Using Legendre polynomials Formulas at Fresnel Integral Diffraction

Abstract

Many advanced scientific phenomena of optics, light, and physics can be represented mathematically in the form of Volterra integral equations. In this paper, we studied the diffraction phenomena of the light beam. Since the most important category of scalar diffraction theories is the Fresnel diffraction integral. These integrals have been used in numerous studies on the propagation effects of structured light beams. These scalar diffraction theories have been widely utilized instudying the propagation of structured light. finally, we applied first-kind shifted Legendrepolynomials to find the interpolate solutions of weakly singular Volterra integral equations of thesecond kind, where the Fresnel integral of diffraction will be involved. Numerical examples havebeen included in order to show the efficiency of the presented method. The exact solution of the represented example is compared to the approximate solution and the absolute error is calculated to illustrate the efficiency of the proposed method.