## 2 Design, Simulation, and Theoretical Analysis
Meanwhile, the corresponding PCR is more than 99.32%. The energy conversion ratio (ECR) can be used to evaluate the energy loss and is defined as [42]:  
\[ECR={{r_{xy}}^{2}+{r_{yy}}^{2}}. \tag{3}\]  
As shown in Figure 2b, the ECR is very close to 100% in the frequency ranges of 9.38-13.36 GHz and 14.84-20.36 GHz. A proportion of the incident wave is dispersed because of the loss from the dielectric spacer, but the majority of the power is reflected. According to the above analysis, the bulk electromagnetic energy of the \(y\)-polarized incident waves is converted into \(x\)-polarized reflected waves in the two conversion frequency ranges. Therefore, the metasurface structure presented here works as a high-performance polarization converter in terms of both efficiency and bandwidth. Table 1 compares the proposed converter and previously reported dual-broadband polarization converters. In the comparison, the relative bandwidth of [46] is relatively large, and our polarization converter delivers superior performance in terms of the polarization conversion efficiency, which is very close to 100% and far higher than that reported by other researchers. To obtain insights into the operating principle of the proposed polarization converter, the \(x\)- and \(y\)-axes are rotated counterclockwise by 45\({}^{\circ}\) to obtain the \(u\)- and \(v\)-axes, respectively. As shown in Figure 3a, the \(y\)-polarized EM waves under regular incidence conditions can be decomposed into two perpendicular components (\(E_{itu}\) and \(E_{iv}\)) along the \(u\)-axis and the \(v\)-axis, respectively, and the illuminating EM waves can be expressed as [42]:  
\[\overrightarrow{E}_{i}=\frac{\sqrt{2}}{2}E_{i}\exp(j\mathrm{\textit{k}}z) \hat{u}+\frac{\sqrt{2}}{2}E_{i}\exp(j\mathrm{\textit{k}}z)\hat{v}. \tag{4}\]  
When an incoming wave is reflected by the polarization converter, the reflected EM wave can be described using the following expression [42]:  
\[\overrightarrow{E}_{r}=\frac{\sqrt{2}}{2}\Big{[}r_{uu}E_{i}e^{i(-kz+\varphi_{ uu})}+r_{tu}E_{i}e^{i(-kz+\varphi_{uv})}\Big{]}\hat{u}+\frac{\sqrt{2}}{2} \Big{[}r_{vv}E_{i}e^{i(-kz+\varphi_{uv})}+r_{uu}E_{i}e^{i(-kz+\varphi_{uu})} \Big{]}\hat{v} \tag{5}\]  
where \(r_{uu}\), \(r_{uv}\), \(r_{vv}\), and \(r_{uu}\) denote the amplitudes of the reflection coefficients of the \(u\)-to-\(u\), \(v\)-to-\(u\), \(v\)-to-\(v\), and \(u\)-to-\(v\) polarization conversions, respectively. In addition, \(\varphi_{uu}\), \(\varphi_{uv}\), \(\varphi_{vv}\), and \(\varphi_{uu}\) are the corresponding reflection phases. When \(r_{uv}=r_{uu}=0\), \(r_{uu}=r_{vv}=r\), and \(\Delta\varphi=\varphi_{uu}-\varphi_{vv}=2n\pi\pm\pi\)\((n\in Z)\), the reflected electric field can be expressed as:  
\[\overrightarrow{E}_{r}=\frac{\sqrt{2}}{2}rE_{i}\exp(-j\mathrm{\textit{k}}z) \{\exp(j\varphi_{uu})\hat{u}+\exp[j(\varphi_{uu}+2n\pi\pm\pi)]\hat{v}\}, \tag{6}\]  
and then a linearly polarized wave will be realized. Figure 3b shows that the values of \(r_{uu}\) and \(r_{vv}\) are approximately zero, demonstrating that the resonant ring structure has no polarization conversion effect and thus the reflected EM wave can be expressed using the same reflectance and phase. Figure 3c \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Ref [43]** & **Ref [44]** & **Ref [45]** & **Ref [46]** & **This Work** \\ \hline Operating & 5.50–8.94 & 6.99–9.18 & 11.50–20.0 & 5.1–11.9 & 9.38–13.36 \\ bandwidth (GHz) & 3.10–15.50 & 11.66–20.40 & 28.8–34.0 & — & 14.84–20.36 \\ Relative bandwidth (\%) & 51.19/16.78 & 27.09/54.52 & 53.97/16.56 & 117/– & 35.00/31.36 \\ PCR (\%) & 90/90 & 90/90 & 94/90 & 85/— & 98.21/99.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of our design with other polarization converters. Figure 2: **(a)** Simulated cross-polarization (\(r_{yy}\)) and co-polarization (\(r_{yy}\)) reflections of the designed polarization converter. **(b)** Spectra of the polarization conversion ratio (PCR) and energy conversion ratio (ECR).