## 2 Materials and Methods
### Design Procedure and Analysis of the FSS Element  
The FSS layers were intentionally engineered by folding and extending the I-shaped conductive elements to optimize the performance across a broad frequency range. By \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value (mm)** \\ \hline \(D\) & 24 \\ \(t\) & 0.46 \\ \(w\) & 0.2 \\ \(L_{1}\) & 2 \\ \(L_{2}\) & 11.6 \\ \(L_{3}\) & 22 \\ \(L_{4}\) & 3.8 \\ \(L_{5}\) & 1.8 \\ \(L_{6}\) & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical parameters of the unit cell. Figure 1: The schematic diagram of the unit cell: (**a**) perspective view, (**b**) top layer, and (**c**) bottom layer. incorporating meandering patterns, the unit cell introduced additional capacitance and inductance along the trace, enhancing its ability to pass a wide range of frequencies with minimal attenuation. In addition, this configuration was meticulously designed by considering the proximity between the FSS layers, separated by an ultrathin dielectric substrate, leading to the manifestation of a parallel LC behavior necessary for achieving the bandpass filter's functionality. This strategy, which employs the miniaturization of the I-shaped unit cell through a meandering approach, aims to enhance the effective electrical length of the unit cell within a confined space, simultaneously maximizing the corresponding Q-factor. The number of meanders was selected to achieve a periodicity of 24 mm, carefully balancing the design to prevent higher-order spurious resonances. Meanderization increases the electrical length of the unit cell without increasing its physical size, allowing for reduced periodicity and lower frequency operation. A large periodicity in FSS can lead to higher-order harmonics and unwanted interference, reducing the overall effectiveness. This approach enhances selectivity, suppresses spurious responses, and improves angular stability, enabling the design of compact FSSs with optimized performance. The design procedure was carried out by applying the bandpass condition through exploiting the transmission line equivalent model (TLM) [39]. Figure 2a synthetizes the schematic representation of the overall double-layer FSS impedance. The impedance seen at the input of this circuit \(Z_{IN}\) can be expressed as:  
\[Z_{IN}=\frac{Z_{FSS}Z_{0}}{Z_{FSS}+Z_{0}} \tag{1}\]  
Consequently, the input impedance in Equation (1) can be substituted in the transmission coefficient expression resulting in:  
\[\tau=\frac{2Z_{0}}{Z_{IN}+Z_{0}}=\frac{2}{2+\
icefrac{{Z_{0}}}{{Z_{FSS}}}} \tag{2}\]  
Since the application herein discussed consists of a bandpass filter, the transmission coefficient should ideally be equal to 1 in the band of interest, and 0 everywhere else. Thus,  
Figure 2: (**a**) Schematic representation of the unit cell FSS impedance. Real (**b**) and imaginary (**c**) parts of \(Z_{FSS}\) obtained from full wave simulations by varying the strip width. it is reasonable to state that the ratio \(Z_{0}/Z_{FSS}\) should be equal to 0 to achieve a transmission coefficient equal to 1. This is formalized in:  
\[\left|\frac{Z_{0}}{Z_{FSS}}\right|\ \to 0 \tag{3}\]  
Considering that the free space impedance is equal to \(Z_{0}=377\Omega\), then the interposed FSS must be an open circuit in the passing band, thus realizing a transparency zone. Generally, this open circuit condition is valid in small frequency intervals, resulting in narrow-band filters. Specifically, to have a transmission coefficient of at least \(-3\) dB, the acceptable \(|Z_{0}/Z_{FSS}|\) ratio must be at most 2, so that \(\tau\approx 0.5\). A very wide transmitting band can be accomplished when the unit-cell Q-factor is maximized throughout the overall band of interest, providing a \(|Z_{0}/Z_{FSS}|\) ratio equal or superior to 2. To this aim, the strip line width is a fundamental parameter to achieve Q-factor maximization. Therefore, it was altered to assess the bandpass filter's impedance variations and evaluate the Q-factor behavior.