Multipole expansion of Bessel and Gaussian beams for Mie scattering calculations

Multipole expansions of Bessel and Gaussian beams, suitable for use in Mie scattering calculations, are derived. These results allow Mie scattering calculations to be carried out considerably faster than existing methods, something that is of particular interest for time evolution simulations where large numbers of scattering calculations must be performed. An analytic result is derived for the Bessel beam that improves on a previously published expression requiring the evaluation of an integral. An analogous expression containing a single integral, similar to existing results quoted, but not derived, in literature, is derived for a Gaussian beam, valid from the paraxial limit all the way to arbitrarily high numerical apertures.