An improved argument principle root-search method for modes of slab waveguides, optical fibers, and spheres

TL;DR

This paper presents an enhanced, globally convergent root-search method based on Cauchy's residue theorem for accurately finding complex roots of transcendental equations in optical waveguide and sphere problems, including leaky modes.

Contribution

The authors extend their root-search method to handle dispersion relations of slab waveguides, resonances of spheres, and improve reliability in challenging regimes, also identifying leaky modes and mode discontinuities.

Findings

Method guarantees locating all roots within a domain.

Successfully applied to slab waveguides, fibers, and spheres.

First identification of a discontinuity across the light line in dispersion plots.

Abstract

We update our root-search method for transcendental equations. Our method is globally convergent and is guaranteed to locate all complex roots within a specified search domain, since it is based on Cauchy's residue theorem. We extend the implementation to treat the dispersion relations of slab waveguides and the resonances of a sphere, in addition to step-index fibers. We also implement other improvements, such as to the contour selection procedure and using non-dimensional search variables, to ensure the method remains reliable even in challenging parameter regimes. We also extend the algorithm to identify leaky modes in terms of propagation constant eigenvalue modes, revealing, to the first time to our knowledge, a discontinuity across the light line in the dispersion plot.