## 2 Design of the Proposed Feed Array
### 4 \(\times\) 4 Feed Array Design  
Figure 4 illustrates the configuration of the proposed feed array for feeding a large-aperture antenna. The feed system has 16 elements with a 4 \(\times\) 4 rectangular array configuration to obtain the uniform near-field distribution on a virtual target surface. The array spacing in this configuration is \(d\), and the separation distance between the feed array antenna and the virtual target surface is \(h\). The area of this virtual target surface is assumed to be 200 \(\times\) 200 mm\({}^{2}\). The input phase of each Vivaldi radiator is adjusted by applying different transmission line lengths to the Vivaldi radiators depending on their positions in the array. For example, the transmission line length of the central Vivaldi radiator is shorter than that of the Vivaldi radiator placed on the outside of the array. This configuration can easily control the input phase distribution in the array without the use of a complex phase shifter. The input phase delay of the outer element compared to that of the central element is expressed as \(\phi_{edge}\). Figure 5 shows the flowchart of the GA to optimize the array design parameters such as the array spacing (\(d\)), the input phase delay of the outer element (\(\phi_{edge}\)), and the separation distance between the feed array and the virtual target surface (\(h\)). In this optimization process, we focused on optimizing only the variables for the array configuration to reduce the time and resources required for optimization. To find the optimum design parameters, a random chromosome is created, and the decoded values from the chromosome are then applied to the design parameters of the feed array model. Next, the array antenna model with these design parameters is simulated using the CST Studio Suite EM simulator to calculate the electric field distribution on a virtual target surface. The cost function for the GA is defined as follows:  
\[Cost=\alpha(E_{ave})+\beta(E_{dev}) \tag{3}\]  
where \(E_{ave}\) is the average value of the electric field distribution used to evaluate the uniformity of the electric field distribution. However, since \(E_{ave}\) alone is not suitable for finding the null point of the electric field, we observed not only the average but also the deviation \(E_{dev}\) of the electric field. \(E_{dev}  
Figure 6 presents the variation in the cost value according to the array spacing (\(d\)) and the input phase delay (\(\phi_{edge}\)) at 3.05 GHz. As can be seen in Figure 6a, when the separation distance between the feed array antenna and the virtual target surface (\(h\)) is 0.4\(\lambda\), the minimum cost value is observed at \(d\) = 0.89\(\lambda\) and \(\phi_{edge}\) = 20\({}^{\circ}\). On the other hand, when \(h\) is 0.6\(\lambda\), the minimum cost value is indicated at \(d\) = 0.7\(\lambda\) and \(\phi_{edge}\) = 0\({}^{\circ}\), as shown in Figure 6b. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Crossover ratio & 0.8 \\ Mutation ratio & 0.1 \\ Populations & 30 \\ Generations & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of GA set up. Figure 5: Flowchart of the GA algorithm. Figure 7 represents the normalized electric field distribution of the feed array antenna at 3.05 GHz. In this analysis, the electric field distribution is observed along a straight line (\(x=0\) and \(-100\) mm \(<y<100\) mm) by varying the height (\(h\)). The optimized near-field distribution with design parameters of \(h=0.4\lambda\), \(d=0.89\lambda\), and \(\phi_{\textit{edge}}=20^{\circ}\) are compared with the near-field field when \(h=0.4\lambda\), \(d=1\lambda\), and \(\phi_{\textit{edge}}=0^{\circ}\). The optimized array has a more uniform field distribution (cost = 0.38) than the other (cost = 10.31), as shown in Figure 7a. When \(h\) is 0.6\(\lambda\), the optimized array (\(d=0.7\lambda\) and \(\phi_{\textit{edge}}=0^{\circ}\)) has a field distribution closest to the uniform (cost = 0.29), as shown in Figure 7b.