## 4 Analysis of Numerical Results
### Distilled Water Resonator  
This subsection evaluates the performance of various geometric shapes for the DW resonator. The \(|S_{11}|\) parameter for the cap-based, rectangular prism-based, cone-based, half-cylinder-based, and trapezoidal prism-based geometries are shown in Figure 4a, Figure 4b, Figure 4c, Figure 5a, and Figure 5b, respectively. Furthermore, the radiation patterns for the cap-based, rectangular prism-based, cone-based, half-cylinder-based, and trapezoidal prism-based geometries are illustrated in Figure 6a, Figure 6b, Figure 6c, Figure 7a, and Figure 7b, respectively. Moreover, Table 4 summarizes key parameter values obtained at the resonant frequency. Regarding cap-based geometries, \(|S_{11}|\) in Figure 4a show that the concave hemisphere and hemisphere geometries exhibit additional notches, indicating multiple impedance matching points. These notches suggest the patch antenna can operate at multiple resonant frequencies, offering enhanced potential for multi-band propagation. Among the cap-based geometries, the cylindrical hemisphere achieves the broadest BW, as detailed in Table 4, offering an extended frequency range for enabling impedance matching. Concerning the radiation pattern, shown in Figure 6a, the concave hemisphere achieves the highest MLM value. At the same time, the cylindrical hemisphere and hollow concave hemisphere also exhibit MLM values exceeding that of the patch antenna without a resonator. Additionally, a significant variation in MLD is observed in the hemisphere geometry, altering MLD values from \(0^{\circ}\) to \(135^{\circ}\) and leading to a performance reduction. As listed in Table 4, all cap-based geometries presented a decrease in HPBW and SLL, with the cylindrical hemisphere being an exception in terms of SLL. Considering rectangular prism-based geometries, the \(|S_{11}|\) shown in Figure 4b demonstrate distinct performances. For instance, the rectangular prism maintains an acceptable \(|S_{11}|\) and provides a BW comparable to the patch antenna without a resonator. In contrast, the hollow rectangular prism exhibits unacceptable \(|S_{11}|\) limits. Concerning the radiation pattern, as shown in Figure 6b, the rectangular prism achieves the highest MLM value, whereas the hollow rectangular prism displays the lowest MLM value compared to the patch antenna without a resonator. Furthermore, as summarized in Table 4, the rectangular prism reduces the HPBW, while the hollow rectangular prism increases it. Additionally, the SLL is maintained for the rectangular prism but reduced for the hollow rectangular prism compared to the patch antenna without a resonator. Regarding the findings for cone-based geometries, all exhibit acceptable \(|S_{11}|\) values, as depicted in Figure 4c. The cone achieves the highest BW, while the inverted and hollow inverted cones display the most favorable \(|S_{11}|\). In terms of the radiation pattern, Figure 6c, the inverted cone achieves the highest MLM value, followed by the hollow inverted cone. Moreover, all cone-based geometries achieve MLM values higher than the patch antenna without a resonator. Both the inverted cone and hollow inverted cone show reductions in HPBW and SLL compared to the patch antenna without a resonator, Figure 6c and Table 4, and, consequently, they enhance the directionality of the patch antenna. Figure 4: Patch antenna with a DW-made resonator: magnitude of the scattering parameter (\(|S_{11}|\)). (**a**) cap-based; (**b**) rectangular prism-based; and (**c**) cone-based. Figure 5: Patch antenna with a DW-made resonator: magnitude of the scattering parameter (\(|S_{11}|\)). (**a**) half-cylinder-based; and (**b**) trapezoidal prism-based. Figure 6: Patch antenna with a DW-made resonator: the radiation pattern plot of the 1D far-field realized gain (\(\phi=0\)). (**a**) cap-based; (**b**) rectangular prism-based; and (**c**) cone-based.