Spectral Analysis of Sierpinski Microwave Resonators

Triangular Sierpinski structures are fractal geometries that have recently been studied for microwave frequency applications and implemented in microstrip and coplanar waveguide configurations. Antennas and planar components can be proposed using the Sierpinski geometry. The main advantage of the above solution is the definition of a building block with increased internal complexity to modulate the operating frequency within a frequency range defined by the triangle's edge length, set as a fixed starting parameter. Increasing the complexity of the internal geometry by the number of sub-triangles inside the original figure helps define a shifted resonance frequency, thereby enabling frequency tuning by geometry and, through coupled structures, enhancing the electrical performance of the single triangle via edge coupling. A wider bandwidth is achieved by coupling Sierpinski triangles with increased internal complexity. The single building block must be appropriately excited to obtain the primary resonance frequency and the higher-order modes expected for the entire, non-subdivided triangle. Then, a spectral analysis is mandatory to get a working frequency close to the natural one and minimally affected by the feeding system. In this work, a systematic theoretical approach to reconstructing the resonator spectrum will be presented based on the edge length, the internal complexity, and the feeding system for microwave band-stop triangular Sierpinski resonators in the X-Band (8-12 GHz), using the equilateral triangle, and comparing the response of an electromagnetically simulated structure with the analytical results. Planar electromagnetic 2D simulations will be presented for silicon-based devices, i.e., metallized structures on high-resistivity silicon substrates with Sierpinski geometries, fed by microstrip lines. A perturbative approach will be used to predict the resonance frequencies of an internally subdivided triangle. Internally stretched triangles will also be presented, to propose a method to modulate the resonator's working frequency as an alternative to those characterised by increased internal complexity.