Abstract

Recently, the beam propagation method (BPM) has been significantly improved after finite-difference schemes replaced the FFT scheme. There are two kinds of commonly used finite-difference schemes. One is the Crank-Nicolson methodfl] in which a tridiagonal matrix has to be found followed by finding its inverse matrix for each propagation step. This scheme is stable and accurate. For a weakly guiding structure, a large step-size can be used with a reasonable accuracy. However, for a three dimensional problem, the matrix becomes complicated. It will not be easy to find the inverse matrix. The other is simply an explicit (centred forward-difference) method[2]. The method gives a very simple formulation for each propagation step, which is good for solving a three dimensional problem. However its stability problem is critical. In order to keep the scheme stable, the propagation step has to be less than a certain size which is usually very small, in some cases, too small to be applicable.

© 1994 Optical Society of America