# Transmissive Polarizer Metasurfaces: From Microwave to Optical Regimes
## 5 Theory of Polarization Conversion  
The basic concept for polarization converters is to use a combination of capacitive and inductive guides. Several such structures have been realized through anisotropic metal sheets [62; 63; 64]. MS-based polarization converters can be used in line with this concept, i.e., transmitted co-polarized and cross-polarized components should have the same amplitude,however, should be 90\({}^{\circ}\) phase distinct. MSs consisting of the meandered pattern are designed in such a way that they behave as inductance if the applied tangential electric field is parallel to the strips and as capacitances if the electric field is perpendicular to the strips. For the detailed operation of the LP-to-CP converter, let us consider a plane incident electric field \(\overrightarrow{E_{xi}}\) and \(\overrightarrow{E_{yi}}\) with x-polarization and y-polarizations, respectively. Ignoring the reflection losses and considering MS as an ideal transmissive polarization converter, \(\overrightarrow{E_{yi}}\) and \(\overrightarrow{E_{yi}}\) waves traveling in '+z' are applied on MS. Transmitted waves \(\overrightarrow{E_{xo}}\) and \(\overrightarrow{E_{yo}}\) can be expressed in terms of transmission coefficients, as shown in Equation (4). Here, '\(\mathbf{xo}\)' and '\(\mathbf{yo}\)' represent outgoing waves corresponding to the incident x-polarized and y-polarized waves, respectively. These transmitted waves can be represented in terms of the transmission matrix 'T' for \(\overrightarrow{E_{xo}}\) and \(\overrightarrow{E_{yo}}\), as shown in Equation (4). \[\left[ \begin{array}{c}\overrightarrow{E_{xo}}\\ \overrightarrow{E_{yo}}\end{array} \right]=T\left[ \begin{array}{c} \overrightarrow{E_{xi}}\\ \overrightarrow{E_{yi}}\end{array} \right]=\left[ \begin{array}{c} \overrightarrow{t_{xx}}&\overrightarrow{t_{xy}}\\ \overrightarrow{t_{yx}}&\overrightarrow{t_{yy}}\end{array} \right]\left[ \begin{array}{c}\overrightarrow{E_{xi}}\\ \overrightarrow{E_{yi}}\end{array} \right] \tag{4}\]  
\[\text{Where, }\overrightarrow{E_{xi}}=E_{xi}\overrightarrow{e_{x}}=E_{o}e^{- jK}\widetilde{e_{x}}^{\rightarrow} \tag{5}\]  
Combining magnitude and phase responses of Equations (4) and (5):  
\[\left[ \begin{array}{c}\overrightarrow{E_{xo}}\\ \overrightarrow{E_{yo}}\end{array} \right]=\left[ \begin{array}{c}|\mathbf{t_{xx} }|e^{j\alpha_{xx}}&|\mathbf{t_{xy}}|e^{j\alpha_{yy}}\\ |\mathbf{t_{yx}}|e^{j\alpha_{yx}}&|\mathbf{t_{yy}}|e^{j\alpha_{yy}}\end{array} \right] \left[ \begin{array}{c}\overrightarrow{E_{xi}}\\ \overrightarrow{E_{yi}}\end{array} \right]=\left[ \begin{array}{c}|\mathbf{t_{xx} }|e^{j\alpha_{xx}}&|\mathbf{t_{xy}}|e^{j\alpha_{yy}}\\ |\mathbf{t_{yx}}|e^{j\alpha_{yx}}&|\mathbf{t_{yy}}|e^{j\alpha_{yy}}\end{array} \right] \left[ \begin{array}{c}E_{o}e^{-jK}\widetilde{e_{x}}^{\rightarrow}\\ |\mathbf{t_{yx}}|e^{-jK}\widetilde{e_{y}}^{\rightarrow}\end{array} \right] \tag{6}\]  
where, \(t_{xx}\) and \(t_{xy}\) are transmission coefficients for co and cross-polarization components, respectively, for incident x-polarized wave; \(\varnothing_{xx}\), \(\varnothing_{xy}\) are the corresponding phases. Similarly, \(t_{yx}\), \(\varnothing_{yx}\) and \(t_{yy}\), \(\varnothing_{yy}\) are the transmission coefficient and phase for the transmitted x- and y-polarized waves for the incident y-polarized waves, respectively. \(|\mathbf{t_{xx}}|\), \(|\mathbf{t_{xy}}|\), \(|\mathbf{t_{yx}}|\), \(|\mathbf{t_{yy}}|\) can be computed as shown in Equation (7):  
\[|\mathbf{t_{xx}}|=\frac{|\mathbf{E_{xo}}|}{|\mathbf{E_{xi}}|},\ |\mathbf{t_{xy}}|=\frac{|\mathbf{E_{yo}}|} {|\mathbf{E_{xi}}|},\ |\mathbf{t_{yx}}|=\frac{|\mathbf{E_{xo}}|}{|\mathbf{E_{yi}}|},\ |\mathbf{t_{yy}}|=\frac{|\mathbf{E_{yo}}|}{|\mathbf{E_{yi}}|} \tag{7}\]  
Here, we are considering that the incident wave is an x-polarized wave.