M. Lohmeyer et al., BOUNDARY-CONDITIONS FOR THE FINITE-DIFFERENCE BEAM-PROPAGATION METHODBASED ON PLANE-WAVE SOLUTIONS OF THE FRESNEL EQUATION, IEEE journal of quantum electronics, 33(2), 1997, pp. 279-285
Each particular implementation of the beam propagation method (BPM) re
quires a special procedure allowing for radiation to leave the computa
tional window. We propose a new approach to constructing the finite di
fference schemes of the BPM at the boundary of the computational windo
w, These schemes are independent of the computed fields and allow for
a similar treatment of both interior and boundary points, The new appr
oach can be further improved by correcting the field values at the bou
ndary points according to Hadley's method. The algorithm is easy to im
plement for both two- and three-dimensional structures. The new method
considerably reduces computation times because the propagation matric
es remain constant in longitudinally invariant sections, thus avoiding
repeated LU-decompositions. The basic idea-establishing the finite di
fference scheme such that locally exact, approximate, or plausible sol
utions are recovered-may be of interest for other efforts to solve par
tial differential equations by the finite difference method.