# Transmissive Polarizer Metasurfaces: From Microwave to Optical Regimes
## 5 Theory of Polarization Conversion
Transmitted waves will have two orthogonal parts: one in phase with the incident wave (\(\mathbf{t_{xx}}\), \(\varnothing_{xx}\)) and the other is orthogonal to it, also called the cross-polarization component here (\(\mathbf{t_{xy}}\), \(\varnothing_{xy}\)). Magnitudes for these transmitted waves will vary over frequency ranges. However, in the band of interest, if these transmission coefficients become comparable in magnitude and \(\pm 90^{\circ}\) apart in their phases, i.e., \(|\mathbf{t_{xx}}|\)\(\approx\)\(|\mathbf{t_{xy}}|\) and \(\varnothing_{d}=\varnothing_{xy}-\varnothing_{xx}=\mathbf{2n\pi}\pm\pi/2\); where \(\mathbf{n}=\mathbf{0},\pm\mathbf{1},\ \pm 2\ldots\) is an integer. A transmitted wave will be called a circularly polarized wave. Transmission conversion performance for the transmitted wave in the microwave and millimeter region is usually described by the axial ratio (**AR**) which can be defined in Equation (8) as [65]:  
\[\textbf{AR}=\left(\frac{\left|\mathbf{t_{xx}}\right|_{2}+\left|\mathbf{t_{xy}}\right|^ {2}+\sqrt{a}}{\left|\mathbf{t_{xx}}\right|_{2}+\left|\mathbf{t_{xy}}\right|^{2}-\sqrt{ a}}\right)^{\mathbf{1/2}} \tag{8}\]  
where '\(a\)' can be calculated from Equation (9) as [65]:  
\[\mathbf{a}=\)\(\left|\mathbf{t_{xx}}\right|_{4}+\left|\mathbf{t_{xy}}\right|^{4}+2|\mathbf{t_{xx}}|^{2}| \mathbf{t_{xy}}|^{2}\textbf{cos}(2\varnothing_{d}) \tag{9}\]As mentioned previously for ideal polarization conversion, \(\left|\mathbf{t}_{\mathbf{x}}\right|=\left|\mathbf{t}_{\mathbf{xy}}\right|\) and \(\varnothing_{\mathbf{d}}=\varnothing_{\mathbf{xx}}-\varnothing_{\mathbf{xy}}=\mathbf{2n\pi}\pm\mathbf{ \pi}/\mathbf{2}\). In this case, the outgoing wave will be perfect circularly polarized and \(\mathbf{\mathrm{AR}}\) will be 1 (0 dB). Most of the systems allow \(\mathbf{\mathrm{AR}}\) in the range of 0-3 dB. However, a higher value of \(\mathbf{\mathrm{AR}}\) indicated that the transmitted wave will be slightly elliptical. For THz and the optical region, the performance of transmitted polarized wave is calculated using stokes parameters, as in Equations (10)-(13) [66]:  
\[\mathbf{I}=\left|\mathbf{t}_{\mathbf{xx}}\right|^{2}+\left|\mathbf{t}_{\mathbf{xy}}\right|^{2} \tag{10}\]  
\[\mathbf{Q}=\left|\mathbf{t}_{\mathbf{xx}}\right|^{2}-\left|\mathbf{t}_{\mathbf{xy}}\right|^{2} \tag{11}\]  
\[\mathbf{U}=\mathbf{2}*\left|\mathbf{t}_{\mathbf{xx}}\right|\left|\mathbf{t}_{\mathbf{xy}}\right|\text {\bf cos}(\varnothing_{\mathbf{d}}) \tag{12}\]  
\[\mathbf{V}=\mathbf{2}*\left|\mathbf{t}_{\mathbf{xx}}\right|\left|\mathbf{t}_{\mathbf{xy}}\right|\text {\bf sin}(\varnothing_{\mathbf{d}}) \tag{13}\]  
As compared to the axial ratio in microwave and millimeter bands, in THz and optics regimes, ellipticity depicts the polarization of the outgoing wave, calculated as \(\mathbf{E}llipticity=\mathbf{V}/\mathbf{I}\). Ideally, ellipticity as '\(-1\)' and '\(+1\)' depicts that the transmitted wave is an RHCP wave and LHCP wave, respectively. However, a value close to \(\pm 1\) is considered to be acceptable. This analysis is essentially the same as the \(\mathbf{\mathrm{AR}}\) analysis for perfect circular polarization conversion. It is essentially just two different terminologies for microwaves and light waves. Traditionally, the design of a polarizer (LP-to-CP converter) involves the equivalent circuits modeling of patterned MSs in combination with electromagnetic simulations. Values of equivalent circuit elements are obtained that fulfill the design conditions for co- and cross-polarizations, whereas EM simulations are used to find the corresponding values of circuit parameters. Of course, this is valid for polarizer designs in the microwave regime only. Another design approach comes into play when the circuit model does not allow standard design procedures. This design involves optimization which is applied to equivalent circuits. Most of the designs consider an incident wave tilted at 45\({}^{\circ}\) concerning cell periodicity. In this way, two independent models for each polarization are considered.