## 3 Operational Principles  
Previously, we analyzed LC THz phase shifters with a homogeneous alignment [2; 5; 6; 8; 9; 10]. Basically, the analysis followed the treatment of [PERSON] and Gu [25]. The medium through which the THz wave propagates is assumed to be homogeneous, non-absorbent, and magnetically isotropic. This is an approximation, as LC and fused silica and the TCE and alignment layers exhibit a small yet non-negligible absorption of THz waves. If we are only interested in the calculation of the phase shift experienced by THz waves, such a straightforward analysis is valuable. This is confirmed from our earlier works cited above and adopted by other groups. For this study, we extended the analysis to a TN cell. In the absence of the external field, the LC molecules in a TN cell were initially in equilibrium and were oriented toward the preferred directions of the alignment layer as shown in Figure 2a. When the root-mean-square (RMS) applied bias voltage with a sinusoidal waveform at 1 kHz, V\({}_{\text{RMS}}\) was greater than the threshold voltage (V\({}_{\text{RMS}}\) > V\({}_{\text{TN}}\)), the LC molecules reoriented themselves toward the direction of the external field, the so-called Freedericksz transition [25]. This is illustrated in Figure 2b. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**LC** & \(\mathbf{k_{1}}\) **(pN)** & \(\mathbf{k_{2}}\) **(pN)** & \(\mathbf{k_{3}}\) **(pN)** & \(\mathbf{\varepsilon_{H}}\) & \(\mathbf{\varepsilon_{\perp}}\) & **Cell Thickness (\(\upmu\))** & **Identity** \\ \hline MDA-00-3461 [8] & 12.6 & 11.2 * & 15.4 & 15.6 & 4.4 & 250 & MDA-250 \\ MDA-00-3461 [8] & 12.6 & 11.2 * & 15.4 & 15.6 & 4.4 & 550 & MDA-550 \\ Mixture 1825 [21] & 12.5 & 7.4 & 32.1 & 21.7 & 4.7 & 250 & Mixture-250 \\ \hline \end{tabular}
\end{table} 
Table 1: Material constants and parameters of the three types of TN- liquid crystal LC terahertz (THz) phase shifters (* value calculated using the ratio \(k_{2}/k_{1}=0.89\)[22]). \(\varepsilon_{0}\) is the vacuum permittivity \(8.85\times 10^{-12}\) F/m. Figure 2: Schematic illustration of the operational principles of a TN-LC THz phase shifter for (**a**) V\({}_{\text{RMS}}\) = 0 and (**b**) V\({}_{\text{RMS}}\) > V\({}_{\text{TN}}\). The threshold voltage for reorienting the LC molecules is given by \(V_{TN}=\frac{V_{T}}{\sqrt{1-\left(\frac{k_{3}-2k_{2}}{k_{1}}\right)\mu^{2}}}=V_{T} \sqrt{[1+\frac{k_{3}-2k_{2}}{k_{1}}\ \frac{\Phi^{2}}{\pi^{2}}]}\). \(V_{T}=\pi-\frac{\sqrt{k_{1}}}{\sqrt{\varepsilon_{0}(\varepsilon_{//}- \varepsilon_{\bot})}}\) is the threshold voltage in a homogenously aligned LC cell and \(\mu^{2}=\left[(1-\frac{k_{3}-k_{2}}{k_{2}})+\frac{k_{1}}{k_{2}}\ \frac{\pi^{2}}{\Phi^{2}}\right]^{-1}\); \(\Phi\) is the total twist angle which is defined by the rubbed direction at the boundaries z = 0 and z = d of a TN-LC cell. Furthermore, \(k_{1}\), \(k_{2}\), and \(k_{3}\) are the splay, twist, and bend elastic constants, respectively, whereas the dielectric constants along the perpendicular axis to the preferred axis are \(\varepsilon_{//}\) and \(\varepsilon_{\bot}\), respectively, for the uniaxial symmetry in a LC cell. All the above-mentioned constants of both types of LCs investigated, namely, MDA-00-3461 and Mixture 1825, are also listed in Table 1. The maximum tilt angle of the LC director, \(\theta_{\text{max}}\), is assumed to be at the mid-point position of the TN-LC cell (i.e., \(z=d/2\)). In this work, the THz wave is assumed to be propagating along the z-direction, normal to the TN-LC cell.