## 2 Unit Cell Design and Properties of LC Materials
### Electromagnetic Properties of LC Materials  
LC materials are primarily classified into thermotropic LCs, solvatropic LCs, and polymeric LCs. LCs are a unique class of polymer compound materials between liquids and solids, combining the fluidity of liquids and the anisotropy of crystals [29]. In this study, thermotropic nematic-phase LCs with dielectric anisotropies are selected as tunable materials for RMS. Nematic phase LCs are composed of uniaxial rod-shaped molecules, which can move between molecular layers, have superior mobility, simple orientation control, and low threshold voltage, which makes them widely used in the field of microwave device tuning. When different bias voltages are applied, the orientation of the nematic LC molecules determines the dielectric constant tensor in various deflection states [30]. Figure 2 demonstrates the deflection characteristics of nematic phase LC molecules under different applied bias voltages [29]. Different cases of LC molecules and dielectric constant variation with applied bias voltage are shown in Figure 2a, b, and c, respectively, where \(V_{max}\) denotes the critical maximum voltage, and \(\varepsilon_{eff}\) is the effective dielectric constant of the LC. \(E\) is the electric field direction of the applied bias voltage. When the applied voltage is below the threshold voltage (\(V_{th}\)), which is insufficient to deflect the LC molecules, the LC molecules align perpendicular to the electric field, with a dielectric constant of \(\varepsilon_{\perp}\) and a corresponding tangent loss of \(\tan\delta_{\perp}\). As the voltage reaches \(V_{max}\), the LC molecules will be parallel to the electric field, with a dielectric constant of \(\varepsilon_{\parallel}\) and a corresponding tangent loss of \(tan\delta_{\parallel}\). The dielectric anisotropy of LC is expressed as \(\Delta\varepsilon_{r}=\varepsilon_{r,\parallel}-\varepsilon_{r,\perp}\) (where \(\varepsilon_{r,\perp}\geq\varepsilon_{\perp}\) and \(\varepsilon_{r,\parallel}\leq\varepsilon_{\parallel}\)). The varying rotational angles of the LC molecules under externally applied bias voltages correspond to different dielectric constants of the LCs. The pointing vectors, \(\overrightarrow{n}\), of these molecules determine the dielectric constant tensor in different deflected states, with \(\theta\) representing the angle of molecular deflection. When \(\overrightarrow{n}=(cos\theta,0,sin\theta)\), its relative dielectric constant can be expressed as [29]:  
\[\underset{\varepsilon_{r}}{\leftrightarrow}=\left( \begin{matrix}\varepsilon _{\perp}+\Delta\varepsilon_{r}cos^{2}\theta&0&\Delta\varepsilon_{r}sin\theta cos \theta\\ 0&\varepsilon_{\perp}&0\\ \Delta\varepsilon_{r}sin\theta cos\theta&0&\varepsilon_{\perp}+\Delta \varepsilon_{r}sin^{2}\theta\end{matrix} \right) \tag{1}\]  
Figure 1: Structure of the proposed reflective MS unit cell involving LC is listed as: (**a**) exploded view and dimension parameters, and (**b**) perspective side view. When no bias voltage is loaded, \(\overset{\rightarrow}{n}\) is perpendicular to the direction of the electric field, the LC molecules are not deflected, \(\theta\) is approximated to be \(0^{\circ}\), and the LC relative dielectric tensor matrix is:  
\[\underset{\epsilon_{r}}{\leftrightarrow}= \begin{pmatrix}\epsilon_{\parallel}&0& 0\\ 0&\epsilon_{\perp}&0\\ 0&0&\epsilon_{\perp}\end{pmatrix}  \tag{2}\]  
When the loaded bias voltage is greater than \(V_{max}\), \(\overset{\rightarrow}{n}\) is parallel to the direction of the electric field, and \(\theta\) is approximated to be \(90^{\circ}\), the relative dielectric tensor matrix of the LC is:  
\[\underset{\epsilon_{r}}{\leftrightarrow}= \begin{pmatrix}\epsilon_{\perp}&0& 0\\ 0&\epsilon_{\perp}&0\\ 0&0&\epsilon_{\parallel}\end{pmatrix}