## 2 Design and Simulation  
The nematic LC is an anisotropic material, which can be explained by its rod-like molecular structure. The electromagnetic properties of the LC, such as relative permittivity and loss tangent, depend on the LC molecules [17]. The orientation of LC molecules can be coordinated by the align layer and the external electric field. In the align layer, the grooves were created along a certain direction by mechanical friction, and the liquid crystal molecules are parallel to the groove under the groove anchoring energy. As shown in Figure 1a, the liquid crystal molecules' orientation is parallel to the y-axis. If an external electric field is added, the orientation of the molecules will be forced parallel to electric field direction. By adjusting the external electric field, the relative permittivity of LC can be continuous controlled and resonant frequency of unit cells simultaneous changed. Many electrically tuned devices based on LC technology have been reported, such as phase shifters [18; 19], absorbers [20; 21], and filters [22; 23; 24]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Frequency (GHz)** & **Voltage (V)** & **Phase Shift** & **Reference** \\ \hline
10 & 20 & 183\({}^{\circ}\) & [10] \\
36 & 20 & 257\({}^{\circ}\) & [12] \\
77 & 20 & 582\({}^{\circ}\) & [2] \\
330 & 25 & 330\({}^{\circ}\) & [16] \\
350 & 40 & 360\({}^{\circ}\) & [9] \\
120 & 10 & 306\({}^{\circ}\) & proposed \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of performances for liquid crystal (LC)-base phase shifter. Figure 1: (a) Schematic of the liquid crystal (LC) based tunable phase shifter; (b) Geometrical dimensions of the unit cell structure. As shown in Figure 1a, the proposed three-dimensional phase shifter. The unit cell is formed by two quartz plates (\(\varepsilon_{r}=3.75\); \(tan\delta=0.002\)), which are separated by the polystyrene microspheres and contain a 45 \(\upmu\)m-thick LC. The copper patches which are printed on the quartz by evaporation method, have a conductivity of 5.8 \(\times\) 10\({}^{7}\) S/m and thickness of 500 nm. As shown in Figure 2, the reflection coefficients of the phase shifter have been simulated at different conductivity. It can be seen that reflection losses will increase as the metal conductivity decreases. Moreover, the thin polyimide (PI) layer is spin-coated on the copper surface to align the direction of unbiased LC molecules. The LC mixture (HFUT-HB01) is used in simulation and design. Its ordinary, extraordinary refractive indices are about 1.58 and 1.92 at 125 GHz, measured by terahertz time domain spectroscopy, as shown in Figure 3. Where its relative permittivity in the unbiased state (\(\varepsilon_{\perp}\)) and the fully biased state (\(\varepsilon_{//}\)) was set to 2.47 and 3.6, respectively [9]. In the simulation, the loss tangent of the LC layer was set to 0.02. In each row all the cells are separated by an isolation slot to realize one-dimensional scanning. Figure 1b shows the geometrical dimensions of the resonant layer. The metal layer was separated by a 127 \(\upmu\)m void at the \(x\)-axis. The list of the unit cell (along the \(x\)-axis) was connected, with the same bias voltage, and provided the same phase shift. The phase of the reflected wave is being adjusted at each row element, and a beam directing in the angle of \(\theta\) can be obtained in the far field as shown in Figure 4.  
Figure 3: (a) Refractive index of liquid crystals at T = 25 \({}^{\circ}\)C; (b) Birefringence of liquid crystals at T = 25 \({}^{\circ}\)C.  
Figure 2: Simulated effect of conductor losses on the reflection loss of the phase shifter. The static electric field, which is distributed among the LC layer, is the most important characteristic of the LC tuned devices. We simulated the electric field's direction and strength using the external electric field.