## 3 Simulation Results of Unit Cell and Theory Analysis of LCPC
And AR < 1 dB in the frequency ranges of 8.28-8.36 GHz and 10.18-12.06 GHz, with FBWs of 0.96% and 16.91%. For \(\varepsilon_{r}=3.9\), the frequency intervals within 8.37-8.43 GHz, 9.01-9.11 GHz, and 9.86-11.83 GHz, AR < 3 dB, and FBWs are 0.71%, 1.1% and 18.17%, respectively. In the frequency range of 10.17-11.45 GHz, AR < 1 dB, and FBW is 11.84%. For \(\varepsilon_{r}=4.6\), AR < 3 dB, and 1.66% and 14.54% for FBWs within the frequency intervals 8.96-9.11 GHz and 9.95-11.51 GHz, respectively, and 7.66% for FBW, AR < 1 dB within the frequency intervals 10.3-11.12 GHz, respectively. In Figure 5b, the polarization conversion ratios (PCRs) exceed 0.98 (greater than 98%) in the following frequency ranges: 8.17-8.45 GHz and 9.9-12.43 GHz for \(\varepsilon_{r}=2.8\), 9.92-11.75 GHz for \(\varepsilon_{r}=3.9\), and 8.98-9.1 GHz and 10.01-11.44 GHz for \(\varepsilon_{r}=4.6\). The surface current distribution of the unitary radiating patch at \(\varepsilon_{r}=2.8\) is investigated under vertical positive feed incidence of y-polarized wave (\(\varphi=90^{\circ}\), \(\theta=0^{\circ}\)) at operating frequencies of 8.24 GHz and 11 GHz, respectively. This analysis reveals the mechanism behind the transition between linear and circular polarization. At a frequency of 8.24 GHz in Figure 6a, the surface current vectors of the primary radiating patch rotate counterclockwise as the resonance phase increases from \(0^{\circ}\) to 270\({}^{\circ}\), indicating the achievement of RHCP. Conversely, Figure 6b demonstrates that at a frequency of 11 GHz, the surface current vector shows clockwise rotation, indicating that the conversion of LHCP is achieved. Typically, for an LP wave, the EM wave can be decomposed into two components along orthogonal directions, denoted as \(u\) and \(v\), as illustrated in Figure 1. These components are symmetric about the y-axis, with angles of \(-45^{\circ}\) and \(45^{\circ}\) relative to the y-axis, respectively. Due to the anisotropic nature of the subsurface, the co-polarized reflection coefficients (\(r_{mu}\), \(r_{\varpi}\)) are independent, yet their magnitudes are nearly equal to 1.0, owing to the low dielectric loss. Therefore, neglecting the minimum dielectric loss, the following equation can be derived: \(r_{\varpi}=r_{mu}e^{-j\Delta\varphi_{mu}}\), and the electric field unit  
Figure 5: Results obtained by calculations at the incidence of a y-polarized EM wave. (**a**) Axial ratio (dB) and (**b**) linear-to-circular polarization conversion ratio (PCR). vectors for the \(x\)- and \(y\)-polarized waves can be expressed using the \(u\) and \(v\)-axis as follows: \(e_{x}=\frac{\sqrt{2}}{2}(e_{u}-e_{v})\) and \(e_{y}=\frac{\sqrt{2}}{2}(e_{u}+e_{v})\)[33], where \(e_{u}\) and \(e_{v}\) are the unit vectors along the \(u\) and \(v\) axes, respectively, and the phase difference, \(\Delta\varphi_{uv}=\varphi_{uu}-\varphi_{vv}\). Assuming a vertically incident \(y\)-polarized wave as an input (\(E^{i}=E_{0}e_{y}=\frac{\sqrt{2}}{2}(e_{u}+e_{v})\)), and using Equations (6) and (7), the reflected EM waves are characterized as follows: \(E^{r}=E^{r}_{u}e_{u}+E^{r}_{v}e_{v}\), which can be simplified as follows:  
\[E^{r}=\frac{E_{0}}{2}[(r_{uu}+r_{vv})e_{y}+(r_{uu}-r_{vv})e_{x}] \tag{11}\]  
where \(r_{yy}\) and \(r_{xy}\) can be summarized as follows: \(r_{yy}=\frac{1}{2}(r_{uu}+r_{vv})=\frac{1}{2}r_{uu}(1+e^{-j\Delta\varphi_{uv}})\) and \(r_{xy}=\frac{1}{2}(r_{uu}-r_{vv})=\frac{1}{2}r_{uu}(1-e^{-j\Delta\varphi_{uv}})\). Then, the reflection coefficient matrix (\(R_{lin}\)) can be converted into:  
\[R_{lin}= \begin{pmatrix}r_{xx}&r_{xy}\\ r_{yx}&r_{yy}\end{pmatrix} =\frac{1}{2}r_{uu} \begin{pmatrix}1+e^{-j\Delta \varphi_{uv}}&1-e^{-j\Delta\varphi_{uv}}\\ 1-e^{-j\Delta\varphi_{uv}}&1+e^{-j\Delta\varphi_{uv}}\end{pmatrix}