## 2 Operation Principles and Analysis Methods
### Sensitivity Analysis and General Design Flowchart  
The impact of the geometric parameters of the FSS on the resonant frequencies, transmission zeros, and bandwidths is illustrated in Figure 9. Figure 9a demonstrates that when the sum of \(l_{x}\) and \(l_{y}\) is held constant, increasing the value of \(l_{x}\) results in a reduction in the frequency ratio between the two passbands, which aligns with the previous analysis. As shown in Figure 9b, the size of the coupling apertures \(a_{x}\) and \(a_{y}\) primarily affects the coupling amount of the two U-shaped rings, i.e., the passband bandwidth and the TZ positions. To examine the design sensitivity to the incident angle, Figure 9c,d displays the transmission coefficients of the FSS under varying incident angles in the \(xoz\) and \(yoz\) planes, respectively. In the \(yoz\) plane, the transmission response remains stable for incident angles ranging from 0 to 50 degrees. When the incident vector is in the \(xoz\) plane, the TM incident wave activates the even mode, leading to dual-band behavior. Nevertheless, the in-band cross-polarization suppression in the lower passband still exceeds 13 dB even at an incident angle of 30 degrees. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{TE Polarization} & C/L 1  & \(C_{u1}\) & \(C_{u2}\) & \(C_{t}\) & \(C_{m}\) & \(L_{g}\) & \(L_{t}\) & \(L_{u}\) & \(L_{v}\) \\ \cline{2-10}  & Value (pF/nH) & 0.093 & 0.511 & 0.008 & 0.38 & 0.125 & 0.638 & 8.91 & 3.44 \\ \hline \multirow{2}{*}{TM Polarization} & C/L & \(C_{u1}\) & \(C_{u2}\) & \(C_{t}\) & \(C_{m}\) & \(L_{g}\) & \(L_{t}\) & \(L_{u}\) & \(L_{v}\) \\ \cline{2-10}  & Value (pF/nH) & 0.061 & 0.170 & 0.008 & 0.05 & 0.028 & 0.638 & 8.04 & 3.44 \\ \hline \end{tabular}
\end{table} 
Table 4: Parameters of the equivalent circuit models. Figure 9: The effect of the geometric parameters and incident angle on FSS transmission magnitudes. (**a**) Different \(l_{x}\) while the sum of \(l_{x}\) and \(l_{y}\) is fixed. (**b**) Different \(a_{x}\) and \(a_{y}\). (**c**) Different incident angles in \(xoz\) plane. (**d**) Different incident angles in \(yoz\) plane. Figure 10 provides general design guidelines for the proposed dual-frequency and polarization-selective meta-surfaces. Detailed design steps of the proposed X-band FSS are given here:  
Step 1: The center frequencies of the dual frequencies are set to 9.7 GHz for TE and 10.7 GHz for TM, which corresponds to an approximate frequency ratio of 1.1. Considering that the adopted resonant cell is based on a microstrip U-ring structure, the achievable 3-dB fractional bandwidth (FBW) is between 3% and 7%; therefore, the target bandwidth requirement is set to 5%. Step 2: Here, we utilize Taconic RF-35 substrates. The relative permittivity is set to 3.5, and the single-layer thickness is 0.508 mm. To reduce the errors introduced by finitely periodical boundaries and to improve the stability under oblique incidence, the recommended cell size is between one-third and one-half wavelength. Therefore, in this case, both \(P_{x}\) and \(P_{y}\) are set to 5 mm. Step 3: Initialize \(l_{c}\) and \(w\) to the same value, 0.8 mm. This value can be initially set to the width corresponding to a 50-ohm microstrip line and optimized for a range of approximately 20 to 120 ohns when input coupling is considered. To satisfy the processing conditions, the grounded via radius is set to 0.2 mm. It is initially assumed that the arm has the same length as the branch, that is, \(l_{y}=2l_{x}\). According to Equation (3), the value of \(l_{y}\) is approximately 3.33 mm. Then, minor optimization in HFSS is used to maintain the center frequencies at 9.7 GHz and 10.7 GHz, respectively. The optimized parameters, which are all close to their initial values, are \(l_{y}=2l_{x}=3.04\) mm, \(w=0.9\) mm, and \(l_{c}=0.8\) mm. Step 4: Build the whole meta-surface model shown in Figure 6 with the values obtained in step 3. The initial coupling aperture size is set to \(a_{x}=2a_{y}=w\).