## 2 Metasurface Design and Simulation  
The schematic design of the proposed converter is presented in Figure 1. The proposed converter is mainly composed of a rectangular patch embedded with a rectangular slot and with two radial stubs separated from a ground plane by a dielectric spacer Roger RT5880 with the property of \(\epsilon_{r}=2.2\), \(h=1.6\) mm and tangent loss of tan \(\delta=0.0009\). The metallic ground plane and proposed unit cell are modeled using a copper sheet with an electrical conductivity of \(\sigma=5.8\times 10^{7}\) S/m and a thickness of 0.017 mm. The other physical parameters are tabulated in Table 1. According to Figure 1, the rectangular slot is inserted in the middle of the rectangular patch at a 45\({}^{\circ}\) rotational angle. We execute a series of simulations using CST Microwave Studio to evaluate the dual-wideband reflective linear to-circular polarization converter efficiency. The XY boundary conditions are constrained by the unit cell, and a Floquet port is excited on the \(z\)-axis. Figure 2 shows the simulation results when a normal-incidence \(y\)- or \(x\)-polarized wave is set as an input. The reflection coefficient's magnitudes of the cross- and co-polarization are \(r_{xy}\) and \(r_{yy}\), associated with a phase difference. The phase difference (\(\Delta\varphi_{yx}\)) between \(r_{yy}\) and \(r_{xy}\) exists due to the anisotropic features of the metasurface. An efficient reflective LTC polarization converter can be designed if \(r_{yy}=r_{xy}\) and \(\Delta\varphi_{yx}=\varphi_{yy}-\varphi_{xy}=\pm 90^{\circ}=2k\pi\pm\pi/2\) (\(k=0,\,1,\,2\,\dots\)), where the sign (\(\pm\)) represents the RHCP (\(-\)) and LHCP (\(+\)) of the reflected wave. According to Figure 2a, the magnitudes of \(r_{yy}\) and \(r_{xy}\) of the dual-band LTC polarization converter based on the rectangular slot reflective metasurface are nearly equal over the frequency ranges of 17.67-21.14 GHz and 28.36-32.91 GHz. According to Figure 2b, the phase difference between the 15-24.22 GHz and 25.50-37 GHz frequency bands is always close to \(-90^{\circ}\), and satisfies the phase relation (\(\Delta\varphi_{yx}=\varphi_{yy}-\varphi_{xy}=\pm 90^{\circ}=2k\pi\pm\pi/2\), \(k=0,\,1,\,2\,\dots\)). The reflected wave can be considered circularly polarized if its axial ratio is below 3 dB. To calculate the bandwidth of the anticipated dual-wideband polarization conversion, we computed the axial ratio of the reflected wave from the succeeding equation, using the data set presented in Figure 2[7]. \[\text{AR}=\sqrt{\frac{\left|r_{yy}\right|^{2}+\left|r_{xy}\right|^{2}+\sqrt{ \left|r_{yy}\right|^{4}+\left|r_{xy}\right|^{4}+2\left|r_{yy}\right|^{2}\left| r_{xy}\right|^{2}\cos\left(2\Delta\varphi_{yx}\right)}}{\left|r_{yy}\right|^{2}+ \left|r_{xy}\right|^{2}-\sqrt{\left|r_{yy}\right|^{4}+\left|r_{xy}\right|^{4}+ 2\left|r_{yy}\right|^{2}\left|r_{xy}\right|^{2}\cos\left(2\Delta\varphi_{yx} \right)}}} \tag{1}\]  
The calculated axial ratio of the proposed converter exists below 3 dB over the dual-frequency wideband of 16.49-23.54 GHz and 26.44-34.56 GHz, as presented in Figure 3a. The 3 dB axial ratio bandwidth within dual-frequency wideband of 16.49-23.54 GHz and 26.44-34.56 GHz is 35.23% and 26.62%, respectively. Additionally, the axial ratio is kept under 1.5 dB, demonstrating that the polarization conversion ratio of the polarization \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(p\) & \(r\) & \(w\) & \(l\) & \(l_{e}\) & \(w_{e}\) \\ \hline
6 & 0.86 & 0.58 & 4.6 & 4.6 & 0.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimized value (in mm) of the reflective metasurface unit cell. Figure 1: Unit cell of the proposed reflective converter. (**a**) Top view of the rectangular-slot polarization converter with physical parameters and (**b**) side view with substrate properties. converter is above 99%.