## 3 Theoretical Formulism
However, in this study, the thickness of the LC layer for the 2\(\uppi\) THz phase shifter was approximately 1.12 mm, which was approximately 100 times thicker than typical LC devices that were designed in the visible frequency range. In such thick cells, LC molecules more than a few tens of microns away from the cell walls will not be exactly parallel to the rubbed direction in the absence of the applied electric field. Besides, almost all the LC molecules can be easily reoriented in the direction, which is parallel to the external field when the LC cell is biased. In Figure 2a, we drew a thick LC cell in the e-ray mode without bias. The LC molecules that are sufficiently away from the substrates, highlighted in yellow, are not oriented toward the rubbed direction. After applying a sufficiently high voltage, e.g., \(V_{\mathrm{max}}=70\) V, we can achieve the o-ray mode for the cell shown in Figure 2b. From the measured phase difference for THz wave propagation through LC cells in these two modes, we find the maximum possible difference in relative phase retardation, \(\Delta\phi(V_{\mathrm{max}})\) can be significantly different from the total relative phase shift in the ideal case, 2\(\pi\)_f_ (\(n_{\mathrm{e}}-n_{\mathrm{o}}\))_d_/_c\({}_{0}\). We propose, therefore, to apply a correction factor for the phase shift,  
\[\alpha=[2\pi\textit{f}\ (n_{\mathrm{e}}-n_{\mathrm{o}})\textit{d}\ /c_{0}]/\Delta\phi(V_{\mathrm{max}}) \tag{4}\]  
If the LC molecules in the middle of the cell are not perfectly aligned with the rubbing direction because of the weak anchoring ability of the thick LC device, \(\alpha>1\). As mentioned above, the effect of pretilt angle on phase shift in our devices cannot be neglected due to the glancing-angle structure of ITO NWs. Figure 2b-d are used to describe the experimental method for measuring the pretilt angle of our devices. We first measured the effective refractive index, \(n_{\mathrm{p}}\), which is the sum of the refractive indices in the o-ray mode, \(n_{\mathrm{o},\ \mathrm{Exp}}\), and the birefringence that is caused by the pretilt angle, \(\Delta n_{\mathrm{E}}\), in the absence of the external field (see Figure 2c). The same procedure was repeated with the cell biased at 70 V (rms), as shown in Figure 2b. The parameters, \(\Delta\phi_{\mathrm{E}}\), and \(n_{\mathrm{p}}\) are then related by  
Figure 2: Configurations for determining the correction factors of the thickness (\(\alpha\)), and pretilt angle (\(\beta\)) of liquid crystal (LC) devices. The cell (**a**) in the e-ray mode without external field, (**b**) in the o-ray mode with high driving voltage, and (**c**) in the o-ray mode without external field. (**d**) Index ellipsoid used to describe the pretilt angle (\(\theta_{0}\)). \[n_{\rm p}(\theta_{0})=\Delta n_{\rm E}+n_{\rm o,\ Exp}=\frac{c_{0}}{2\pi fd}\Delta \phi_{\rm E}+n_{\rm o,\ Exp}, \tag{5}\]  
where \(\theta_{0}\) is the pretilt angle. On the other hand, \(n_{\rm p}\) can also be determined by using the index ellipsoid shown in Figure 2d,  
\[n_{\rm p}(\theta_{0})=\left[\frac{\sin^{2}(\theta_{0})}{n_{\rm e}^{2}}+\frac{ \cos^{2}(\theta_{0})}{n_{\rm o}^{2}}\right]^{-1/2}. \tag{6}\]  
The pretilt angle can then be determined by the above procedure and applying Equations (5) and (6). Taking into account of the thick cell and effect of the pretilt angle mentioned above, we obtain an expression for the effective phase shift,  
\[\Delta\phi_{\rm eff}=(\alpha)(\beta)(\Delta\phi), \tag{7}\]  
where the compensation factor for pretilt, \(\beta\), is given by,  
\[=\frac{\pi}{2}\frac{V}{V_{\text{th}}},\]0  
For the case of \(\beta<1\), the LC cell exhibits a pretilt angle in the absence of the bias electric field.