Robust location of optical fiber modes via the argument principle method

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.