## 3 Resonators: Materials and Geometries
### Geometric Shapes
\begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**Feature** & **Cap-Based** & **Rectangular Prism-Based** & **Cone-Based** & **Half-Cylinder-Based** & **Trapezoidal Prism-Based** \\ \hline \multirow{4}{*}{Field Confinement} & Strong due to curved shapes, minimizing surface currents. & Strong, confined throughout the solid volume; highly dependent on material properties. & Strong near the apex, with an efficient energy focus. & Strong along the curved surface; excellent near the apex of the curve. & Concentrated in specific regions due to tapering and slanted sides. \\ \hline \multirow{4}{*}{Mode Control} & Supports TE, TM, and WG modes with good isolation; susceptible to higher-order modes. & Good for TE and TM modes; requires optimization to suppress unwanted modes. & Efficient mode isolation; higher-order modes are naturally suppressed by tapering shape. & Improved suppression of higher-order modes due to curved surface design. & Excellent suppression of higher-order modes due to the non-uniform cross-section. \\ \hline \multirow{4}{*}{Bandwidth} & Moderate to wide, depending on geometry (e.g., hollow concave geometries after the broadest BW). & Moderate; limited by material losses in solid structures. & Wide, especially with hollow designs. & Moderate to wide, depending on fabrication and material choices. & Broad, enhanced by tapering sides and minimized reflections. \\ \hline \multirow{4}{*}{Gain} & Improved for concave and hollow concave designs; uniform for hemispheres. & Moderate; enhanced by compact designs and high-dielectric materials. & High, particularly for hollow cones focusing energy efficiently. & High, with directional improvements due to curved surfaces. & High, with enhanced statically and tapering structure. \\ \hline \multirow{4}{*}{Radiation Efficiency} & Enhanced for hollow and concave geometries due to reduced losses. & Moderate; heavily influenced by dielectric or metallic material properties. & High, due to focused field confinement near the apex. & High, particularly inverted configurations that improve coupling. & High, with improved sampling from the slanted sides. \\ \hline \multirow{4}{*}{Compactness} & Compact, especially for dielectric-based designs. & Compact and straightforward; solid structures are space-efficient. & Compact due to tapering, hollow structures reduce size further. & Compact, particularly fast standard half-ylinders; slightly larger for hollow designs. & Compact, with efficient utilization of space due to slanted sides. \\ \hline \multirow{4}{*}{Fabrication Complexity} & Moderate to high, especially for hollow and concave geometries requiring precision. & Moderate to high, with tapering and machine or mold. & Moderate; requires with tapering and hollow structures requiring precision. & Moderate to high; precise fabrication of saturated sides and tapering required. \\ \hline \multirow{4}{*}{Thermal Stability} & Requires low thermal expansion materials for high-frequency stability. & Good for solid designs: dependent on material properties. & Dependent on material selection; hollow designs may introduce stability concerns. & Requires low-expansion materials for consistent performance. & Requires low-expansion materials for consistent frequencies. \\ \hline \multirow{4}{*}{Applications} & Wideband systems (e.g., hollow concave), gain-critical designs (e.g., concave), or narrowband designs. & Narrowband filters, high-Q resonators, and compact antenna designs. & Wideband, high-gain and systems requiring decision. & Gain-critical systems, wideband applications (inverted or hollow), and compact designs. & Waideband, high-gain systems, and compact or weight-sensitive designs. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison between the chosen geometries for designing RF resonators applied to improve patch antennas.