Expansion of Arbitrary Electromagnetic Fields in Terms of Vector Spherical Wave Functions

TL;DR

This paper introduces a novel analytical method to expand arbitrary electromagnetic fields using vector spherical wave functions by canceling radially-dependent terms, enabling efficient calculations for complex incident fields.

Contribution

It provides the first analytical expressions for beam shape coefficients for arbitrary incident fields, enhancing computational methods in electromagnetic scattering.

Findings

Derived analytical expressions for Bessel beams.

Extended the expansion method to modes of metallic waveguides.

Facilitated faster calculations of radiation forces in optical trapping.

Abstract

Since 1908, when Mie reported analytical expressions for the fields scattered by a spherical particle upon incidence of an electromagnetic plane-wave, generalizing his analysis to the case of an arbitrary incident wave has proved elusive. This is due to the presence of certain radially-dependent terms in the equation for the beam-shape coefficients of the expansion of the electromagnetic fields in terms of vector spherical wave functions. Here we show for the first time how these terms can be canceled out, allowing analytical expressions for the beam shape coefficients to be found for a completely arbitrary incident field. We give several examples of how this new method, which is well suited to numerical calculation, can be used. Analytical expressions are found for Bessel beams and the modes of rectangular and cylindrical metallic waveguides. The results are highly relevant for speeding up calculation of the radiation forces acting on small spherical particles placed in an arbitrary electromagnetic field, for example in optical tweezers.