## 2 Materials and Methods  
### Antenna Design  
FR4 material is used in the proposed Vivaldi antenna design, and this antenna has a loss tangent value of 0.002 and a dielectric constant of 4.4. Of note, 0.035 micron copper was used on both surfaces of the FR4 material. A 50 \(\Omega\) impedance female coaxial connector is used as the feeding port in the FR4 material. The FR4 thickness is 1.6 mm. Antenna design and simulation were performed using CST MWS. The antenna size was reduced to 34.6 mm \(\times\) 33 mm using this program. The Vivaldi antenna was manufactured using a printed circuit board machine. The exponential equation used for antenna design calculates the edge taper of the antenna as follows [81]:  
\[y=C_{1}\mathrm{e}^{Rx}+C_{2} \tag{1}\]  
Here, \(R\) is the angle ratio \(y\) skew. Using the geometric parameters of the antenna, \(C_{1}\) and \(C_{2}\) can be calculated:  
\[C_{1}=\frac{y_{2}-y_{1}}{\mathrm{e}^{Rx_{2}}-\mathrm{e}^{Rx_{1}}} \tag{2}\]  
\[C_{2}\ \frac{y_{1}\mathrm{e}^{Rx_{2}}-y_{2}\mathrm{e}^{Rx_{1}}}{\mathrm{e}^{Rx_ {2}}-\mathrm{e}^{Rx_{1}}} \tag{3}\]  
The first points of the curves in the antenna are \(y_{1}\) and \(x_{1}\); the last points are \(y_{2}\) and \(x_{2}\). The geometric configurations of the Vivaldi antenna are shown in Table 1. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Vivaldi Antenna Value (mm)** \\ \hline \(W\) (Antenna width) & 34.6 \\ \(L\) (Antenna length) & 33 \\ \(Wx\) (Spacing width) & 26.6 \\ \(Ly\) (Flap length) & 22.4 \\ \(W\)s (Slot width) & 11.7 \\ \(Wa\) (Port width) & 2.75 \\ \(Wb\) (Port width) & 6.25 \\ \(La\) (Port length) & 6 \\ \(a\) (Elliptical edge) & 0.085 \\ \(b\) (Circular diameter) & 11.25 \\ \hline \end{tabular}
\end{table} 
Table 1: Geometric configurations of opposite-polarity Vivaldi antennas. As seen in Figure 1, the Vivaldi antenna was designed with 3-slot pairs. The antenna operates between 3.60 and 13 GHz frequency. The measurement results for parameter S11 of the proposed Vivaldi antenna are as shown in Figure 2.  
The S11 parameter represents the frequency operating range of the antennas [82]. The low values of the return loss curves indicate that the antenna is operating stably [83; 84]. As shown in Figure 2, the curves in the operating band range dropped to -19, -20.5, and -23 dB levels. This shows that the antenna is stable. The maximum real gain of the realized Vivaldi antenna at 9.5 GHz frequency is 9.1 dB, the gain is 9.27, and the directivity is 11 dB. Figure 3 shows the gain, real gain, and directivity results. Figure 1: Vivaldi antenna configuration: (**a**) geometrical configuration; (**b**) antenna width (mm); and (**c**) antenna length (mm). Figure 2: S11 reflection coefficient measurement curve of the proposed opposite-polarity Vivaldi antenna. The relationship between the radiation directivity of the frequency points in the antenna operating band and the object (electrical properties of the breast tissue) was investigated. The simulation results for the radiation patterns for the E plane of the Vivaldi antenna at 4.5 GHz, 7 GHz, 9 GHz, and 10.5 GHz are shown in Figure 4. The Vivaldi antenna provides a directional radiation pattern in the simulation process without the breast phantom. A wider beam was observed without the breast model. It is shown that the majority of the power at the upper frequencies of the proposed directional antenna is radiated towards the object (breast model). A comparison of the proposed antenna with some previously reported antennas is given in Table 2.  
Figure 4: Radiation patterns of the Vivaldi antenna: (**a**) 4.5 GHz without a tumor; (**b**) 7 GHz without a tumor; (**c**) 9 GHz without a tumor; (**d**) 10.5 GHz without a tumor; (**e**) 4.5 GHz with a tumor; (**f**) 7 GHz with a tumor; (**g**) 9 GHz with a tumor; (**h**) 10.5 GHz with a tumor. (The red lines are the radiation directivity and the blue lines are the angle width.) Figure 3: Vivaldi antenna performance: (**a**) real gain; (**b**) gain; and (**c**) directivity.