Numerical methods for solving the eigenvalue problem for a positive branch confocal unstable resonator

Abstract

The resonator problem for a positive branch confocal unstable resonator reduces to a Fredholm homogeneous integral equation of the second kind, whose numerical solution here is based on a sequence of algebraic eigenvalue problems. We compare two algorithms for the solution of an optical resonator problem. These are obtained by (i) successive degenerate kernel approximation by Taylor polynomials of the Fredholm kernel and (ii) Nyström’s method with Simpson’s rule as the subordinate numerical integration method. The numerical results arising from these routines compare well with other published results, and have the added advantage of simplicity and easy adaptability to other resonator problems.