# Electrically Tunable Propagation Properties of the Liquid Crystal-Filled Terahertz Fiber
## 2 Design of the THz LC-MOF  
The cross-section of the designed LC-MOF is shown in Figure 1. We constructed the fiber cladding by arranging circular holes (with hole diameter \(d\)) in a triangular lattice pattern (with lattice constant \(\Lambda\)) in the background of Topas COC. The cladding holes were infiltrated with 5 CB. As the refractive indices of 5 CB (\(n_{o}\approx\) 1.58, \(n_{e}\approx\) 1.77, 25 \({}^{\circ}\)C) [17] are higher than the refractive index of background material Topas (\(n_{b}\) = 1.5258) [11] in the THz frequency range, the propagation mechanism of the designed LC-MOF is based on the PBG effect.  
Several factors should be considered to design suitable fiber structural parameters. Firstly, an appropriate value of \(d\) and \(\Lambda\) should be adopted to insure the cladding of the LC-MOF has PBGs in the THz frequency range (0.2 \(\sim\) 3 THz) and the unfilled index guiding MOF (with a triangular air-hole lattice cladding) is endlessly single-mode, i.e., the hole diameter to pitch ratio (\(d/\Lambda\)) should be less than 0.45 [23]. Secondly, the number of the cladding hole rings is also an important structural parameter. Too many cladding hole rings increase the outer diameter of the fiber to several millimeters, and reduce both the flexibility and the tuning response rate of the THz LC-MOF, as \(\Lambda\) is comparable to the guiding wavelength. However, the high-index inclusions (i.e., the LC cylinders embedded in the low-index Topas background) are responsible for the PBG guiding mechanism and confinement loss [24]. Decreasing the number of the cladding hole rings diminish the PBG effect and lead to a significant increase in confinement loss [16]. Considering the compromise between the flexibility and confinement loss of the fiber, three rings of cladding holes with \(d/\Lambda\) = 0.4 were adopted for the optimized THz LC-MOF. The designed structure is relatively simple and can be fabricated by the extrusion-stretching technique [25], which is suitable for mass production.  
Local orientation for the rodlike molecules of the nematic liquid crystal 5 CB is described by an unit vector \(n\) (as shown in Figure 1), i.e., the director, which is parallel to the direction of the average orientation of the molecules. Experimental results of [PERSON] et al. [26] show that the director of LC filled in the cladding holes (with diameter of several microns) of MOF is parallel to the fiber axis when temperature is below \(T_{c}\) and without external static electric (or magnetic) field across the fiber. Dielectric tensor of the nematic LC takes the form \(\varepsilon_{\text{LC}}=\text{diag}\left(n_{o}^{2},n_{o}^{2},n_{e}^{2}\right)\). However, the homogeneous alignment of LC filled in the cladding holes (with diameter about tens to hundreds of microns) of THz MOF could be achieved under the influence of the appropriate anchoring conditions [17].  
When an external electric field is applied, the director can be reoriented along the applied field [27; 28]. Almost uniform orientation of the director in the x-y plane can be controlled by two orthogonal sets of electrodes with sufficiently strong voltage as shown in Figure 1. So, the nonhomogeneity of the director was neglected in the following analysis for simplicity of calculation. Dielectric tensor of the nematic LC takes the form of:  
\[\varepsilon_{r}=\left[\begin{array}{ccc}\varepsilon_{xx}&\varepsilon_{xy}&0 \\ \varepsilon_{yx}&\varepsilon_{yy}&0\\ 0&0&\varepsilon_{zz}\\ \end{array}\right] \tag{1}\]where  
\[\varepsilon_{xx}(r)=\varepsilon_{o}(r)\sin^{2}\varphi+\varepsilon_{e}(r)\cos^{2}\varphi\]  
\[\varepsilon_{xy}(r)=\varepsilon_{yx}(r)=[\varepsilon_{e}(r)-\varepsilon_{o}(r) ]\sin\varphi\cos\varphi\]  
\[\varepsilon_{yy}(r)=\varepsilon_{o}(r)\cos^{2}\varphi+\varepsilon_{e}(r)\sin^{ 2}\varphi\]  
\[\varepsilon_{zz}(r)=\varepsilon_{o}(r)\]  
\(\varphi\) is the rotation angle of the director with the x-axis, and it can be arbitrarily controlled by the external static electric field.