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A theoretical study on a long-period liquid-crystal (LC) fiber grating is presented by means of an anisotropic waveguide analysis and the coupled-mode theory with the discretization method. Effective propagation and coupling constants are found for a hollow-core fiber filled with nematic liquid crystal, the optic axis of which is in the direction of the fiber axis. The results show that when uniaxial perturbations of the liquid-crystal director exist in the core by external electric fields, the mode couplings can occur between modes with different azimuth variations as well as between modes with the same. The cross couplings between the fundamental core mode and the transverse-magnetic cladding modes are most strong compared even to the self-coupling of the fundamental core mode, and thus the spectral shift with the growth of the director modulation becomes negligible in the weak-modulation regime. Using the derived theoretical results, we discuss numerical examples of long-period LC fiber gratings with respect to the modulation depth of the director and the polarization states of the fundamental core mode. The characteristics of the LC fiber can be modified in many ways based on the LC material, and the dynamic controllability of LC fiber gratings can have many applications in fiber-optic systems. Thus this theoretical analysis could be utilized for them.
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