## 2 Operation Principles and Analysis Methods
### Sensitivity Analysis and General Design Flowchart
Then, the element values in the equivalent circuit are fitted to the full wave simulation results. In practice, only the value of \(C_{a}\), \(L_{a}\), and \(L_{g}\) must be fitted. The value of the remaining elements can be calculated from the transmission line models or resonance models. Due to the introduction of the cross-coupling, the self-resonant frequency of the resonator will be affected, and a small correction to \(C_{u}\) and \(L_{u}\) is required. During our design, this correction range is within 25%. Step 5: Modify the simulation model based on the equivalent circuit obtained in step 4. Adjust the values of \(l_{x}\) and \(l_{y}\) for resonance tuning, as well as the values of ax and ax for  
Figure 10: General flowchart of the proposed frequency\(-\) and polarization\(-\)selective surfaces design procedure. the coupling amount, i.e., the values of \(C_{a}\), \(L_{a}\), and \(L_{g}\). Numerical optimization is possible using Equations (7) and (8). Four or five iterations for each polarization will bring the simulation results closer to the desired value. Step 6: Minor parameter optimization in HFSS, where the value ranges are all within 3%, is carried out to ensure good in-band flatness and the required bandwidth. The final physical dimensions and elements values have been listed in Tables 2 and 4, respectively. To further demonstrate this design method, additional simulation results for different operating frequency bands are presented in Figure 11. In both of these design examples, the topology structure shown in Figure 6 and the equivalent circuits shown in Figure 7 are still used, and the dimensions corresponding to these structures are listed in Table 5. For the S-band design shown in Figure 11a, the center frequencies are approximately 2.25 GHz and 2.55 GHz, while the polarization isolation is around 77 dB and 75 dB, respectively. The center frequencies and polarization isolation for the K-band design depicted in Figure 11b are approximately 18.7 GHz and 20.7 GHz, and 46 dB and 43 dB, respectively. Noting that the insertion losses in these designs at S-, X-, and K-bands are all less than 0.4 dB, the proposed polarization and frequency selective meta-surfaces are applicants for use in modern wireless communication systems. Far-field conditions are ensured by the separation between the transmitting antenna and the prototype. To calibrate the measured results, the transmission coefficients without FSS under various incident angles are first measured. Figure 13 presents the transmission coefficients measured in _xoz_ and _yoz_ planes. The center frequencies of the two measured passbands are 9.68 GHz and 10.73 GHz, with the worst insertion losses of 0.9 dB and 1.1 dB, respectively. The 3-dB FBW of each passband is around 3.5%, and the minimum and maximum in-band cross-polarization suppressions under normal incidence are 30.4 dB and 41.7 dB, respectively. Due to the TZs introduced on both sides of the passband, the out-of-band rejection reaches 20 dB for both polarizations. Importantly, the response of the prototype remains stable with the incident angle varying within the range of 0\(\sim\)30 degrees in the _xoz_ plane and 0\(\sim\)50 degrees in the _yoz_ plane. Note that the measured operating frequency, the insertion loss, and the depth of transmission zeros are all slightly higher than their simulated values. This discrepancy between simulation and measurement may be attributable to the substrate dielectric constant imprecision, fabrication error, and measurement error. Table 6 shows the comparison between the fabricated design and the dual-band PSS described in other published works. The structure proposed in this paper has a low profile and design difficulty, a small frequency ratio, high cross-polarization isolation, and out-of-band suppression in comparison to existing literature. Figure 12: The measurement of the fabricated X-band design. (**a**) The FSS prototype with the dimension listed in Table 2. (**b**) The measurement setup.