# _Photonics_
## 3 Results and Discussion  
Now we are in a position for the analysis of the polarization features of the alleged polarizer. The polarization is analyzed in terms of the reflection of the co- and cross-component. The reflected waves, after the initial incidence of the electromagnetic waves on the metasurface, decompose into two components: \(r_{yy}\) representing a co-polarized field and \(r_{xx}\) representing the same polarization as the incident field, and a cross-polarized field coefficient, which is orthogonal to the incident field, is represented as \(r_{xy}\) and \(r_{yx}\). The co- and cross-polarized coefficients for a y-polarized wave are \(r_{yy}=E_{yr}/E_{yi}\) and \(r_{xy}=E_{xt}/E_{yi}\), respectively; on the other hand, the corresponding coefficients for an x-polarized wave are \(r_{xx}=E_{xt}/E_{xi}\) and \(r_{yx}=E_{yt}/E_{xi}\). Figure 2 shows the optimization of the proposed polarizer. In this investigation, we are examining the TE polarization; however, it is showing equal effectiveness for the TM polarization because of the diagonal symmetry of the studied geometry. Cross-polarization conversion is seen to be taking place when the cross-polarized coefficient rises and there is a reduction in the co-polarized component coefficient. Figure 2a shows the first stage of the design of the polarization conversion. It is noticed that the cross-component of the polarizer reaches 80% at 15 GHz, and remains above 80% up to 18 GHz, and a further increase in the operating frequency reduces the cross-component. In the second stage, another stair-like resonator is added with a broken symmetry as shown in Figure 2b. Noticeably, the cross-polarization reflection coefficient is considerably increased, and it remains above 80% from 13 to 25 GHz. Further, another stair-like resonator is added in the unit cell with a broken symmetry. It is observed that the cross-polarization component is further increased, and the co-polarized reflection coefficient is reduced. However, adding another component resonator reduces the co-polarized coefficient. Therefore, Figure 2c shows the optimized results of the polarizer. Figure 2d shows the polarization conversion ratio (PCR); noticeably,  
Figure 1: Schematic of the proposed polarization converter: (**a**) front view; (**b**) side view. this remains greater than 90% from 12.94 to 26 GHz, except for a small dip from 16.55 to 17.53 GHz. The PCR is determined by the following formula:  
\[PCR=\frac{\left|r_{xy}\right|^{2}}{\left|r_{xy}\right|^{2}+\left|r_{yy}\right|^{ 2}}\]  
In this formula, \(\left|r_{xy}\right|^{2}\) represents the squared magnitude of the reflection coefficient for the cross-polarized component, while \(\left|r_{yy}\right|^{2}\) denotes the squared magnitude of the reflection coefficient for the co-polarized component. This ratio quantifies the efficiency of polarization conversion, with higher PCR values indicating a more effective transformation of the incident polarization into the desired output state. Further, we introduce a coordinate system of u- and v-axes to examine the anisotropy of the proposed structure, and with respect to the x-axes and y-axes, the axes are rotated at 45\({}^{\circ}\). For investigation, the electromagnetic waves that are examined are the y-polarized incident electromagnetic waves. Figure 3 shows the anisotropic geometry of the polarization converter, which includes mutually perpendicular symmetric u- and v-axes. When the metasurface is hit with a y-polarized wave, the y-polarized wave splits into two components: \(E_{iy}=\hat{u}E_{in}e^{i\varphi}+\partial E_{in}e^{i\varphi}\) along the u- and v-axes. The two reflected components of the affecting waves after being combined together, resulting in an x-polarized wave, i.e., \(E_{xr}=\hat{u}E_{n}e^{i\varphi_{u}}+\hat{v}E_{r\varphi}e^{i\varphi_{v}}\) where \(\varphi_{u}\) and \(\varphi_{v}\) are the \(r_{u}\) and \(r_{v}\) phases, respectively.