# Transmissive Polarizer Metasurfaces: From Microwave to Optical Regimes
## 3 History and Modeling of MSs
Using the homogeneous sheet model of MSs described by [36], we can retrieve r\({}_{1}\), r\({}_{2}\), and 't'. Using transmission line modeling of two-port passive and reciprocal devices, MS can be represented by an equivalent T-circuit with two series impedances \(Z_{1}\), \(Z_{2}\), and a shunt admittance Y. Using r\({}_{1}\), r\({}_{2}\), and 't', \(Z_{1}\), \(Z_{2}\), and Y can be calculated for the MS [14]. Detailed modeling work has already been carried out [37]. Modeling is out of the scope of this review, therefore we will not discuss its details. For incident oblique waves, if MS has non-zero electromagnetic susceptibility, susceptibilities \(\overrightarrow{\overline{X_{ee}}}\), \(\overrightarrow{\overline{X_{em}}}\), and \(\overrightarrow{\overline{X_{mm}}}\) are independent of the incident angles. In this case, three equivalent circuit parameters \(Z_{1}\), \(Z_{2}\), and admittance Y can be represented in terms of scalar susceptibilities [14]. Figure 2: MS structured as an interface between two homogeneous isotropic media. Metasheets are a class of MSs based on uni-layered substrate. They can be further classified into electric metasheets and magneto-electric metasheets. Electric metasheets are those metasheets which only have electric polarization when an incident electric field is applied. Such sheets are single metallic patterned over the substrate and/or have metal ground. They can be described through a few equivalent parameters. For example, a metasheet based on metal gratings can be described by grid impedance, whereas a metasheet based on planar array patches over the ground can be described as surface impedance [38]. Applied electric field on such sheets does not produce induced current loops inside them, being extremely thin layered dielectric sheets to act as an equivalent sheet of magnetic surface current. Examples are graphene sheets, single metallic layered sheets, such as dense wires and strip grids, etc. As the incident electric field produces only electric dipole moments, FSS can also be considered as a metasheet because FSS also has just an electric response, as they require having frequency-selective transmission and reflection. FSS can be categorized as the patterns which can be equalized as inductances, such as inductive thin wires or strips; capacitances, such as an array of patches, and the one which has both inductances and capacitive behavior. If MSs have a non-zero but still negligible thickness as compared to the wavelength, electric current loops arise inside the MSs even if they are not closed. Usually, MS has both electric and magnetic currents. Induced electric currents on the MS are comprised of two parts: symmetric and asymmetric currents concerning the middle plane. The asymmetric electric current in each unit cell gives rise to magnetic surface currents. In spite of advanced functionalities in MSs, control over transmission and reflection properties still need to be explored. This needs the induction of independent electric and magnetic currents. In bi-anisotropic MSs, incident electric or magnetic polarization will provide induction of both electric and magnetic fields. Most generic relations for these induced electric and magnetic fields are [14]:  
\[\left[ \begin{array}{c}\mathbf{J_{e}}\\ \mathbf{J_{m}}\end{array} \right]=\left[ \begin{array}{cc}\mathbf{Y_{ee}}&\mathbf{Y_{em}} \\ \mathbf{Y_{me}}&\mathbf{Y_{mm}}\end{array} \right]\cdot\left[ \begin{array}{c}\mathbf{E_{ inc}}\\ \mathbf{H_{inc}}\end{array} \right] \tag{3}\]  
where \(\mathbf{J_{e}}\) and \(\mathbf{J_{m}}\) are equivalent electric and magnetic current densities, \(\mathbf{Y_{ee}}\), \(\mathbf{Y_{mm}}\), \(\mathbf{Y_{em}}\) and \(\mathbf{Y_{me}}\) are corresponding admittances to \(\overline{\mathbf{\chi_{ee}}}\), \(\overline{\mathbf{\chi_{mm}}}\), \(\overline{\mathbf{\chi_{em}}}\) and \(\overline{\mathbf{\chi_{me}}}\). As mentioned previously, of the three types of MSs, reflection-based MSs--also called impenetrable MS--are described by surface admittance.