# Cascaded Frequency Selective Surfaces with Matryoshka Geometry for Ultra-Wideband Bandwidth
## 2 Proposed Matryoshka Design  
The matryoshka geometry is inspired by the Russian dolls of the same name, in which a larger doll contains a smaller doll, which contains another smaller doll, and so on, but all occupying only the volume of the larger doll [38]. Its design starts with concentric rings with width w and lengths \(L_{x1}\) and \(L_{y1}\) for the outer ring and \(L_{x2}\) and \(L_{y2}\) for the inner ring on a unit cell with length \(W_{x}\) and \(W_{y}\). Then, the concentric rings receive slits of length \(g\) at the same point and are connected to each other with two strips of the same width \(w\) as the rest of the two rings, making one continuous line through the whole geometry, as shown in Figure 2.  
As the rings are interconnected, the effective length of the matryoshka ring increases without increasing the occupied area, limited to the area of the outermost concentric ring, presenting features like miniaturization and multiband operation. This geometry is polarization-dependent, with the connecting part being displayed only on the right side of the design. For an equivalence in circuits, the base of the concentric rings is used, such as a pair of square rings, so the equivalent circuit behaves as a dual circuit of inductors and capacitors, as shown in Figure 3, with each square ring being equivalent to one inductor and one capacitor in series being presented in parallel because of their position as concentric squares. By connecting these two rings, it is possible to obtain a matryoshka geometry. The connections between them introduce more inductor and capacitor elements both in series and in parallel, as shown in Figure 3. In more complex geometries with more rings and connections, such as examples with independent polarization, numerical methods are needed to generate an accurate equivalent circuit [38; 39; 40]. The matryoshka geometry [40] was the base for the geometry developed in this paper, with independence of polarization and a more complex design. This paper presents the use of the geometry displayed in Figure 4 for one of the FSS and its negative equivalent for another. This modified matryoshka geometry has connections between the rings on each side of the squares, making it a more symmetric design. The connections between the three rings on each side make the geometry horizontally and vertically symmetric and independent of polarization. The dimensions used in the project are shown in Table 3. Usually, \(L_{x1}=L_{y1}=L_{1}\) for square rings. Hence, \(L_{xi}=L_{yi}=L_{i}\) with \(i=1,\ 2,\ 3\). Figure 2: Development of a matryoshka geometry from (a) a pair of concentric square loops to (b) the opened and patched square loops to (c) the matryoshka geometry. Figure 4: Matryoshka geometry independent of polarization with three square rings. Figure 3: Matryoshka geometries and equivalent circuits for (**a**) a pair of concentric square loop geometry and (**b**) a square matryoshka geometry. \begin{table}
\begin{tabular}{c c} \hline \hline
**Dimension** & **Measurement (mm)** \\ \hline Cell horizontal size (\(W_{x}\)) & 24 \\ Cell vertical size (\(W_{y}\)) & 24 \\ Outer square ring (\(L_{1}\)) & 22 \\ Middle square ring (\(L_{2}\)) & 16 \\ Inner square ring (\(L_{3}\)) & 10 \\ Gap (\(g\)) & 1 \\ Stripe width (\(w\)) & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dimensions of the proposed matryoshka geometry.