Two Dimensional Richardson Extrapolation for Optical Waveguide Problems

Abstract

Light propagation in optical waveguides is studied in the paraxial approximation using Richardson extrapolation. Highly accurate solutions have been efficiently obtained using Richardson extrapolation and the mid-step Euler finite different method [1]. In Richardson extrapolation, numerical solutions with lesser accuracy are extrapolated to zero step size to obtain a more accurate solution. Richardson extrapolation is a simple algebraic procedure that can be used with any numerical scheme to improve the accuracy of the solution [2]. Richardson extrapolation stabilizes the unstable mid-step Euler method and also allows one to use Hadley’s transparent boundary conditions. When solving the paraxial wave equation using finite difference methods, both the propagation and transverse dimensions are discretized. Discretization errors arising from both the transverse and the propagation dimension have to be minimized in order to obtain an accurate solution. We solve this problem by applying Richardson extrapolation in both directions.