Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory II Off-axis beams

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Abstract
Generalized Lorenz–Mie theory describes electromagnetic scattering of an arbitrary light beam by a spherical particle. The localized approximation is an analytical function that accurately models the beam-shape coefficients that give the decomposition of a focused Gaussian beam into partial waves. A mathematical justification and physical interpretation of the localized approximation is presented for a focused off-axis Gaussian beam that propagates parallel to but not along the z axis.