# Liquid Crystal-Embedded Hollow Core Fiber Temperature Sensor in Fiber Ring Laser
## 2 Working Principle and Experimental Setup  
The schematic diagram of the optical fiber sensor is shown in Figure 1. The structure consists of a hollow optical fiber (HCF) containing a liquid crystal (E7, King Optronics Co., Ltd., Suzhou, China). Both ends are coreless fiber with an equal length of 1.5 mm. Single mode fiber (SMF), no core fiber (NCF), and HCF are coaxially connected by commercial fusers (FSM-60S, Fujikura, Tokyo, Japan). The HCF has an outside diameter of 125 \(\upmu\)m and an inside diameter of 80 \(\upmu\)m with a pure silicon capillary length of 10 mm. Then, through the set process and parameters, the splicing parameter conditions are modified to avoid the excessive collapse of pores and ensure the appropriate strength. Accurate length cutting of NCF and HCF with vernier calipers and fiber cutters. As shown in Figure 1, the first segment of the NCF, which acts as a beam splitter, outputs part of the light channeled in the SMF to the higher-order mode, since the lead-in of SMF, NCF, and HCF has a circular cross and the fusion splice between them is aligned accurately. When light is emitted into HCF, due to the mode field mismatch, a portion of the fundamental mode (LP01) beam is coupled to the higher order mode, and then the beam is further emitted into HCF. As the refractive index of the core mode and the cladding mode are different, the interference occurs when the two beams recombine. The proposed interferometer involves the interference of the fundamental mode and the higher mode. The two-mode interference model is used to simplify the analysis of the transmitted light intensity:  
\[I=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}COS\Delta\varphi \tag{1}\]  
where \(I\) is the transmitted light intensity of NHN structure, and \(I_{1}\) and \(I_{2}\) are the intensity of core mode and cladding mode, respectively. \(\Delta\varphi\) is the phase difference between two beams and can be represented as:  
\[\Delta\varphi=\frac{2\pi l}{\lambda}(n_{2}-n_{1})=\frac{2\pi l}{\lambda}\Delta n \tag{2}\]where \(\lambda\) is the wavelength of free space light, L is the length of HCF, \(n_{1}\) and \(n_{2}\) are the RI of the air cavity mode and the silicon mode, respectively, and \(\Delta n\) is the RI difference of the two interference beams. Liquid crystal molecules are generally rigid rod-like molecules. Due to the different molecular groups connected to the head, tail, and side of the molecule, the liquid crystal molecules have different properties in the two directions of the long axis and the short axis, and become polar molecules. Ordered by intermolecular forces, the long axis of liquid crystal molecules is always parallel to each other, or has a preferred direction, and the center of mass is free. When the light propagates in the liquid crystal, the optical refractive index anisotropy, namely birefringence, occurs, and the refractive index (\(n\)) is affected by the molecular structure of the liquid crystal. When light passes through a uniaxial crystal, the refractive index of unusual light (\(n_{\mathrm{e}}\)) is greater than that of ordinary light (\(n_{0}\)). The results show that the propagation velocity of light in liquid crystal is ve < v0; the change of temperature field leads to the change of \(\Delta n\) value, resulting in the deviation of interference peak. It shows that there are different propagation velocities for light vibration in different directions. NHN structure is sensitive to temperature, and thermophonic effect reflects the change of RI of liquid crystal. It is worth noting that the liquid crystals are almost completely thermally induced. The transition between different liquid crystal states within a certain concentration range is called an intermediate phase. In this regard, LC is an ideal choice for implementing a tunable hollow fiber (LC-HCF) temperature sensor. As shown in Figure 2, the anisotropic optical properties of E7 can be described by the special and ordinary refractive indices of \(n_{e}\) and \(n_{o}\).