A numerically stable rational approximant for the split-step Padé propagator

Abstract

The split-step Pade algorithm due to Collins [J. Acoust. Soc. Am. 93, 1736-1742 (1993)] provides a fast and accurate method for solving the parabolic equation (PE). The formal solution to the PE propagator, which involves the pseudo-differential operator (1+X)^1/2, is replaced by an [n/n]-Pade rational approximant. This approximant can be expanded either as a product or a sum of rational-linear terms, each term leading to a tridiagonal system of equations in X which is readily solved numerically. To ensure adequate suppression of undesirable contributions from the evanescent part of the spectrum (X<-1), stability constraints must be imposed. In contrast to this approach, we follow the suggestion of Lu and Ho [Optics Lett. 27, 683-685 (2002)] and examine the use of an [n-1/n]-Pade approximant that inherently dampens these evanescent components of the spectrum. An algorithm for generating the necessary coefficients is described. Transmission losses computed using both rational approximants are compared for a typical shallow-water configuration.