## 3 Resonators: Materials and Geometries
### Geometric Shapes
Conversely, the rectangular prism is well-suited for antenna designs requiring simplicity, small size, reduced energy loss, and a high \(Q\) factor, particularly in narrowband applications [19; 35; 36]. * Cone-based geometries: Cone-based geometries utilize conical shapes to effectively confine and control electromagnetic energy. These geometries have an intrinsic ability to suppress and mitigate unwanted higher-order modes due to the natural tapering of the cone, which reduces interference and enhances frequency selectivity. The field confinement near the cone apex is excellent, making this geometry ideal for achievinghigh \(Q\) factors and precise frequency responses. Furthermore, cone-based designs result in compact resonators, which are well-suited for improving radiation efficiency and BW. Despite these advantages, cone-based geometries present challenges in fabrication. The precision required to manufacture the tapering shape is critical, particularly for high-frequency applications. Additionally, materials with low thermal expansion coefficients are essential to maintain frequency stability under varying environmental conditions. In this work, we focus on designing resonators using the following cone-based geometries: Figure 2g inverted cone, Figure 2h hollow inverted cone, and Figure 2i cone. Each geometry serves specific performance goals. The hollow inverted cone is particularly beneficial when gain, BW, and weight reduction are critical; however, it is more complex to design and fabricate due to its hollow structure. The inverted cone is advantageous for improving BW and efficiency, as its geometry provides better coupling with the patch antenna. Lastly, the cone offers a simple and compact resonator design suitable for applications with sufficient moderate performance improvements [22; 37]. * Half-cylinder-based geometries: Half-cylinder-based geometries are highly effective for enhancing the performance of patch antennas, particularly in terms of gain, BW, and efficiency. The curved surface of the half-cylinder provides excellent field confinement, especially near the apex, while the flat surface improves coupling and integration with the patch antenna. This combination enhances the radiation efficiency and directional properties of the antenna, making these geometries suitable for a wide range of applications. Additionally, their compact form factor makes them ideal for space-constrained designs. These geometries support TE and TM modes and offer better mode isolation and control than other designs. The resonance characteristics are determined by the physical dimensions of the half-cylinder, which directly influence its performance. As a result, they are well-suited for wideband patch antennas, where BW and efficiency are critical. However, there are challenges associated with half-cylinder-based geometries. Achieving precise dimensions and smooth surfaces is crucial to maintaining consistent resonant characteristics, especially at high frequencies, where minor deviations can degrade performance. Furthermore, materials with low thermal expansion coefficients must ensure frequency stability under varying environmental conditions. In this work, we focus on designing resonators using the following half-cylinder-based geometries: Figure 2j inverted half-cylinder, Figure 2k hollow inverted half-cylinder, and Figure 2l half-cylinder. The inverted half-cylinder is handy when strong coupling and wide BW are required, balancing performance and moderate fabrication complexity. A hollow inverted half-cylinder is the most suitable choice for applications with critical gain, BW, and weight reduction. However, it poses more significant challenges in fabrication due to its intricate structure. Lastly, the half-cylinder is a cost-effective option for narrowband applications requiring moderate performance improvements and simplicity in design [19]. * Trapezoidal prism-based geometries: Trapezoidal prism-based geometries are highly appealing for achieving effective field confinement, mode control, and design adaptability.