## 2 Operation Principles and Analysis Methods
### Frequency Selection and Cross-Polarization Suppression Introduced by Coupling Apertures  
To meet the requirements for dual polarization isolation and signal bandwidth in wireless communication systems, the structure under discussion must incorporate multiple resonances and cross-coupling structures to achieve a flat transmission coefficient in the passband and multiple TZs outside of the passband. Figure 6 depicts the structure proposed in this paper with dual-frequency polarization selection characteristics. Two Taconic RF-35 substrates with a shared ground layer compose this structure. Each layer possesses a U-shaped, uniformly sized resonant ring that is grounded. On the ground layer, two identical coupling apertures are etched close to the open ends of the arms, where the electric field  
Figure 5: Full-wave simulation validation of the parametric study. (**a**) Resonance frequency tuning by adjusting the stub length \(l_{x}\). (**b**) Resonance frequency tuning by adjusting the arm length \(l_{y}\). (**c**) Resonance frequency ratio tuning by adjusting the width of the stub and arms. (**d**) TE resonance frequency tuning by adjusting the via radium. is at its strongest. In both the _xoy_ and _yoz_ planes the entire structure is symmetrical. The addition of the coupling aperture generates dual-port transmission characteristics and two additional out-of-band TZs. The physical parameters of this structure are listed in Table 2.  
Table 3 presents the electric field and surface current distributions of the U-shaped resonator and ground under the TE wave incidence at 9.54 GHz and the TM wave incidence at 10.56 GHz, respectively. Under TE polarization, an odd-mode resonance is observed, whereas under TM polarization, an even-mode resonance is supported. The equivalent circuit model with a uniform topology, as shown in Figure 7, can therefore be utilized to analyze the design in both polarizations. In Figure 7, all material-caused losses are disregarded, and the free space is represented by two infinite transmission lines with a characteristic impedance of \(Z_{0}=377\)\(\Omega\). Each dielectric substrate is equivalent to a short transmission line with a characteristic impedance of \(Z_{s}=Z_{0}/\sqrt{\varepsilon_{r}}\) and a length of \(l_{s}=h\), where \(\varepsilon_{r}=3.5\). The U-shaped resonator is replaced by the series connection of an inductance \(L_{u}\) and two capacitors \(C_{u1}\) and \(C_{u2}\). The parallel combination of an inductance \(L_{a}\) and a capacitance \(C_{a}\) represents the equivalent elements of the two symmetrical apertures for coupling. The capacitance \(C_{m}\) represents the electrical coupling between the two U-shaped resonators, whereas the coupling between the U-shaped resonator and the ground is relatively small and insignificant. In addition, metallic \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Parameter** & \(l_{x}\) & \(l_{y}\) & \(l_{c}\) & \(P_{x}\) & \(P_{y}\) & \(\varpi\) & \(r_{\varphi}\) & \(l_{a}\) & \(a_{x}\) & \(a_{y}\) \\ \hline Value (mm) & 1.45 & 3.22 & 0.8 & 5.0 & 5.0 & 0.9 & 0.2 & 3.19 & 1.8 & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometric parameters of the double layers FSS. Figure 6: The proposed FSS unit cell. (**a**) The three\(-\)dimensional structure. (**b**) The top layer. (**c**) The ground plane with two etched apertures. Since the equivalent circuit is symmetrical at the \(A_{1}A_{2}\) plane, even-odd-mode analysis can be utilized once more. Significantly, each short transmission line can be replaced by a series inductor with a value of \(L_{t}=\mu_{0}\mu_{t}h\) and a shunt capacitor with a value of \(C_{t}=\varepsilon_{0}\varepsilon_{t}h\), where \(\mu_{0}\) and \(\varepsilon_{0}\) are the permeability and the permittivity of the free space, respectively, and \(\mu_{r}\) is the relative permeability of the substrate. For the even-mode resonance, an equivalent magnetic wall is inserted at the symmetry plane \(A_{1}A_{2}\).