  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.
