## 5 Theoretical Analysis  
To evaluate the polarization conversion mechanism of the proposed structure the \(u\)- and \(v\)-axes are initiated, as illustrated in Figure 7. The proposed structure is explained by reinterpreting the \(xoy\) as the \(uov\) coordinate system, by rotation 45\({}^{\circ}\) anticlockwise. As illustrated in Figure 7, the \(u\)- and \(v\)-axes are \(\pm 45^{\lx@math@degree}\) with respect to the \(y\)-axis, respectively, and the \(y\)-polarized incident wave can be decomposed into \(u\) and \(v\) components:  
Figure 6: The simulated results for the parametric variations: (black dashed line indicates 3 dB line) (**a**) AR (h = 1.55, 16.0, 1.65 mm); (**b**) incident angle (\(\theta=0^{\lx@math@degree}\) to \(30^{\lx@math@degree}\)); (**c**) material properties (\(\varepsilon_{r}\) = 2.0, 2.1 and 2.2); (**d**) phase difference (h = 1.55, 1.60, 1.65 mm). Figure 7: Decomposition of \(UV\) polarized geometrical configuration. \[ \begin{array}{l}\overset{\rightarrow}{E}_{i}=\hat{y}\overset{\rightarrow}{E}_{i \eta}e^{jkz}=\left(\hat{u}E_{u}^{i}+\hat{v}E_{v}^{i}\right)e^{jkz}\\ =\left(\sqrt{2}/2\right)E_{y}^{i}(\hat{u}+\hat{v})e^{jkz}\end{array}  \tag{9}\]  
So, the equation for the reflected waves is as follows:  
\[ \begin{array}{l}\overset{\rightarrow}{E}_{r}=(\hat{u}E_{u}^{r}+\hat{v}E_{r }^{r})e^{-jkz}=\left(\hat{u}r_{u}E_{u}^{i}+\hat{v}r_{v}E_{v}^{i}\right)e^{-jkz} \\ =\left(\sqrt{2}/2\right)E_{y}^{i}(\hat{u}|r_{u}|e^{j\varphi_{u}}+\hat{v}|r_{v}| e^{j\varphi_{v}})e^{-jkz}\end{array}  \tag{10}\]  
where \(r_{u}\) and \(r_{v}\) stand for the reflection coefficients of the co-polarized wave for the \(u\)- and \(v\)-polarized incidence waves, and the corresponding reflection phases of the reflected waves are indicated by \(\varphi_{u}\) and \(\varphi_{v}\). The phase difference between them is the main factor in the polarization conversion. The metasurface symmetry along the \(u\)- and \(v\)-axes results in extremely low cross-polarization and significant co-polarization reflection. The phase difference is typically described as \(\Delta\varphi_{uv}=\varphi_{u}-\varphi_{v}\). Reflected electromagnetic waves of various polarization types can be obtained when the phase and magnitude of the reflected wave along the \(u\)- and \(v\)-axes satisfy the requirements. The following equation represents condition 1:  
\[|r_{u}|\approx|r_{v}|\text{ and }\Delta\varphi_{uv}=180^{\circ}\pm n\pi \tag{11}\]  
where \(n\) is a constant, and the reflected wave is an LP wave. The following equation expresses condition 2:  
\[|r_{u}|\approx|r_{v}|\text{ and }\Delta\varphi_{uv}=\pm 90^{\circ}\pm 2n\pi \tag{12}\]  
where Equation (12) represents a CP reflected wave. Figure 8a,b display the amplitude of the reflection coefficients, and the reflection phase curves of the proposed metasurface for the \(u\)- and \(v\)-polarized incidence waves, whereas Figure 8c shows the phase difference curve. In the distinct frequency ranges of 10.09-11.83, 16.34-23.90, 28.87-31.72, and 33.68-38.69 GHz, as anticipated, \(|r_{u}|\approx|r_{v}|\), and the phase difference is approximately \(\pm 180^{\circ}\). It satisfies the above-analyzed Equation (11) and accomplishes the conversion of the \(y\)-polarized wave into the \(x\)-polarized wave, or CPC. The conversion from an LP wave to a CP wave is achieved in the distinct frequency ranges of 10.01-11.59, 15.33-25.15, 27.18-32.15, and 33.03-38.56 GHz; where \(|r_{u}|\approx|r_{v}|\), the phase difference is approximately an odd integral multiple of \(90^{\circ}\), as shown in Figure 8c. Finally, the AR of the reflected wave is calculated from the above data following the phase difference (\(\Delta\varphi_{uv}\)) obtained in Figure 8c, and the resulting findings are displayed in Figure 8d. The derived AR essentially agrees with one of the phase differences (\(\Delta\varphi_{yx}\)), as shown in Figure 8d.