Since 1908, when Mie reported analytical expressions for the fields scatteredby a spherical particle upon incidence of an electromagnetic plane-wave,generalizing his analysis to the case of an arbitrary incident wave has provedelusive. This is due to the presence of certain radially-dependent terms in theequation for the beam-shape coefficients of the expansion of theelectromagnetic fields in terms of vector spherical wave functions. Here weshow for the first time how these terms can be canceled out, allowinganalytical expressions for the beam shape coefficients to be found for acompletely arbitrary incident field. We give several examples of how this newmethod, which is well suited to numerical calculation, can be used. Analyticalexpressions are found for Bessel beams and the modes of rectangular andcylindrical metallic waveguides. The results are highly relevant for speedingup calculation of the radiation forces acting on small spherical particlesplaced in an arbitrary electromagnetic field, for example in optical tweezers.
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