## 3 Theoretical Formulism  
Expected performances of the phase shifters can be analyzed using the approach that we reported earlier [3]. In the absence of the external field, the LC molecules were initially in equilibrium and were oriented more or less parallel to the substrates (Figure 1a). Because of the Freedericksz transition, when the applied bias was greater than the threshold voltage, the LC molecules will reorient themselves toward the direction of the external field. The threshold voltage for reorienting the LC molecules is given by \(V_{\text{th}}=\pi\cdot(k_{1}/(\varepsilon_{0}\cdot\Delta\varepsilon))^{1/2}L/d\)[23], where \(\varepsilon_{0}=8.854\)\(\times\)\(10^{-12}\) F\(\cdot\)m\({}^{-1}\), \(\Delta\varepsilon=\varepsilon_{//}-\varepsilon_{\perp}=11.2\), \(k_{1}=12.6\)\(\times\)\(10^{-12}\) N, and \(L\) and \(d\) are the free-space permittivity, dielectric anisotropy, style elastic constants, distance between two electrodes, and thickness of the LC layers, respectively. The maximum tilt angle of the LC director, \(\theta_{\text{max}}\), is assumed to be found at the mid-point position of the LC cell, i.e., z = \(d/2\). Here, the THz wave is assumed to be propagating along the z-direction, normal to the LC cell. The relationship between \(\theta_{\text{max}}\) and \(V\) can be determined by solving [23]:  
\[\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{\left(1+\zeta\sin^{2}\theta_{ \text{max}}\sin^{2}\delta\right)\left(1+\rho\sin^{2}\theta_{\text{max}}\sin^{ 2}\delta\right)}}{\sqrt{\left(1-\sin^{2}\theta_{\text{max}}\sin^{2}\delta \right)\left(1+\rho\sin^{2}\theta_{\text{max}}\sin^{2}\delta\right)}}d\delta \tag{1}\] \[=\frac{\pi}{2}\frac{V}{V_{\text{th}}},\]  
where \(\theta\) is the tilt angle of the LC molecule, sin\(\delta\) = sin\(\theta\)/sin\(\theta_{\text{max}}\). We define \(\eta\) = sin\(\theta_{\text{max}}\), then sin\(\theta\) (\(z\)) = sin\(\theta_{\text{max}}\cdot\) sin\(\delta^{\prime}\)(\(z\)), \(\delta^{\prime}\)(\(z\)) being the derivative of the integrating variable. In Equation (1), \(\zeta\) = \((k_{3}-k_{1})/k_{1}\)\(\approx\) 0.22, \(k_{3}\) = 15.4 \(\times\)\(10^{-12}\) N, \(\rho\) = \((\varepsilon_{//}-\varepsilon_{\perp})/\varepsilon_{\perp}\)\(\approx\) 2.55, \(\varepsilon_{\perp}\) = 4.4 for the LC that we used. \(V\) is the applied voltage. Equation (1) can be rewritten and used to find \(\theta\) at every \(z\) position in the LC cell, as follows:  
\[\int_{0}^{\delta^{\prime}}\frac{\sqrt{\left(1+\zeta\eta^{2}\sin^{2}\delta \right)\left(1+\rho\eta^{2}\sin^{2}\delta\right)}}{\sqrt{\left(1-\eta^{2}\sin ^{2}\delta\right)\left(1+\rho\eta^{2}\sin^{2}\delta\right)}}d\delta=\frac{\pi} {2}\frac{E\cdot z}{V_{\text{th}}}, \tag{2}\]  
where \(E\) is the electric field applied to the device. When considering the effective birefringence of the LC, the total phase shift experienced by the THz wave propagating through the device, \(\Delta\phi\), is given by  
Figure 1: **(a)** Schematic diagrams of three different configurations of indium–tin–oxide-nanowhiskers (ITO-NWhs)-based phase shifters, type A, B, and C, respectively. The cross-sectional images of (**b**) the substrate of ITO NWhs, (**c**) ITO NWs coated with polyimide, and (**d**) ITO NWhs treated with the rubbing process. \[\Delta\phi(V)=\int_{0}^{d}\frac{2\pi f}{c_{0}}\left[\left(\frac{\cos^{2}\theta}{n_{ \mathrm{o}}^{2}}+\frac{\sin^{2}\theta}{n_{\mathrm{e}}^{2}}\right)^{-1/2}-n_{o} \right]dz, \tag{3}\]  
where_f_ and _c\({}_{0}\)_ are the frequency and speed, respectively, of the THz wave in vacuum. The extraordinary and ordinary refractive indices of MDA-00-3461 in a frequency range of 0.3-1.4 THz at 25 \({}^{\circ}\)C are essentially frequency-independent: \(n_{\mathrm{e}}=1.74\) and \(n_{\mathrm{o}}=1.54\), respectively [24]. The aforementioned model is suitable for a thin LC cell (thickness of approximately 20 \(\upmu\)m or less).