On the series expansion of high-aperture, vectorial diffraction integrals

In this paper we compare four series expansions for computing the integrals of high-aperture, vectorial diffraction theory. We present a detailed analysis of the convergence and run time of the expansions and show that, contrary to common belief, direct integration produces the fastest numerical evaluation. Explicit solutions for the focal distributions along both the axial and radial principal axes are also presented.