (solid lines) and an analytical calculation (markers).