## 3 Theoretical Analysis  
The anticipated reflective converter performs a dual-frequency wideband LTC polarization conversion with high efficiency and as an ultrawideband. In the following analysis, we conduct a thorough theoretical investigation to achieve a complete physical configuration of the fundamental root. There would not be any cross-polarized reflecting factors under \(u\)- and \(v\)-polarized waves because it is a symmetric configuration with two mutually orthogonal symmetric axes. The reflection matrix (\(\mathbf{R}_{\text{lin}}\)) is due to a linearly polarized (LP) incident wave in the \(uv\)-local coordinate system, and can be written as [7]:  
\[\mathbf{R}_{\text{lin}}= \begin{pmatrix}r_{uu}&0\\ 0&r_{vv}\end{pmatrix}  \tag{6}\]  
Since the metasurface structure is anisotropic, the reflection coefficients of the co-polarization, i.e., \(r_{uu}\), \(r_{vv}\), will be independent of one another, but the magnitudes of co-polarization reflection coefficients will be relatively near to 1.0 because of the minimal dielectric loss. Thus the succeeding equation can be acknowledged without considering the minimal dielectric loss: \(r_{vv}=r_{uu}e^{-j\Delta\varphi_{uv}}\). The reflection coefficients \(r_{uu}\) and \(r_{vv}\) have a phase difference which can lie between \(-180^{\circ}\) and \(+180^{\circ}\). Moreover, the symmetrical axes \(v\)- and \(u\)- are oriented in the same direction as the \(y\)-axis and have a \(\pm 45^{\circ}\) tilt angle. The \(x\)- and \(y\)-polarized waves can be characterized using the \(uv\)- coordinates as \(\overset{\wedge}{\boldsymbol{e}}_{x}=\frac{\sqrt{2}}{2} \begin{pmatrix} \boldsymbol{\hat{e}}_{u}-\overset{\wedge}{\boldsymbol{e}}_{v}\end{pmatrix} \) and \(\overset{\wedge}{\boldsymbol{e}}_{y}=\frac{\sqrt{2}}{2} \begin{pmatrix} \boldsymbol{\hat{e}}_{u}+\overset{\wedge}{\boldsymbol{e}}_{v}\end{pmatrix} \), respectively. In the \(xy\)-coordinate system, the \(\mathbf{R}_{\text{lin}}\) can be computed via Equation (6). If a normal incidence \(y\)-polarized wave is supposed as an input (\(\boldsymbol{E}^{i}=E_{0}\overset{\wedge}{\boldsymbol{e}}_{y}=\frac{\sqrt{2}}{2 } \begin{pmatrix}\overset{\wedge}{\boldsymbol{e}}_{u}+\overset{\wedge}{ \boldsymbol{e}}_{v}\end{pmatrix} \)), the reflected wave can be differentiated as follows: \(\left(\boldsymbol{E}^{r}=E_{u}^{r}\overset{\wedge}{\boldsymbol{e}}_{u}+E_{v}^ {r}\overset{\wedge}{\boldsymbol{e}}_{v}\right)\), via Equation (6). Following algebraic simplification, the reflected wave can be expressed as \(\boldsymbol{E}^{r}=\frac{E_{0}}{2} \begin{pmatrix}(r_{uu}+r_{vv})\overset{ \wedge}{\boldsymbol{e}}_{y}+(r_{uu}-r_{vv})\overset{\wedge}{\boldsymbol{e}}_ {x}\end{pmatrix} \). The reflection coefficients of co-(\(r_{yy}\)) and cross- (\(r_{xy}\)) polarization for the normal-incidence \(y\)-polarized wave can be written as \(r_{yy}=\frac{1}{2}(r_{uu}+r_{vv})=\frac{1}{2}r_{uu}\big{(}1+e^{-j\Delta\varphi _{uv}}\big{)}\) and \(r_{xy}=\frac{1}{2}(r_{uu}-r_{vv})=\frac{1}{2}r_{uu}\big{(}1-e^{-j\Delta\varphi _{uv}}\big{)}\). After a similar computation for the \(x\)-polarized wave, the final form of \(\mathbf{R}_{\text{lin}}\) in the \(xy\)-coordinate system can be derived using [7], as shown below:  
\[\mathbf{R}_{\text{lin}}= \begin{pmatrix}r_{xx}&r_{xy}\\ r_{yx}&r_{yy}\end{pmatrix} =\frac{1}{2} \begin{pmatrix}r_{uu}+r_{vv}&r_{uu}-r_{vv }\\ r_{uu}-r_{vv}&r_{uu}+r_{vv}\end{pmatrix} =\frac{1}{2}r_{uu} \begin{pmatrix}1+e^{- j\Delta\varphi_{uv}}&1-e^{-j\Delta\varphi_{uv}}\\ 1-e^{-j\Delta\varphi_{uv}}&1+e^{-j\Delta\varphi_{uv}}\end{pmatrix}