Boundary conditions for the finite difference beam propagation method based on plane wave solutions of the Fresnel equation

Each particular implementation of the beam propagation method (BPM) requires a special procedure allowing for radiation to leave the computational window. We propose a new approach to constructing the finite difference schemes of the BPM at the boundary of the computational window. These schemes are independent of the computed fields and allow for a similar treatment of both interior and boundary points. The new approach can be further improved by correcting the field values at the boundary points according to Hadley´s method. The algorithm is easy to implement for both two- and three-dimensional structures. The new method considerably reduces computation times because the propagation matrices remain constant in longitudinally invariant sections, thus avoiding repeated LU-decompositions. The basic idea-establishing the finite difference scheme such that locally exact, approximate, or plausible solutions are recovered-may be of interest for other efforts to solve partial differential equations by the finite difference method