We extend the continuity relations of field derivatives across an abrupt interface to arbitrary orders for transverse electric and magnetic waves in slab structures. Higher-order finite-difference formulation is then obtained by combining the systematically-obtained interface conditions with Taylor series expansion. Generalized Douglas scheme is also adopted to further enhance the convergence of truncation errors by two orders. We apply the derived finite-difference formulation, up to nine-points in this paper, to solve the guided modes in simple a slab waveguide and multiple quantum well waveguides. The results shows the truncation error is much higher, up to tenth order, as expected. Using those higher-order schemes, accurate results are obtained with much fewer sampled points, and hence with tremendously less computation time and memory.
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| http://dx.doi.org/10.1364/OE.18.004088 | DOI Listing |
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