## 2 Operation Principles and Analysis Methods
### Dual Modes U-Shaped Resonator with One Grounded Metallic Via  
Figure 1a depicts the fundamental U-shaped resonator structure used in this article. The full-wave analysis of this structure is performed using the High-Frequency Structure Simulator (HFSS). Using the periodic boundary condition (PBC) in HFSS, the electric field of the TM polarization incident Floquet mode is along the positive \(x\)-axis, whereas the electric field of the TE polarization is along the positive \(y\)-axis. Due to its wide frequency response, the U-shaped resonant ring composed of a microstrip line structure has been widely used in antenna and feeding structure design over the past few decades [35; 36; 37; 38]. Nonetheless, its application in FSS and PSS has received limited consideration [39; 31; 40]. Figure 1b demonstrates that the metallic portion of the U-shaped ring consists of two open-ended arms and a stub with a long edge along the \(y\)-axis. The specific geometric dimensions of the adopted U-shaped resonator are detailed in Table 1. Our designs employ Taconic RF-35 substrates with a dielectric constant of 3.50 and a loss tangent of 0.002. Figure 2a illustrates the relationship between the reflection coefficient of a U-shaped resonator without a grounded via and the operating frequency for TE and TM polarized incident waves. The metallic surface current distributions under different incident polarizations are depicted in Figure 2b,c, respectively. When the incident polarization is TM, the incident electric field is along the \(x\)-axis direction, which results in an even-mode current flowing along the stub and two anti-phase currents in the two arms. On the other hand, under TE polarization, the incident electric field is along the \(y\)-axis direction, which excites two in-phase currents in the two arms and an odd-mode distribution along the stub. Therefore, the transmission line model depicted in Figure 3 is applied to analyze the response of this structure. The characteristic impedance of the microstrip line, where the source line width is \(l_{c}\) and \(w\), is denoted as \(Z_{stub}\) and \(Z_{arm}\)[41; 42], respectively. The propagation constants corresponding to these transmission lines are denoted as \(\beta_{stub}\) and \(\beta_{arm}\). A \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Parameter** & \(\mathbf{l_{x}}\) & \(\mathbf{l_{y}}\) & \(\mathbf{l_{c}}\) & \(\mathbf{P_{x}}\) & \(\mathbf{P_{y}}\) & \(\mathbf{h}\) & \(\mathbf{w}\) & \(\mathbf{r_{p}}\) \\ \hline Value (mm) & 1.45 & 3.22 & 0.8 & 5.0 & 5.0 & 0.508 & 0.9 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric parameters of the adopted U-shaped unit cell. Figure 1: Structure of the U-shaped resonator used in our design. **(a)** The three-dimensional (3-D) structure and simulation setup in HFSS. **(b)** The top view and relevant parameter notations. short transmission line with a length of \(l_{open}\) is introduced as the equivalent compensation amount due to the open-ended condition [42]. The influence of bending and chamfering is disregarded here and can be corrected with minor optimization later on. For the TM polarization, the symmetry plane \(AA_{1}\) is the equivalent magnetic wall and the resonance condition can be expressed as follows [42]:  
\[\frac{Z_{arm}}{\tan\left(\frac{l_{y}+l_{open}}{\beta_{arm}}\right)}-Z_{stub} \tan\left(\frac{l_{x}}{\beta_{stub}}\right)=0 \tag{1}\]  
As for the TE polarization, the symmetry plane \(AA_{1}\) is the equivalent electric wall and the resonance condition is:  
\[Z_{arm}\tan\left(\frac{l_{y}+l_{open}}{\beta_{arm}}\right)-Z_{stub}\tan\left( \frac{l_{x}}{\beta_{stub}}\right)=0 \tag{2}\]  
In the frequency range of 7 to 13 GHz, as depicted in Figure 2a, this structure exhibits a resonance characteristic at 10.2 GHz that is exclusive to the TM polarization, as confirmed by Equation (1).