# A Low-Profile Wideband Linear-to-Circular Polarization Conversion Slot Antenna Using Metasurface
## 2 Design and Analysis of the Proposed Antenna
### Mechanism of Circular Polarization  
An equivalent circuit of the metasurface unit cell is shown in Figure 3a. When there are no parasitic squares at the diagonal corners of the cross, the orthogonal components will have the same impedance because of the symmetrical structure. \(\mathbf{E}_{1}\) and \(\mathbf{E}_{2}\) treat the metasurface as an equivalent RLC circuit, as shown in Figure 3b, the impedance of which can be written as Equation (1):  
\[Z=2R+j\omega(2L)+\frac{1}{j\omega\mathbb{C}}, \tag{1}\]  
where \(R\) and \(L\) denote the resistance and inductance, respectively. \(\mathbb{C}\) represents the capacitance created by the cross gaps. \(\mathbf{E}_{1}\) and \(\mathbf{E}_{2}\) can be seen as two different impedances \(Z_{1}\) and \(Z_{2}\). When the corners are truncated, the cross gaps increase, resulting in \(Z_{1}\) being more capacitive than \(Z_{2}\). Therefore, the phase difference between \(Z_{1}\) and \(Z_{2}\) can be varied by changing the size of the parasitic squares. When \(|Z_{1}|=|Z_{2}|\) and \(\angle Z_{1}-\angle Z_{2}=90^{\circ}\), then \(|\mathbf{E}_{1}|=|\mathbf{E}_{2}|\) and \(\angle\mathbf{E}_{1}-\angle\mathbf{E}_{2}=90^{\circ}\), and circular polarization can be realized [8]. Therefore, the corner-cut square metasurface unit cell of Figure 3b was arranged in a \(2\times 2\) layout to form the basic metasurface of Figure 3c. According to the basic theory of electromagnetic fields, circularly polarized waves can be decomposed into two linearly polarized waves that are orthogonal to each other, having the same amplitude and a phase difference of \(90^{\circ}\). Therefore, these two conditions must be met to achieve linear-to-circular polarization conversion. As shown in Figure 3b, the linearly polarized wave is incident along the \(\mathbf{E}\) direction, and the angle between this and \(\mathbf{E}_{1}\) is \(\theta\). When \(\theta=45^{\circ}\) or \(\theta=135^{\circ}\), the linear polarization wave can be decomposed into two orthogonal linear polarization components  
Figure 3: The mechanism of generating circularly polarized waves from linearly polarized waves for the corner-cut square metasurface (**a**) an equivalent circuit of the metasurface unit cell; (**b**) an electric field distribution of the metasurface unit cell; (**c**) the basic metasurface. Figure 2: Antenna polarization at different rotation angles. with equal amplitude. If \(\theta=45^{\circ}\) is used as an example for analysis, the electric field of the incident wave can be expressed as Equation (2) [18; 19]:  
\[\overset{\rightarrow}{E}=\left|\overset{\rightarrow}{E_{1}}\right|\text{e}^{i \theta_{1}}+\left|\overset{\rightarrow}{E_{2}}\right|\text{e}^{i\theta_{2}}= \frac{\sqrt{2}}{2}\left(\overset{\rightarrow}{E_{1}},\overset{\rightarrow} {E_{2}}\right)\left( \begin{array}{c}1\\ 1\end{array} \right). \tag{2}\]  
Let \(\varphi_{1}\) and \(\varphi_{2}\) be the reflection phases of the unit in directions \(\mathbf{E}_{1}\) and \(\mathbf{E}_{2}\). If \(|\varphi_{1}-\varphi_{2}|=90^{\circ}\), then Equation (2) can be rewritten as Equation (3):  
\[\overset{\rightarrow}{E_{\text{c}}}=\frac{\sqrt{2}}{2}\left(\overset{ \rightarrow}{E_{1}},\overset{\rightarrow}{E_{2}}\right)\left( \begin{array}[] {c}1\\ \pm j\end{array} \right). \tag{3}\]  
Furthermore, the electric field can be rewritten in terms of the metasurface of the two circularly polarized components \(\mathbf{E}_{1}\) and \(\mathbf{E}_{2}\); thus, the axial ratio (AR) can be easily determined according to Equation (4) [18]:  
\[AR_{\text{dB}}=20\log\left(\frac{\left|\overset{\rightarrow}{E_{1}}\right|+ \left|\overset{\rightarrow}{E_{2}}\right|}{\left|\overset{\rightarrow}{E_{1} }\right|-\left|\overset{\rightarrow}{E_{2}}\right|}\right). \tag{4}\]  
The above analysis explains how circularly polarized waves are generated. From the basic metasurface with four parasitic square cross gaps shown in Figure 4a, we inferred that circularly polarized performance of the antenna could be enhanced by adding more metasurface unit cells of Figure 3b.