Iterative calculation of complex propagation constants of modes in multilayer planar waveguides

A Newton-Raphson complex root-finding algorithm, based on the thin-film transfer matrix analysis, is described. This second-order convergent algorithm, which requires only O(n) calculations per iteration, was presented to find the complex propagation constants of any mode in an n-layer planar waveguide. The number of calculations per iteration is reduced below those required by competitive schemes through application of the Cauchy-Riemann relations. The method demonstrates higher accuracy and improved efficiency over other iterative techniques, making this approach highly suitable for the analysis and design of planar structures consisting of a large number of layers.>