## 2 Design, Simulation, and Theoretical Analysis  
Figure 1 shows the schematic diagram of the unit cell structure of the designed polarization converter with a three-layered metal-dielectric-metal structure. The top metallic structure consists of two symmetrical double slot metallic open rings and one criss-cross structure, which is placed 45\({}^{\circ}\) forward with respect to the \(+x\)-axis. The dielectric spacer is F4B-2 (\(\varepsilon_{r}=2.65-0.002\)j) [38] with a thickness \(t_{\mathrm{s}}\) of 3 mm and the bottom layer is a metal plate. The two metal layers are composed of copper films with electrical conductivity of \(5.8\times 10^{7}\) S/m [38] and a thickness of 35 \(\upmu\)m. The optimized geometrical parameters are as follows: \(a=1.5\) mm, \(b=1\) mm, \(R=4.32\) mm, \(r=3.5\) mm, \(w_{1}=0.19\) mm, \(w_{2}=0.44\) mm, \(p=10\) mm, and \(\theta=110^{\circ}\). Figure 1: Schematic diagram of the designed linear polarization converter (yellow: metal; blue: dielectric spacer material). The design is simulated using the RF-Module software from Comsol Multiphysics [39], which enables the use of the finite element method (FEM). The periodic boundary conditions of a unit element are used to simulate an infinite periodic array on the \(x\)-\(z\) and \(y\)-\(z\) planes. Because of the geometrical symmetry, the reflection coefficients of cross-polarization and co-polarization of the \(y\)-polarized incident waves are similar to those of the \(x\)-polarized incident waves. Therefore, the reflection coefficients for the \(y\)-polarized incoming wave are studied. The wave vector \(k\) is oriented in the negative direction along the \(z\)-axis. Additionally, the effective surface current excited by a Gaussian beam impresses onto the metasurface and allows radiation using Comsol Multiphysics [39]. The reflected wave is composed of both cross- and co-polarized components because of the anisotropy of the unit cell structure. Co-polarization indicates that the polarization direction of the reflected wave coincides with that of the incident wave, while cross-polarization indicates that the polarization direction of the reflected wave is perpendicular to that of the incident wave. The definition of the reflection coefficient in this study is as follows [1]:  
\[r_{yy}=\frac{\left|E_{yr}\right|}{\left|E_{yi}\right|},r_{xy}=\frac{\left|E_{ xr}\right|}{\left|E_{yi}\right|} \tag{1}\]  
where \(r_{yy}\) and \(r_{xy}\) represent the reflection coefficients of the co-polarization (\(y\)-to-\(y\)) and the cross-polarization (\(y\)-to-\(x\)), respectively. \(\left|E_{yr}\right|\) and \(\left|E_{yi}\right|\) are the magnitudes of the reflected and incident wave electric fields in the \(y\)-direction, respectively. \(\left|E_{xr}\right|\) is the magnitude of the reflected wave along the \(x\)-axis in the polarization direction. In addition, the PCR can be used to describe the polarization converter performance and is defined as follows [40]:  
\[PCR=\frac{{{r_{xy}}^{2}}}{{{r_{xy}}^{2}+{r_{yy}}^{2}}} \tag{2}\]  
where \(r_{xy}\) and \(r_{yy}\) are the reflection coefficients of the cross-polarization and the co-polarization, respectively. If the PCR is equal to 1, complete conversion between the linear \(x\)- and \(y\)-polarized EM waves is realized [41]. Figure 2 shows the simulated amplitude reflection spectra of the cross-polarization \(r_{xy}\) and co-polarization \(r_{yy}\) as well as the PCR in the 6.5-20.8 GHz frequency range. Figure 2a shows that the designed polarization converter has two conversion frequency ranges. The co-polarization reflectance \(r_{yy}\) is less than \(-\)10 dB and the cross-polarization reflectance \(r_{xy}\) is more than \(-\)1 dB in the frequency range between 9.38 and 13.36 GHz. Accordingly, there are two resonance frequencies at 10.67 and 12.22 GHz in the same frequency range. The PCR in the first frequency band is almost 98.21%, as determined from Figure 2b. The other frequency band is between 14.84 and 20.36 GHz and two resonance frequencies (16.18 and 19.51 GHz) exist in this band.