# Triangular Sierpinski Microwave Band-Stop Resonators for K-Band Filtering
## 1 Introduction  
Sierpinski triangles are fractal complex configurations obtained by defining empty sub-triangles inside a full triangle. Since the seminal work in [1], several contributions have been published regarding 2D and 3D configurations and their applicability in different fields. It must be stressed that the merit of Sierpinski geometry is to give a mathematical formulation of the problem, even if the same geometries are easily found for decorative purposes in ancient monuments, such as the floors of many churches or even columns in cloisters, and studied from cultural heritage and mathematical perspectives [2, 3]. In the case of microwave planar resonators, to obtain a Sierpinski structure means to include empty triangles inside a metal triangular full patch. Some examples of the multi-resonant response of these configurations and their properties for high-frequency signal processing have been published, especially regarding antenna applications [4, 5, 6, 7]. Still, resonators inspired by the same geometry have been considered [7, 8]. A preliminary study on some Sierpinski resonators and their metamaterial nature was also published by us [9]. Our contribution aims to study a specific fractal geometry and its high-frequency response using a coplanar waveguide configuration for a planar resonator. Many research articles on the Sierpinski geometry have been focused on free propagation structures and their utilization for multiband purposes, but few contributions are available for planar signal processing. Among others, we also selected recent studies [10, 11]. The results from [10] mainly focused on traditional coupled line filters mixed with Sierpinski triangles to fine-tune the filter response. In the paper, the results consisted of the multi-band response and some improvement in the component footprint. Additional properties wereconsidered for wideband applications, such as the lowering of the operative frequencies and better electrical matching, linking the resonant narrowband response of the triangles to the wideband performance of the coupled line filter. In [11], a completely different geometry was considered, still inspired by Sierpinski geometry, and focused on a resonant narrowband response but in a gasket configuration, potentially to be manufactured on a suspended membrane. Our paper presents a complete study of triangular equilateral resonators, with a systematic experimental approach to the single and coupled structures inspired by Sierpinski geometry. The obtained results should be considered the basis for a heuristic approach to interpret the response of fractal microwave Sierpinski resonators in coplanar configuration. We give evidence for what can be easily determined and the difficulties in efficiently exciting the Sierpinski configurations or predicting their resonance frequencies compared to microstrip-fed structures. In the following sections, we analyze some Sierpinski triangular structures designed and manufactured on oxidized high-resistivity silicon wafers and measured on-wafer. We compare the measurement with the theoretical expectations obtained using electromagnetic 2.5D commercial software. Simulations were performed using the AXIEM environment of the AWR-CADENCE Microwave Office software release 22.1, and the tests were carried out with the on-wafer measurement system available at CNR-IMM. The devices were manufactured using the technological facilities of FBK. The operating frequencies \(F\) originally selected for the studied resonators were 20 GHz and 26 GHz, respectively. They lie in the K-band (18-27 GHz), currently, one of the frequency ranges most used for satellite telecommunications. The Ku (K-under, 12-18 GHz) and Ka (K-above, 27-40 GHz) bands were also used for the same purposes. Typical requirements for transmit/receive (TX/RX) signals, i.e., for uplink (towards the satellite) and downlink (back to the earth) operations, are around 30 GHz and 20 GHz, respectively.