## 2 Antenna Design Procedure  
The planar bow-tie antenna element is studied in terms of the flare angle through a simulation first. With a fixed flare radius, \(r\), of 50 mm, various bow-tie antennas are designed with flare angles between 40\({}^{\circ}\) and 160\({}^{\circ}\), in increments of 40\({}^{\circ}\). The nomenclature for each antenna follows its flare angle--bow-tie-40 (Figure 1a), bow-tie-80 (Figure 1b), bow-tie-120 (Figure 1c), and bow-tie-160 (Figure 1d). As is evident, the vertical length of the bow tie decreases as the flare angle of the bow tie increases with the fixed radii. The simulated \(S_{11}\) (matched with a characteristic impedance of 50 \(\Omega\)), \(R_{in}\), and \(X_{in}\) for each of the aforementioned bow ties are shown in Figure 2a-c, respectively. The \(-10\) dB bandwidth, resonant frequency (defined as where \(X_{in}\) is \(\Omega\) (Figure 2c)), and \(R_{in}\) (Figure 2b) at the resonant frequency, are examined for each bow tie. Bow-tie-40 and bow-tie-80 exhibit similar performances, with bandwidths of 20% and 22%, respectively. As the flare angle is increased from 40\({}^{\circ}\) to 80\({}^{\circ}\), there is a minor upward shift in the resonant frequency (1.16 GHz to 1.18 GHz) and a slight drop in \(R_{in}\) (56.7 \(\Omega\) to 41.2 \(\Omega\)). As the flare angle is further increased, \(R_{in}\) at the resonant frequency becomes smaller and smaller, resulting in an increase in the \(S_{11}\) values. For instance, bow-tie-120 has an \(R_{in}\) of 18.4 \(\Omega\) at its resonant frequency of 1.36 GHz and, as a result, does not reach \(-10\) dB in \(S_{11}\). Bow-tie-160 has a more significant reduction in \(R_{in}\), with a value of 4.1 \(\Omega\), at its resonant frequency of 1.72 GHz, resulting in poor matching with the characteristic impedance of 50 \(\Omega\). The resonant frequency also shifts upward at a faster pace as the flare angle becomes larger and larger. This behavior is explained by a decreasing effective length, as the flare angle increases and the physical length of the antenna decreases. In Table 1, the aforementioned characteristics of each bow tie are summarized. A further investigation into the effect on the input impedance is provided here by observing the \(R_{in}\) and \(X_{in}\) of each bow tie in the angle study (Figure 1) at a single frequency, denoted by the vertical, black-dashed line in Figure 2b,c. The single frequency chosen for this study is 1.16 GHz--this is the resonant frequency of bow-tie-40, where \(X_{in}\) is equal to 0 \(\Omega\). As the flare angle increases at 1.16 GHz, both \(R_{in}\) and \(X_{in}\) decrease, and the bow-tie antenna becomes capacitive, similar to the behavior of a short dipole, due to the decrease in the vertical effective length of the antenna. The results for each flare angle studied are summarized in Table 2.  
Similar angle studies performed for other frequency ranges demonstrate that the same antenna characteristics apply regardless of the frequency range. Therefore, the antenna can be scaled up or down for operation in any frequency range. To demonstrate the impact of the flare angle on the current distribution, Figure 3 shows the current distributions on each of the bow-tie antennas from the flare angle study (shown in Figure 1), at each of the resonances listed in Table 1 (1.16 GHz, 1.18 GHz, 1.36 GHz, and 1.72 GHz). The colored geometry depicts the current magnitude at each point along the bow-tie antenna, and the red arrows indicate the direction of the flow of the current, obtained from the simulation results. These arrows are only shown on the right side since the antenna design is symmetrical with respect to the z-axis. The white arrows are included to demonstrate how much of the center of the antenna's geometry is covered by strong (red) currents with a value greater than 3 A/m.