New formulation of the beam propagation method based on the slowly varying envelope approximation

We combine the theoretical results of some recent publications on the beam propagation method based on the finite difference method (FDBPM) and the slowly varying envelope approximation (SVEA). The use of the SVEA leads to a considerable reduction of the computation time but introduces a fundamental error in the phase velocity of a given mode and also to an error due to the finite step size along the propagation direction. A new, effective scheme is given, which results in a considerable reduction of these errors, without much increase of the computational effort. An error analysis and a few results of applications are given.