## 3 Resonators: Materials and Geometries
### Geometric Shapes
A brief description of the aforementioned geometric shapes is provided below. Figure 2: Geometric shapes for the resonator considered in this study: (**a**) concave hemisphere; (**b**) hollow concave hemisphere; (**c**) cylindrical hemisphere; (**d**) hemisphere; (**e**) rectangular prism; (**f**) hollow rectangular prism; (**g**) inverted cone; (**h**) hollow inverted cone; (**i**) cone; (**j**) inverted half-cylinder; (**k**) hollow inverted half-cylinder; (**l**) half-cylinder; (**m**) inverted trapezoidal prism; (**n**) hollow inverted trapezoidal prism; and (**e**) trapezoidal prism. * Cap-based geometries: Cap-based geometries are highly effective for confining and controlling electromagnetic energy due to their curved shapes, which minimize energy losses by reducing surface currents. These geometries facilitate the design of compact resonators with high-quality factor \(Q\), characterized by sharp and selective frequency responses. Additionally, cap-based resonators can enhance antenna BW and support multiple modes, including TE, TM, and whispering gallery (WG) modes, making them versatile for various RF applications. Cap-based resonators can be constructed from dielectric or metallic materials, offering a favorable trade-off between performance and size. Dielectric materials provide strong field confinement and compact designs, while metallic structures ensure a high \(Q\) factor by minimizing energy dissipation. However, the performance of these resonators is sensitive to minor geometric deviations, which can lead to frequency shifts and degraded \(Q\) factor. Materials with low thermal expansion coefficients are essential to maintain frequency stability, particularly in high-precision applications. In this work, we focus on designing resonators using the following cap-based geometries: Figure 2a concave hemisphere, Figure 2b hollow concave hemisphere, Figure 2c cylindrical hemisphere, and Figure 2d hemisphere. Each geometry is tailored to specific performance needs. The cylindrical hemisphere offers superior coupling and broader BW, making it ideal for applications requiring enhanced interaction with the patch antenna. The concave hemisphere improves gain and radiation directionality, making it suitable for gain-critical designs. The hollow concave hemisphere achieves the best trade-off between maximum gain, efficiency, and BW due to its strong field confinement and reduced material losses, although it presents greater fabrication complexity. Lastly, the hemisphere provides a simple, cost-effective solution for narrow-band applications, where moderate gain and efficiency are sufficient [33; 34]. * Rectangular prism-based geometries: Rectangular prism-based geometries are widely used for their simplicity, robust design, and predictable performance. These geometries confine electromagnetic waves, making them ideal for compact and reliable resonators. They support TE and TM modes, with the resonant frequency determined by their physical dimensions. Their fabrication is straightforward, utilizing standard machining or dielectric molding methods. Typically, they are constructed from high-dielectric-constant materials, which enable a significant reduction in resonator size. Furthermore, these geometries exhibit a high quality factor (\(Q\)), resulting in minimal energy loss and ensuring sharp and selective frequency responses. Despite these advantages, rectangular prism-based resonators have some drawbacks. They require materials with low thermal expansion coefficients to maintain frequency stability and their design must carefully isolate the desired mode to avoid interference from unwanted modes. In this work, we focus on designing resonators using the following rectangular prism-based geometries: Figure 2e rectangular prism and Figure 2f hollow rectangular prism. Note that the hollow rectangular prism helps provide better mode control, broader BW, high efficiency, and reduced weight but is more complex to fabricate and requires careful design to suppress unwanted modes.