Robust Location of Optical Fiber Modes via the Argument Principle Method

TL;DR

This paper presents a MATLAB implementation of a robust, globally convergent argument principle method for locating all complex roots of transcendental equations, specifically applied to optical fiber mode analysis, accommodating singularities and branch points.

Contribution

The authors develop a simple, adaptable MATLAB implementation of the argument principle method that handles singularities and branch points, improving robustness for optical fiber mode calculations.

Findings

Successfully locates all complex roots in the specified domain.

Effectively handles singularities and branch points within the search domain.

Applied to step index fiber dispersion relation and permittivity eigenvalue problems.

Abstract

We implement a robust, globally convergent root search method for transcendental equations guaranteed to locate all complex roots within a specified search domain, based on Cauchy's residue theorem. Although several implementations of the argument principle already exist, ours has several advantages: it allows singularities within the search domain and branch points are not fatal to the method. Furthermore, our implementation is simple and is written in MATLAB, fulfilling the need for an easily integrated implementation which can be readily modified to accommodate the many variations of the argument principle method, each of which is suited to a different application. We apply the method to the step index fiber dispersion relation, which has become topical due to the recent proliferation of high index contrast fibers. We also find modes with permittivity as the eigenvalue, catering to recent numerical methods that expand the radiation of sources by eigenmodes.