Abstract

The accuracy of finite-difference computations is limited by the errors inherent in the classical three-point formula for ${\partial }_x^2$ (of order $h^2$) and in the discretization of discontinuities in the refractive index (of order from $h^0$ to $h^2$). Various improvements are reviewed and analyzed, both for normal-mode derivations and for the beam propagation method (BPM). For the BPM it seems that sophisticated discretizations for discontinuities have a limited interest, whereas using a $h^4$-accurate three-point formula for ${\partial }_x^2$ often allows the use of fewer grid points and hence faster computations; also it makes the operation of transparent boundaries with Bérenger layers considerably more efficient. Finally, the permittivity should always be averaged mesh by mesh.

© 1997 Optical Society of America

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