Modified finite-difference beam propagation method based on the generalized Douglas scheme for variable coefficients

The accuracy of the implicit finite-difference beam propagation method (FD-BPM), in which the phase term is not split, is improved using the generalized Douglas scheme. The propagation error of the fundamental mode in two- and three-dimensional waveguides is evaluated by the mode-mismatch loss calculation. It Is demonstrated that the truncation error is reduced to O(Δx)4 in the transverse direction, even when the parabolic wave equation contains variable coefficients, The computational time is almost identical to the conventional FD-BPM based on the Crank-Nicholson scheme