## 2 Design of the Proposed Feed Array  
### Single-Element Design  
Figure 1 presents the geometry of a single element of the proposed feed array for feeding a large-aperture antenna. To obtain a uniform electric field distribution in the near-field region on a virtual target surface, the proposed element consists of a Vivaldi radiator and the parasitic rings. The parasitic rings are placed on top of the Vivaldi radiators, and the gap between the Vivaldi radiator and the parasitic rings is \(g\), as shown in Figure 1a. The Vivaldi radiator is directly connected to the SMA connector, and the parasitic rings are coupled-fed from the Vivaldi radiator. The Vivaldi radiator and the parasitic rings have different radiating modes, and, in particular, the near-field pattern of the parasitic rings can be varied by adjusting their radii. The Vivaldi radiator and parasitic rings are stably fixed to the ground plane with a resin support structure. This resin structure is fabricated using the stereo lithography apparatus (SLA) 3D printing method. The Vivaldi radiator has two flares and a coupled-fed structure, and they are printed on an FR-4 substrate (\(\epsilon_{r}\) = 4.6, tan\(\delta\) = 0.018, thickness = 1.6 mm). The width and length of the flares are \(w_{f}\) and \(l_{f}\), respectively, as shown in Figure 1b. The curvature of the inner flares is determined using the function \(f(z)\) to achieve broadband characteristics as follows [22; 23]:  
\[f(z)=a\times e^{k(z-(l_{1}-l_{f}))} \tag{1}\]  
\[k=\frac{1}{l_{f}}\ln(\frac{1}{a}(w_{f}-w_{g})+1) \tag{2}\]  
where \(w_{g}\) is the gap between the two flares, and \(a\) is the coefficient of the exponential curve line. On the opposite side of the flares, a microstrip line with a radial stub is printed to feed the antenna through the electromagnetic coupling. The Vivaldi radiator has a circular cavity with a radius of \(r_{c}\), which is located at a distance \(h_{c}\) from the ground surface to further improve the antenna bandwidth, as shown in Figure 1c [23]. The parasitic rings consist of inner and outer rings, and the radii of these two rings are \(r_{1}\) and \(r_{2}\), respectively, as shown in Figure 1d. The required near-field distribution on the virtual target surface can be achieved by optimizing the radii of the parasitic rings. The design parameters of the proposed single element have been optimized using a CST Studio Suite 2020 EM simulator [24], and they are listed in Table 1. In addition, the simulation settings are shown in Table 2.  
Figure 2 shows the reflection coefficient of the single element of the proposed feed array antenna. The Vivaldi radiator has the advantage of easily achieving broadband characteristics, which exhibits a matching bandwidth of 1.78 GHz (2.21 GHz to 3.99 GHz, \(\left|\Gamma\right|_{\text{dB}}<-10\) dB) with a fractional bandwidth of 59%. In addition, the average reflection \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline \(g\) & 24 mm \\ \(w_{f}\) & 32.15 mm \\ \(w_{2}\) & 85 mm \\ \(w_{1}\) & 2 mm \\ \(l_{1}\) & 85 mm \\ \(l_{f}\) & 69.5 mm \\ \(r_{1}\) & 12 mm \\ \(r_{2}\) & 15 mm \\ \(r_{c}\) & 4.4 mm \\ \(r_{s}\) & 7 mm \\ \(l_{k_{c}}\) & 8.65 mm \\ \(l_{m}\) & 15.5 mm \\ \(t\) & 1.6 mm \\ \(a\) & 0.01 \\ \(w_{g}\) & 0.7 mm \\ \(l_{p}\) (Central element) & 0 mm \\ \(l_{p}\) (Outer element) & 1.8 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the proposed single element. Figure 1: Geometry of the single element. (**a**) Isometric view. (**b**) Front view of the Vivaldi radiator. (**c**) Back view of the Vivaldi radiator. (**d**) Top view of the parasitic rings. coefficient in the operating frequency band is \(-18\) dB. There are some discrepancies between the simulated and measured reflection coefficients due to manufacturing and assembly tolerances during the antenna fabrication process. In particular, a connection point with soldering between the transmission line and SMA connector in the Vivaldi radiator is very sensitive to the antenna's performance.