Effective method for calculating modes of multilayer waveguide

A fast and efficient numerical method for finding the modes of a multilayered waveguide is proposed. Using the complex vector of the Riemann-Silberstein has resulted in a reduction of the order of the differential equations describing the passage of light through the different layers, as well as a double reduction of the searched variables. The approximation of the differential equations is made by the method of Galyorkin by a suitable choice of the base functions. The calculation of the effective indices and their corresponding wave configurations is realized by the inverse-shifting power method with Rayleigh's quantity. The method was successfully applied waveguide systems, generating values close in the effective indices.