The first-order paraxial approximation is used to obtain the distributions of the electric and magnetic fields for the core and cladding hybrid fiber modes. The coupling coefficients of these modes are found for fibers subject to twist. The longitudinal electric field component determines the mode coupling in twisted fibers. It is shown that in the first-order paraxial approximation the cladding hybrid modes propagating in a twisted fiber rotate along the direction of the twist at the same rate as the core mode, independently of the azimuthal and radial mode numbers. Four hybrid modes constituting one linearly polarized mode have different longitudinal components, and the corresponding cladding-mode resonances of a long-period fiber grating undergo different shifts owing to different mode self-coupling coefficients. This results in the removal of mode degeneracy and splitting of resonances of long-period gratings in twisted fibers.
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