# Additive Manufacturing of Variable Density Lenses for Radio Frequency Communications in X-Band
## 3 Horn Antenna and Gradient Index Lens Design
### Electromagnetic Design
#### 3.1.2 Gradient Index Lenses  
The design of the lens covered several stages. First of all, some parameters were chosen in order to meet the sensible size and weight of the whole set. Then, an adjoint method, see [32], was used to obtain the permittivity values depending on the radius. The focal point \(r_{0}\) was chosen such that \(f/D=1\), indicating that the focal length is equal to the diameter of the lens. For practical purposes, the diameter of the lens was \(D=15\) cm, and its width \(H=2.5\) cm. That would provide a realizable lens with a low cost AM printer. However, the achievable range of permittivity values was constrained by the available materials and the limitations of the additive manufacturing process. Specifically, printing densities below 20% were avoided to maintain the structural integrity of the lens. Consequently, the feasible permittivity values ranged from \(\varepsilon_{\max}\) to \(0.2(\varepsilon_{\max}-1)+1\), where \(\varepsilon_{\max}\) represents the relative dielectric constant of the material at 100% density. Figure 8: Mechanical design dimensions of the X-Bend Antenna Assembly: (**a**) horn elevation view; (**b**) horn plan view; (**c**) support profile view; (**d**) plan view of the X-Bend Lens. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameter** & **Dimension (mm)** & **Parameter** & **Dimension (mm)** \\ \hline C1 & 29.74 & C2 & 7.07 \\ D1 & 56 & D2 & 23.40 \\ D3 & 48 & D4 & 56 \\ D5 & 2.8 & D & 160 \\ L1 & 33 & L2 & 25 \\ L3 & 80 & H & 25 \\ T & 2 & f & 131.6 \\ R & 12.5 & g & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of mechanical design of the X-Bend Antenna Assembly. On the other hand, not all possible ranges of permittivity could be considered, as the design was constrained by the available materials and the limitations of the printers. A printing density lower than 20% would compromise the structural integrity of the lens. So the possible values ranged from \(\varepsilon_{max}\) to \(0.2(\varepsilon_{max}-1)+1\), where \(\varepsilon_{max}\) is the relative dielectric constant of the material printed at 100 % density. The optimization process determined the optimal distribution of \(\varepsilon_{r}(x,y)\) within a rectangular domain of height \(D\) and width \(H\). This domain was divided into 11 regions, resulting in 11 levels of freedom where \(\varepsilon_{r}\) was constant within each region. Thus, the lens was modeled via the Lippmann-Schwinger integral equation in 2D, as in Equation (3). \[E(\mathbf{r})=E_{\text{inc}}(\mathbf{r})+k_{0}^{2}\int_{V}G(\mathbf{r},\mathbf{ r}^{\prime})\,\chi(\mathbf{r}^{\prime})\,E(\mathbf{r}^{\prime})\,\mathrm{d}V^{ \prime}, \tag{3}\]  
with contrast \(\chi(\mathbf{r})=\varepsilon_{r}(\mathbf{r})-1\) and free-space Green's function \(G\). The design objective was the field intensity at the focal point \(\mathbf{r}_{f}\), Equation (4). \[J=\big{|}E(\mathbf{r}_{f})\big{|}^{2}, \tag{4}\]  
By using the adjoint field \(E^{\ddagger}\) and satisfying Equation (5),  
\[E^{\ddagger}(\mathbf{r})=G(\mathbf{r},\mathbf{r}_{f})+k_{0}^{2}\int_{V}G( \mathbf{r},\mathbf{r}^{\prime})\,\chi(\mathbf{r}^{\prime})\,E^{\ddagger}( \mathbf{r}^{\prime})\,\mathrm{d}V^{\prime}, \tag{5}\]  
the gradient with respect to each annular permittivity \(\varepsilon_{r}^{(i)}\) was given by Equation (6). \[\frac{\partial J}{\partial\varepsilon_{r}^{(i)}}=2k_{0}^{2}\Re\Big{\{}\!\int_ {V_{i}}E^{\ddagger}(\mathbf{r})\,E(\mathbf{r})\,\mathrm{d}V\Big{\}}, \tag{6}\]  
A quasi-Newton optimizer then updated the 11 discrete \(\varepsilon_{r}^{(i)}\) until convergence (\(\Delta J/J<10^{-5}\) or 100 iterations). The sketch representing the lens with sectors and corresponding values of permittivity is presented in Figure 8d. It is worth mentioning that the optimized and manufactured lenses presented a central cylindrical sector with a radius of 12.5 mm (which corresponds to the permittivity \(\varepsilon_{r}^{\theta}\)) and then the five successive concentric rings with a thickness of 12.5 mm.