Interest of improved three-point formulas for finite-difference modeling of optical devices

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Abstract
The accuracy of finite-difference computations is limited by the errors inherent in the classical three-point formula for ∂x2 (of order h2) and in the discretization of discontinuities in the refractive index (of order from h0 to h2). 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 h4-accurate three-point formula for ∂x2 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.