Solution of the eigenvalue problem of an integral equation with the help of its associated differential equation applied to the calculation of diffraction losses in confocal resonators

Abstract

A problem in optics is chosen in order to develop a general method for solving the eigenvalue problem of a homogeneous integral equation with the help of the eigenfunctions of the associated differential equation. The problem chosen is that of the modes of optical resonators with circular, confocal mirrors which are given by the solutions of a homogeneous Fredholm integral equation which can be derived from Kirchhoff ’s diffraction formula. This integral equation can be converted into a hyperspheroidal differential equation supplemented by appropriate boundary conditions. The solutions and eigenvalues of this differential equation are studied in detail for both small and large values of a parameter called the Fresnel number. These eigenfunctions are then used for the computation of the eigenvalues of the original integral equation which measure the diffraction loss in the resonator. Throughout the same general perturbation method is used, and our emphasis is on the solution of the general problem of solving the eigenvalue problem of the homogeneous integral equation together with that of its related differential equation.