Two approximate methods for the determination of the fundamental mode of an optical waveguide with rectangular core cross section and step refractive-index profiles are presented and analyzed thoroughly. Both methods are based on Galerkin's method. The first method uses Hermite-Gauss basis functions and the second uses the guided and nonguided slab waveguide solutions as basis functions. The results are compared with results from an accurate circular harmonic analysis. Both methods provide values of the normalized propagation constant with errors less than 0.1% for practical rectangular single-mode waveguides. The slab waveguide method is the fastest, and even when only one slab waveguide mode is used the propagation constant for the fundamental mode can be calculated with an error of less than 1%. The slab waveguide method also gives very accurate results for the propagation constant for higher order modes.
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Rasmussen, T., Povlsen, J. H., Bjarklev, A. O., Lumholt, O., Pedersen, B., & Rottwitt, K. (1993). Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide. Journal of Lightwave Technology, 11(3), 429-433. https://doi.org/10.1109/50.219576