## 2 Theoretical Basis  
In the terahertz frequency range, LC can be treated as a lossy and dispersive media [23; 34]. We employ the polynomial model to fit the refractive index (\(n\)) and absorption coefficient (\(\alpha\)) of E7, which is used as the LC material throughout the paper. As for the dielectric properties of E7, it has a positive dielectric anisotropy (\(\Delta\varepsilon>0\)). The ordinary refractive index (\(n_{o}\)) and extraordinary refractive index (\(n_{e}\)) of E7 are 1.56 and 1.70, respectively, at the operating frequency of 1 THz and temperature of 26 \({}^{\circ}\)C [35; 36]. With the electric field \(E>7\) kV/m, the E7 reaches a maximum value that is the extraordinary refractive index (\(n_{e}\)) of 1.70. The refractive index of E7 is a value that can be controlled between 1.56 and 1.70, depending on the external electric field. Its dielectric properties have been studied experimentally in reference [35]. Fitting the dielectric properties of E7 in the terahertz range by the polynomial model, the lossy and dispersive characteristics of the LC presented in this paper would be accurate and reasonable. The polynomial model is employed and shown below:  
\[n_{E}(\omega) = \sum_{i=0}^{M_{\mathrm{g}}}k_{i,E}\omega^{i} \tag{1}\] \[\alpha_{E}(\omega) = \sum_{i=0}^{M_{\mathrm{g}}}h_{i,E}\omega^{i} \tag{2}\]  
where \(n_{E}\) and \(\alpha_{E}\) are the refractive index and absorption coefficient of the LC under the bias electric field \(E\), respectively; \(M_{n}\) and \(M_{\alpha}\) are the order of the polynomials for \(n_{E}\) and \(\alpha_{E}\), respectively; \(k_{i,E}\) and \(h_{i,E}\) stand for the coefficients in the fitted polynomials for the refractive index and absorption coefficient, respectively. Owing to the simplicity and effectiveness of the polynomial model, it was chosen to model the dielectric properties of E7 in the terahertz range. Figure 1 presents our fitted results based on Equations (1) and (2) (\(M_{n}\) = 5, \(M_{\alpha}\) = 16), which are in good agreement with the experimental ones in reference [35]. Throughout the paper, the bias electric field of the LC is selected from 0 kV/m, 3 kV/m, 5 kV/m or 7 kV/m, which can tune the dielectric properties of the LC. The external electric field is set in the range of 0 to 7 kV/m, since when \(E>7\) kV/m the refractive index of E7 has already reached the maximum value that is the extraordinary refractive index (\(n_{e}\) = 1.70) [35]. Due to the numerous parameters in the polynomial fit, the fitting parameters of E7 with order \(M_{n}\) equals to 5 and \(M_{\alpha}\) equals to 16 are listed in Tables 1 and 2 of Appendix A.  
Sandwiched by the upper and lower metal plates, the main channel forms a planar waveguide where electromagnetic (EM) waves propagate. Figure 2 shows the schematic of the MIM waveguide. Across the upper and lower metal plates, a voltage is applied to tune the dielectric properties of the LC. The voltage induces an electric field in the main channel and tilts the LC molecules, thereby changing the dielectric properties of the LC. When the strength of the electric field applied to the main channel varies, the main body of the LC molecules tends to tilt with different angles, leading to different permittivities of the LC and hence different propagation characteristics of EM waves in the main channel. Under the geometry shown in Figure 2, the TM mode dominates in the MIM waveguide. The propagation constant of a planar waveguide is determined by the width of the main channel as well as the dielectric property of the LC and metal forming the channel. Taking the dielectric property  
Figure 1: Fitted dielectric parameters of E7 in terahertz frequency range: (**a**) Refractive index; (**b**) Absorption coefficient. The fitted data are in good agreement with those in reference [35]. Figure 2: Geometry of the metal-insulator-metal (MIM) waveguide with the infiltration of the liquid crystal (LC).