Reformulation for the beam-propagation method

Splitting scalar fields into a sum of counterpropagating waves leads to formulation of the Helmholtz equation as two coupled parabolic partial differential equations. This rigorous formalism naturally leads to the beam-propagation method (BPM) equation when the backward-propagating field is neglected, but it also permits discussion of various corrections. Two problems specifically are analyzed: wide-angle propagation in near-uniform systems and propagation through lenslike systems, with a strongly discontinuous refractive index. Finally, a BPM modeling is proposed for fiber microlens operation.