# Ultrawideband Cross-Polarization Converter Using Anisotropic Reflective Metasurface
## 4 Decomposed U\(-\)V Incident Analysis  
Some mathematical formulations are presented here to better understand the polarization conversion mechanism. Generally, \(x-\) and \(y-\)polarized incident waves are mathematically represented as \(\overrightarrow{E}_{xi}=\overrightarrow{E}_{xi}\overrightarrow{e}_{x}\) and \(\overrightarrow{E}_{yi}=\overrightarrow{E}_{yi}\overrightarrow{e}_{y}\), where (\(\overrightarrow{e}_{x},\overrightarrow{e}_{y}\)) are unit vectors along the \(x-\) and \(y-\)axes. Furthermore, as we discussed earlier, the co\(-\) and  
Figure 3: Polarization conversion response of proposed metasurface: (**a**) co\(-\) and cross\(-\) polarization component \(r_{yy}\) and \(r_{xy}\), (**b**) polarization conversion ratio (_PCR_), (**c**) magnitude of orthogonal components \(r_{u}\) and \(r_{v}\), and (**d**) phase difference.  
cross\(-\)polarized coefficients can be defined as \(\left|r_{yy}\right|=E_{yr}/E_{yi}\) and \(\left|r_{xy}\right|=E_{xr}/E_{yi}\), respectively, whereas the analogous coefficients for an \(x-\)polarized wave are \(\left|r_{xx}\right|=E_{xr}/E_{zi}\) and \(\left|r_{yx}\right|=E_{yr}/E_{zi}\). To analyze the anisotropy of the structure, a new coordinate system of \(u-\) and \(v-\)axes are defined, and, for that, the axes are rotated at \(45^{\circ}\) with respect to the \(x-\) and \(y-\)axes.  
We only consider the \(y-\)polarized incidence EM wave for analysis in order to keep it simple. Figure 4 depicts the polarization converter anisotropic geometry, including mutually perpendicular symmetric \(u-\) and \(v-\)axes. When the \(y-\)polarized wave illuminates the metasurface, it gets decomposed into two components along the \(u-\) and \(v-\)axes: \(E_{yi}=\hat{\alpha}E_{ui}e^{i\varphi}+\hat{\psi}E_{vi}e^{i\varphi}\). When the reflected components of the impinging waves are composed together, it leads to an \(x-\)polarized wave: \(E_{xr}=\hat{\alpha}E_{ru}e^{i\varphi_{x}}+\hat{\psi}E_{rr}e^{i\varphi_{v}}\), where \(\varphi_{u}\) and \(\varphi_{v}\) are the phases of \(r_{u}\) and \(r_{v}\), respectively. \(E_{yi}\) and \(E_{xr}\) can also be written as Equations (12) and (13).  
\[E_{yi}=E_{ui}\exp(j\mathrm{k}z)e_{u}+E_{yi}\exp(j\mathrm{k}z)e_{v}= \tag{12}\] \[E_{yi}cos(45^{\circ})\exp(j\mathrm{k}z)(e_{u}+e_{v}). \tag{13}\]  
We considered minimum dielectric losses for the structure because it is backed with copper, and most of the energy is considered as reflected, so the amplitudes of \(r_{u}\) and \(r_{v}\) are approximately equal--\(r_{u}=r_{v}=1\)--as shown in Figure 3c. The same method can be applied for the \(x-\)polarized incident wave, and the relation can also be considered for co\(-\)polarized coefficients along the \(u-\) and \(v-\)axes: \(\left|r_{u}\right|=E_{ru}/E_{iu}\) and \(\left|r_{v}\right|=E_{ru}/E_{iy}\). The \(u-\) and \(v-\)polarized coefficients (\(r_{u}\) and \(r_{v}\)) are independent because of the anisotropic nature of the unit cell. If we consider the ideal scenario and neglect the loss tangent of copper, the magnitude of \(r_{u}\) and \(r_{v}\) is unity (\(r_{u}=r_{v}=1\)), and phase difference \(\Delta\varphi_{uv}=\varphi_{u}-\varphi_{v}\)= \(180^{\circ}\), then the reflected field \(\vec{E}_{r}\) will be oriented along the \(x-\)axis, validating the cross\(-\)polarization conversion.  
To verify the amplitude and phase conditions of \(r_{u}\)and \(r_{v}\), simulation results are plotted in Figure 3c. It can be observed that the amplitudes of these components are approximately equal to unity, and that there is a \(180^{\circ}\) phase difference between them for a wide frequency band ranging from 13 GHz to 26 GHz, conforming to the broadband polarization conversion.  
Figure 4: Illustration of the decomposition of orthogonal components \(u\) and \(v\) in reflection mode.