Mathematical modeling of piezo-electro-luminescent effect and diagnostics of pressure distribution along fiber optic sensor

The algorithm of finding the distribution function of pressure along the three-phase fiber optic sensor based on the results of light intensity proceeding from a fiber optic phase measured on the edge section of the sensor is developed for a case of nonlinear "function of a luminescence" which is a dependence between the intensity of light and voltage acting on the electrophosphor. The problem is reduced to the solution of the Fredholm integral equation of the 1st kind with the differential kernel depending on the calculated effective parameters of the sensor and on the derivative set function of a luminescence of an electrophosphor. The analytical solution has been obtained for the function of probabilities density of pressure distribution for a special case when the kernel is expressed by the delta-function; and the Fredholm integral equation is reduced to an algebraic one. "Direct" and "reverse" problems of the Fredholm integral equation for a case of nonlinear function of a luminescence of an electrophosphor are solved as an illustration of the algorithm. The light intensity derivative at the optical fiber output of control voltage for the set uniform law of probability-density function with regard to pressure distribution is found in a direct problem. As for the reverse problem, the probability-density function is determined in comparison with the known exact solution using the direct problem solution for the derivative of light intensity. The numerical solution of the reverse problem is carried out in different approximations in which the distribution of nodal points in intervals and required nodal function values of probability-density pressure are found from a condition of minimizing summary discrepancies based on the values of the light derivative intensity which have been set and calculated on each step based on control voltage at the output of the optical fiber.