# Transmissive Polarizer Metasurfaces: From Microwave to Optical Regimes
## 3 History and Modeling of MSs  
Before further discussion, we clarify here that those planar 2D arrays whose periodicity is not smaller than operating wavelengths will not be called MSs here, because the term 'MS' is associated with cells whose periodicity is smaller than a wavelength. Here, the wavelength is a term used for \(\lambda_{\text{freespace}}\). Mesh and wire-based structures have been extensively explored for antenna systems to realize polarizers, which were based on averaged boundary conditions and, later, the first homogenization theory of artificial electromagnetic surfaces emerged. Frequency selective surfaces (FSS) are another type of planar array which exhibit resonant type transmission and reflection characteristics. Dipole and aperture based FSS have been employed for microwave filters operating with plane waves. MSs were termed nano-islands in the starting period of their explorations in the optical region. Their modeling was carried out using oblate spheroids [21; 22; 23] to study their different characteristics, such as frequency and polarization selectivity. For most of the applications, the only electric response was not sufficient, therefore, an accompanying magnetic response was required to be considered in MS [24; 25]. The homogenization model of MS presented by [26; 27; 28] does not cover bi-anisotropic MS. Such MSs have been targeted for polarization controlling applications for chiral [29; 30] and omega type anisotropicity [31]. Chiral bi-anisotropic MSs for polarization conversions take advantage of inclusions which are polarized by electric or magnetic fields. For such MSs, simultaneous parallel electric or magnetic fields can exist. Another type is the omega type inclusions, where electric or magnetic dipoles are directly orthogonal to each other due to applied electric or magnetic fields only. These inclusions also help in achieving polarization conversion. Unlike the reviews [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34] which described several MSs for different applications, we aim to create a link between single and multiband MSs for polarization conversion applications in microwave and optical regimes. MS excitation can be modeled as an incident plane wave with wave numbers K\({}_{1,2}\), and wave impedance \(\eta_{1,2}\). Modeling of MS can be carried out using two approaches, the surface susceptibility approach and the equivalent circuit approach. The former approach has a disadvantage, as in some bi-anisotropic MSs' surface susceptibility has no physical meaning [14]. Whereas the latter provides an inside to physical procedures. Nonetheless, the common thing is both models characterize MS as a homogeneous sheet. In the surface susceptibility approach, MS can be represented by electric and dipole moments [35]. \[\overrightarrow{P}=\epsilon\overrightarrow{\overline{X_{ee}}}\overrightarrow{ E_{av}}+\sqrt{\epsilon\mu\overline{X_{em}}}\overrightarrow{H_{av}} \tag{1}\]  
\[\overrightarrow{M}=\overrightarrow{\overline{X_{mm}}}\overrightarrow{H_{av}}+ \sqrt{\epsilon\mu\overline{X_{me}}}\overrightarrow{E_{av}} \tag{2}\]  
where \(\overrightarrow{\overline{X_{ee}}}\), \(\overrightarrow{\overline{X_{mm}}}\), \(\overrightarrow{\overline{X_{em}}}\) and \(\overrightarrow{\overline{X_{me}}}\) are the electric/magnetic susceptibilities describing the response to electric/magnetic excitations, and 'av' denotes average fields on both sides of MSs. The above equations can be used to solve many boundary value problems for MSs. As discussed before, the solution for MSs under the susceptibility approach has some shortcomings, such as no physical insight in some cases. Alternatively, MSs can be modeled using an equivalent circuit approach. Let us consider a plane wave striking MS either from medium 1 or medium 2, as shown in Figure 2. We can simulate/measure r\({}_{1}\), r\({}_{2}\), and 't', assuming that 't' remains the same for an incident wave from medium 1 or medium 2 as MSs are reciprocal.