Accurate solution of light propagation in optical waveguides using Richardson extrapolation

Light propagation in optical waveguides is studied in the paraxial approximation using Richardson extrapolation and the mid-step Euler finite difference algorithm. Highly accurate solutions can be efficiently obtained using this combined approach. Since discretization errors in both the transverse and propagation dimensions are greatly reduced, accuracies on the order of 10-11or better are obtained. Richardson extrapolation allows us to use transparent boundary conditions, which normally can only be used with the Crank-Nicholson method. Richardson extrapolation also allows us to stabilize the mid-step Euler method which is explicit and thus use it in place of Crank-Nicholson method which is implicit. Implicit schemes do not vectorize well on the CRAY machines which we are using while explicit schemes do. Consequently, the approach presented here is competitive in CPU cost to the Crank-Nicholson method while generating results of significantly larger accuracy. To illustrate this approach we apply it to the study of a straight, integrated optical waveguide and a y-junction and compare the results to the results from a Crank-Nicholson approach.