## 3 Resonators: Materials and Geometries
### Geometric Shapes  
A resonator can take on various shapes, each impacting the performance of the embedded patch antennas. Furthermore, altering the size of these geometric shapes influences key characteristics of the patch antennas, including resonant frequency, impedance matching, radiation pattern, BW, and mechanical flexibility [32]. These effects result from the interaction between geometric shapes and electromagnetic fields, as each shape supports resonant electromagnetic modes. Note that the resonant electromagnetic modes, namely TE and TM, refer to types of electromagnetic waves propagating in different geometries. TE mode is characterized by the electric field propagating in a direction transverse to the wave propagation, while the magnetic field has a longitudinal component. In contrast, TM mode is defined by the magnetic field propagating transversely to the wave direction, while the electric field has a component along the direction of wave propagation [19]. Depending on the geometry, dimensions, and arrangement of the resonators, they may operate in TE or TM mode due to boundary conditions specific to each geometry, impacting the radiation efficiency of the antenna and frequency tuning capabilities [19]. Figure 2 illustrates fifteen geometric shapes for designing resonators to improve patch antenna performance. The shapes encompass cap-based geometries, Figure 2a-d; rectangular prism-based geometries, Figure 2e,f; cone-based geometries, Figure 2g-i; half-cylinder-based geometries, Figure 2j-l; and trapezoidal prism-based geometries, Figure 2m-o. Note that the cylinder- and rectangular prism-based geometries offer two possible alignment directions: along the width of the patch antenna (WPA) or along the length of the patch antenna (LPA). Most of these geometric shapes are versatile, supporting either TE or TM modes. However, unlike other geometric shapes, the rectangular prism primarily supports the TE mode [19]. A comparison of these geometric shapes for a resonator is important \begin{table}
\begin{tabular}{p{71.1 pt} p{71.1 pt} p{71.1 pt} p{71.1 pt}} \hline \hline
**Feature** & **Distilled Water** & **Glycerin** & **Graphite** \\ \hline Dielectric Properties & High permittivity, low conductivity; suitable for capacitive loading. & High permittivity, slightly higher losses than DW. & Moderate permittivity, high conductivity. \\ \hline Loss Tangent & High loss tangent, especially at microwave frequencies. & Higher loss tangent than DW. & High loss tangent, excellent for conductive applications. \\ \hline Thermal Stability & Poor; properties change significantly with temperature. & Poor; viscosity and permittivity are temperature-sensitive. & Excellent thermal stability; properties remain stable at high temperatures. \\ \hline Field Confinement & Limited due to higher losses; useful in liquid-based tunable designs. & Better than DW but still limited by higher losses. & Excellent for field confinement due to high conductivity. \\ \hline Bandwidth & Moderate; losses limit high-frequency performance. & Moderate; broader range than DW due to better viscosity management. & Wide BW due to stable electromagnetic properties. \\ \hline Fabrication Complexity & Simple; easy to integrate into liquid-based resonators. & Simple; liquid handling requires care to avoid contamination. & Complex; requires precise shaping and integration. \\ \hline Weight & Lightweight, dependent on liquid containment. & Heavy liquid; more challenging to handle in portable systems. & Heavy; requires structural considerations. \\ \hline Thermal Expansion & High; significant expansion with temperature changes. & High similar issues as DW. & Low; minimal thermal expansion. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison between DW, glycerin, and graphite for designing RF resonators applied to improve patch antennas. because each offers unique benefits for meeting specific design requirements and modifying relevant performance parameters (i.e., gain, BW, efficiency, and radiation pattern).