Modified local one-dimensional beam-propagation method based on the Douglas scheme

A finite-difference algorithm that combines the local one-dimensional approximation and the Douglas scheme is suggested to solve the three-dimensional non-linear Schrödinger equation. The truncation error in this scheme is reduced to O(Δx)4 in the transverse direction, whereas that in the conventional local one-dimensional approximation is O(Δx)2. Using this scheme, the computational accuracy and computational efficiency can be improved drastically without the introduction of additional programming works and computational time. Numerical results obtained for an elliptic Gaussian beam propagating in non-linear medium verify the advantages of this method.