# Cascaded Frequency Selective Surfaces with Matryoshka Geometry for Ultra-Wideband Bandwidth
## 1 Introduction
### Contributions  
The original matryoshka geometry is portrayed in [38] and has been studied by different researchers. Among variations of the original design, the polarization-independent circular matryoshka [39] was developed, which then inspired the square version used in [40] and in this work. The most recent application was the square matryoshka geometry in a defective ground structure (DGS) [41]. In this paper, we propose the use of the square matryoshka geometry and its negative geometry in a cascaded FSS structure.  
This paper presents the development and analysis of the cascaded FSS with square matryoshka geometry to provide a bandwidth that, depending on the variation in the gap between the FSSs, can be greater than 1 GHz in the range from 1.8 GHz to 3.8 GHz. When compared with other papers on the subject, their projected structures do not have frequency ranges below 2.4 GHz and bandwidth of 2 GHz with a two-layer 12 mm spaced FSS. Other designs that present larger bandwidths require more layers and greater spacing between them or have bandwidths above 3.0 GHz. Thus, the design in the present paper has the advantages of using less spacing between layers, fewer layers, being easier to manufacture, and reaching lower frequencies, as shown in Tables 1 and 2.  
In this paper, we present the following contributions:  
* Project, simulation, and validation of a cascaded FSS using matryoshka geometry, which is more efficient in cell dimensions in comparison with concentric square rings.
* Frequency response evaluation of the proposed structure in the band around 1.8 GHz.
* Dual-band operation with bandwidths that depend on the spacing between the FSS boards, one with a 1.2 GHz bandwidth in the 2 to 3.2 GHz range, for a spacing equal to 4 mm, and another with a 2 GHz bandwidth in the range from 1.8 to 3.8 GHz, for a spacing of 12 mm.
* Use of the negative geometry of a matryoshka geometry and its effect with the matryoshka in cascade.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Reference** & **Bandwidth** & **Geometry** \\ \hline
[16] & 2.4–2.5 and 4.98–5.825 GHz & Fractal \\
[17] & 2.48–6.13 GHz & Square ring (4 layers) \\
[18] & 3.10–10.60 GHz & Dual square rings (2 layers) \\
[19] & 3.10–10.60 GHz & Patch with slots (3 layers) \\
[20] & 1.8–2 GHz and 2–2.8 GHz & 3D square rings \\
[21] & Around 10 GHz & 4-pointed flower \\
[22] & 0.1 THz to 3.78 THz & Patch (3-layer) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Bandwidths and geometries of the reference designs.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Reference** & **Contribution of the Cascaded FSS with Matryoshka Geometry** \\ \hline
[16] & Higher bandwidth and reaches lower frequencies \\
[17] & Fewer layers, easier to fabricate materials \\
[18] & Four times less spacing between layers \\
[19] & Fewer layers and less spacing in total between layers \\
[20] & Higher bandwidth and easier to fabricate \\
[21] & Easier to fabricate with lower frequencies reach \\
[22] & Easier to fabricate, fewer layers and application on the GHz range \\ \hline \hline \end{tabular}
\end{table}
Table 2: Contributions of the present paper compared to the listed references.