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

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.