Mathematical Model for Diagnosing Strains by an Optical Fiber Sensor with a Distributed Bragg Grating According to the Solution of a Fredholm Integral Equation

Abstract

A mathematical model for the diagnostics of strain distribution along a optical fiber sensor (light guide) with a base distributed weakly reflecting fiber Bragg grating is developed. In the initial, undeformed state, the base Bragg grating has a constant period along the light guide. The base uniform Bragg grating is transformed to a nonuniform one as a result of emergence of nonuniform axial strains along the light guide. These changes in the period of Bragg grating lead to corresponding informative changes in reflection coefficients for various lengths of the waves entering into the optical fiber. As a result, the problem on finding the required density distribution function of axial strains along the optical fiber is reduced to solution of the Fredholm integral equation of the 1st kind by using the measured values of derivatives of reflection coefficients at the exit of the light guide. The results of numerical modeling of the spectra of coefficients of reflection from the nonuniform deformed Bragg grating are presented for various model and real distribution laws of diagnosed strains along the light guide.