# Resonant Coupling Effects in a Double-Layer THz Bandpass Filter
## 2 Theory and Experiments  
The double-layer FSS structure consists of periodically arranged cross apertures perforated on 10 \(\upmu\)m Al foils as shown in Figure 1a. Two FSS layers are stacked parallel with no dielectric substrate in between and without lateral shift. Structural parameters of the cross unit include arm length \(l\), arm width \(w\), thickness \(t\), period \(p\), and double-layer spacing \(d\). The equivalent inductance \(L\) and capacitance \(C\) are different for FSSs with different shapes and structural parameters. For some commonly used FSS elements, such as rectangles, holes, squares, crosses, etc., the values of \(L\) and \(C\) can be calculated using theoretical formulas [26; 27; 28]. However, the accuracy of \(L\) and \(C\) values obtained using formulas is low, and the error is magnified further after multi-layer stacking. Therefore, we used an inversion method, which employs FDTD (FDTD Solutions, Lumerical Inc., Vancouver, BC, Canada) to calculate the equivalent \(L\) and \(C\) of the single-layer FSS, in order to concentrate on the coupling effects of the multi-layer stacking. In the FDTD simulation, we used periodic boundary conditions for X and Y directions, and perfect matching layers along the \(Z\) direction. The material is Al with a plasma frequency of \(2.24\times 10^{16}\) rad/s, and a damping coefficient of \(1.12\times 10^{14}\) rad/s. The polarization of the terahertz waves is along the X direction. Firstly, the transmission coefficient and reflection coefficient of the single-layer FSS were calculated by FDTD. Because the single-layer FSS is a reciprocal symmetric network, we have S\({}_{12}\) = S\({}_{21}\) (transmission coefficient), S\({}_{11}\) = S\({}_{22}\) (reflection coefficient). In this way, all the elements in the S matrix of the single-layer FSS can be obtained. Then, the impedance matrix of single-layer FSS was calculated by Equation (1) according to the conversion relationship of the parameters for a two-port network [29; 30]:  
\[Z_{FSS}=Z_{0}\frac{(1+S_{11})(1-S_{22})+S_{12}S_{21}}{(1-S_{11})(1-S_{22})-S_{ 12}S_{21}} \tag{1}\]  
Figure 1: **(a)** Schematics of the double-layer cross frequency selective surface (FSS) unit structure. \(l\), \(w\), \(t\), \(p\), and \(d\) represent the arm length, arm width, thickness, period, and spacing of the double-layer FSS structure, respectively; **(b)** equivalent circuit model of the double FSS structure. where \(Z_{0}\) is the characteristic impedance of free space. For a slot structure, it is equivalent to the inductance \(L\) and capacitance \(C\) connected in parallel. The admittance can be expressed as  
\[Y_{FSS}=\frac{1}{Z_{FSS}}=\mathrm{j}2\pi fC+\frac{1}{\mathrm{j}2\pi fL} \tag{2}\]  
Assuming that the values of inductance \(L\) and capacitance \(C\) are stable within the frequency range, by selecting two frequency points \(f_{a}\) and \(f_{b}\) in the sub-wavelength region (\(\lambda>p\)) and using Equation (2), the equivalent parameters \(L\) and \(C\) of the circuit can be calculated as follows:  
\[L=\frac{(f_{b}/f_{a}-f_{a}/f_{b})}{j2\pi[f_{b}Y_{FSS}(f_{a})-f_{a}Y_{FSS}(f_{b} )]} \tag{3}\]  
\[C=\frac{(Y_{FSS}(f_{a})/f_{b}-Y_{FSS}(f_{b})/f_{a})}{j2\pi(f_{a}/f_{b}-f_{b}/f _{a})} \tag{4}\]  
By cascading the \(LC\) oscillation circuits corresponding to the upper and lower layers, we obtained the ECM of the double-layer FSS as shown in Figure 1b. One can see that at \(Z_{inp1}\) port, the electromagnetic wave passes through only one FSS layer and the interlay air. At \(Z_{L2}\) port, the electromagnetic wave has passed through two FSS layers, and \(Z_{L2}\) contains the coupling effect between the two FSS layers.