## 3 Theoretical Analysis
 \tag{7}\]  
From Equation (7), the succeeding equation can be derived as:  
\[\frac{r_{yy}}{r_{xy}}=\frac{r_{xx}}{r_{yx}}=\frac{1+e^{-j\Delta\varphi_{uv}} }{1-e^{-j\Delta\varphi_{uv}}}=\frac{1+\cos(\Delta\varphi_{uv})-j\sin(\Delta \varphi_{uv})}{1-\cos(\Delta\varphi_{uv})+j\sin(\Delta\varphi_{uv})}=\frac{j \sin(\Delta\varphi_{uv})}{1-\cos(\Delta\varphi_{uv})} \tag{8}\]  
Moreover, using Equation (8), the magnitudes of co-polarized reflection coefficients \((r_{xx},\ r_{yy})\) and cross-polarized reflection coefficients \((r_{xy},\ r_{yx})\) can be written as follows:  
\[|r_{yy}|=|r_{xx}|=\frac{1}{2}|r_{uu}|\big{|}1+e^{-j\Delta\varphi_{uv}}\big{|} =\frac{1}{2}|1+\cos(\Delta\varphi_{uv})-j\sin(\Delta\varphi_{uv})|=\sqrt{(1+ \cos\Delta\varphi_{uv})/2}\]  
\[|r_{xy}|=|r_{yx}|=\frac{1}{2}|r_{uu}\big{|}1-e^{-j\Delta\varphi_{uv}}\big{|} =\frac{1}{2}|1-\cos(\Delta\varphi_{uv})+j\sin(\Delta\varphi_{uv})|=\sqrt{(1- \cos\Delta\varphi_{uv})/2} \tag{9}\]  
The reflection coefficients of \(r_{xx},r_{yy}\) and \(r_{yx},r_{xy}\)-polarization with \(x\)- and \(y\)-polarized incidence waves are pure imaginary numbers. It is always supposed that the \(\Delta\varphi_{yx}\) between them is \(\pm 90^{\circ}\). It clarifies that in Figure 2b, the \(\Delta\varphi_{yx}\) is always \(-90^{\circ}\) for the dual-frequency bands of 15-24.22 GHz and 25.50-37.0 GHz. When the \(\Delta\varphi_{uv}\
eq 0^{\circ}\) and \(\pm 180^{\circ}\), the polarization reflection coefficients of the \((r_{yy},r_{xx})\) and \((r_{xy},r_{yx})\) will not be zero, resulting inan elliptical polarized reflected wave, with a phase difference between the \((r_{yy},r_{xx})\) and \((r_{xy},r_{yx})\) remaining \(\pm 90^{\circ}\). The 3 dB axial ratio of the reflected wave using the magnitude ratio of \((r_{yy},r_{xy})\) and \((r_{xx},r_{yx})\) can be found from [7] and is given below:  
\[\text{AR}=\left\{ \begin{array}{ll}\frac{|\sin(\Delta\varphi_{uv})|}{1-\cos( \Delta\varphi_{uv})}&if\ |\sin(\Delta\varphi_{uv})|\ \geq 1-\cos(\Delta\varphi_{uv})\\ \frac{1-\cos(\Delta\varphi_{uv})}{|\sin(\Delta\varphi_{uv})|}&if\ |\sin(\Delta \varphi_{uv})|<1-\cos(\Delta\varphi_{uv})\end{array} \right. \tag{10}\]  
Moreover, Equation (8) has been applied to the relationship \(\frac{r_{yy}}{r_{xy}}=\frac{r_{xx}}{r_{yx}}\), indicating that the ratio of the cross (\(y\)- to -\(x\) and \(x\)- to -\(y\))-polarization reflection coefficient under \(y\)- and \(x\)-polarized incidences is always equal, which concludes that an isotropic metasurface converts a normal incidence \(y\)-polarized wave to right- and left-hand circular polarized reflected waves. When \(\Delta\varphi_{uv}=0^{\circ}\), \(|r_{yx}|=|r_{xy}|=0\), Equation (9) implies that the polarization condition of the incident and reflected wave will be equal. When \(\Delta\varphi_{uv}=180^{\circ}\), however, \(|r_{yy}|=|r_{xx}|=0\) to achieve a perfect cross-polarization. Additionally, when \(\Delta\varphi_{uv}\
eq 0^{\circ}\) or \(\pm 180^{\circ}\), the reflected wave will be elliptically polarized, and the reflected wave's axial ratio can be determined using the succeeding equation:  
\[\text{AR}=\sqrt{(1\pm\ \cos\Delta\varphi_{uv})/(1+\cos\Delta\varphi_{uv})} \tag{11}\]  
Here, the (\(+\)) and (\(-\)) signs ensure that the axial ratio will never be less than 1.0, according to Equation (11). The axial ratio of the reflected wave is equal to 1.0 at \(\Delta\varphi_{uv}=90^{\circ}\), which indicates a perfect LTC reflective polarization converter, according to Equations (10) and (11). When \(\Delta\varphi_{uv}=90^{\circ}\), the perfect LTC reflective polarization conversion can be realized. The incident wave and reflected wave under \(x\)- and \(y\)-polarization can be considered a composite wave with equal magnitude \(u\)- and \(v\)-polarized components to negligible dielectric loss. Both orthogonal components appear to be in phase with one another in the incident wave, but the \(\Delta\varphi_{uv}\) between them will change in the reflected wave.