# Electrically Tunable Propagation Properties of the Liquid Crystal-Filled Terahertz Fiber
## 3 Results and Discussion
### Polarization Dependence of the Electrically Tunable Bandgaps  
Once an external electric field is applied across the fiber, the director can be reoriented along the applied field, which is an inherent property of nematic LC [34]. For instance, if the applied field is along the \(x\)-axis (\(V_{x}=V_{0}\), \(V_{y}=0\)), then the rotation angle of the director \(\varphi\) is 0, and the dielectric tensor of the LC takes a form \(\varepsilon_{\rm LC}={\rm diag}\big{(}n_{e}^{2},n_{o}^{2},n_{e}^{2}\big{)}\)[21; 22]. The alignment of the LC director is actually not homogeneous even though the external electric field is high enough, since it is dependent on the geometry of the LC director and the effect of strong boundary conditions [35]. For simplicity in numerical analysis, the nonhomogeneity of the director was neglected in this paper. The bandgap characteristics of the infinite cladding structure (\(d/\Lambda=0.4\)) with an external electric field applied along the x-axis were studied. There exist two effective polarization-independent total bandgaps in which both the x-polarized fundamental modes HE\({}_{\rm 11x}\) and the y-polarized fundamental modes HE\({}_{\rm 11y}\) can be guided, as shown in Figure 3a. However, the modal-dispersion curves for both the HE\({}_{\rm 11x}\) and HE\({}_{\rm 11y}\) modes exceed the polarization-independent total bandgaps. So, the polarization-independent total bandgaps are insufficient to describe the propagation properties of the LC-MOF when external  
Figure 2: Bandgaps of cladding structure for the designed THz LC-MOF without external static electric field across the fiber, \(\varepsilon_{\rm LC}={\rm diag}\big{(}n_{o}^{2},n_{o}^{2},n_{e}^{2}\big{)}\).  
electric field is applied across the fiber, and it is necessary to investigate the polarization-dependent bandgaps [22].  
The index contrast between LC and the background material (\(n_{o}/n_{b}\), \(n_{e}/n_{b}\)), and the anisotropy of LC (\(n_{e}/n_{o}\)) were not significantly large; therefore, the coupling effect between two polarization components of the electric field vector could be neglected [21]. The x- and y-polarization bandgaps can be evaluated by using the cladding structure with \(\epsilon_{\rm LC}=\rm diag(n_{e}^{2},n_{e}^{2},n_{o}^{2})\) and \(\epsilon_{\rm LC}=n_{o}^{2}\), respectively. Figure 3b shows the polarization-dependent bandgaps obtained by neglecting the coupling effect. The pink and green shaded regions show the x- and y-polarization bandgaps, respectively. Comparing Figures 2 and 3, it can be found that the external electric field causes bandgap splitting, and shifts the x-polarization bandgaps towards a lower frequency due to the high difference between \(\rm n_{o}\) and \(\rm n_{e}\). The dispersion curves of the x- and the y-polarized fundamental modes are coincidental with the x- and y-polarization bandgaps, respectively. Thus, the x(y)-polarized modes are supported by the corresponding x(y)-polarization bandgaps. The overlap regions of the x- and  
Figure 3: Bandgaps and modal dispersion curves for polarized fundamental modes \(\rm HE_{11x}\) (solid red line) and \(\rm HE_{11y}\) (dotted blue line) of the THz LC-MOF (\(d/\Delta=0.4\)) with rotation angle of the LC director \(\varphi=0\). (**a**) Polarization-independent total bandgaps, (**b**) polarization-dependent bandgaps. Bottom: mode profiles and transverse electric-field distributions (arrow diagram).  
y-polarization bandgaps in Figure 3b represent the polarization-independent total bandgaps, which are consistent with that in Figure 3a. Therefore, ignoring coupling terms yields little change to the bandgaps. The polarization-dependent bandgap splitting caused by the transverse electric field can give rise to some interesting characteristics, such as electrically tunable high birefringence or single-mode single polarization propagation of THz wave.