Steepest-descent approximation theory for guided modes of weakly guiding optical waveguides and fibers

The steepest-descent approximation theory (SDAT) is developed for the calculation of dispersion relations and field distribution functions in optical waveguides and fibers. Based on the functional-extremum principle, this new method improves the form of the trial functions for the modal field distributions directly, instead of optimizing parameters as in conventional variational approaches. It does so by making use of the concept of the functional derivative of Rayleigh’s quotient and the steepest-descent technique. The SDAT also provides iterative procedures for generating better approximations for both the propagation constants and the modal fields. Several low-order modes of an optical fiber with a Gaussian refractive-index profile are studied, demonstrating the usefulness of this new method. Improved analytical expressions for the field distributions are obtained with improved values for the propagation constants.