# Electrically Tunable Propagation Properties of the Liquid Crystal-Filled Terahertz Fiber
## 3 Results and Discussion
### Structural Parameter Dependence of Bandgaps  
Bandgaps of the infinitely cladding structure (periodic triangular lattice of LC inclusions in a background of Topas COC) without external static electric field across the fiber (\(V_{x,y}=0\) and director is parallel to the \(z\)-axis) were calculated. Figure 2 shows the first bandgap (boundary lines) of the cladding structure with different \(d/\Lambda\). The vertical axis is effective mode index \(n_{\rm eff}\), and the horizontal axis is normalized frequency \(\rm F=f\Lambda/c\). For a certain \(d/\Lambda\), the area between the corresponding boundary lines represents the bandgap in which the cladding modes are forbidden and the core modes are guided. The blue shaded region shows the bandgap of the cladding structure with \(d/\Lambda=0.4\). The horizontal core-line corresponding to refractive index of Topas represents the boundary between states that are propagating or evanescent in the cladding. The bandgap can be characterized by the highest bandgap frequency \(\rm F_{H}\) (where the high frequency bandgap boundary line crosses the core-line) and the lowest bandgap frequency \(\rm F_{L}\) (corresponding to the low-frequency edge at the bottom of the PBG, as shown in Figure 2). The maximum transmission bandwidth and the central frequency of the transmission band can be expressed by \(\rm\Delta F=F_{H}-F_{L}\) and \(\rm F_{c}=(F_{H}+F_{L})/2\), respectively. Then, the relative bandwidth is defined as: \(\rm\Delta F/F_{c}\).  
It can be seen that the increment of the \(d/\Lambda\) drastically shifts the bandgap towards a lower frequency. The cladding structure with \(d/\Lambda=0.4\) has lower central frequency (\(\rm F_{c}=2.9\)) than the  
Figure 1: Cross section of the liquid-crystal microstructured optical fiber (LC-MOF) and diagram for arbitrarily controlling the orientation of the nematic director in the x–y plane by two orthogonal sets of electrodes with voltage, \(V_{x}\) and \(V_{y}\).  
cladding structures with \(d/\Lambda=0.3\) (F\({}_{\rm c}=3.8\)), which means the former has a smaller fiber diameter and thus better flexibility than the later with the same central frequency. According to the structural parameter dependence of bandgaps, as shown in Figure 2, the bandgap can be moved to certain frequency in the range of 0.2-1.5 THz (in which the material dispersion of Topas and 5 CB is negligible) by scaling and tailoring the structural parameters of the fiber. For example, if we need a 5 CB-filled Topas MOF that has a bandgap centered at 1 THz, it is possible to set structural parameters \(d=348\) um, and \(d/\Lambda=0.4\) or \(d=342\) um and \(d/\Lambda=0.3\). As the bandgaps are formed based on antiresonances of the individual LC inclusions, so are the positions of the bandgaps strongly dependent on the cladding-hole diameter and weakly on the lattice constant [31].  
The cladding structure with \(d/\Lambda=0.4\) has a wider relative bandwidth (\(\Delta\)F/F\({}_{\rm c}=0.97\)) than the cladding structures with \(d/\Lambda=0.5\), 0.6 and 0.7. In addition, the unfilled index-guiding MOF with \(d/\Lambda=0.4\) (less than 0.45) is endlessly single-mode [23]. For applications such as polarizers [28], filters [32], and switches [33], only a short piece of LC-MOF is required. The LC-MOF is always a short section of the entire fiber, and the main origin of the insertion loss is the coupling loss between the index-guiding section and the bandgap-guiding section [19; 20; 26]. Besides the inherent advantage of single-mode propagation, the endlessly single-mode property of the unfilled MOF can reduce the coupling loss between the unfilled section and the filled section in a wide frequency range as a result of the well-confined characteristic of the fundamental mode. So, the structure with \(d/\Lambda=0.4\) is adopted for the optimized THz LC-MOF.