Approximate expansion of a narrow Gaussian beam in spherical vector wave functions

The expansion of electromagnetic sources is fundamental to the analysis of field propagation and scattering. Of these electromagnetic sources, one of the most commonly used optical sources is the narrow Gaussian beam. It is, therefore, useful to find an expansion of the narrow Gaussian beam that could be used in applications such as scattering and propagation studies in free-space and materials. The approximate method of expansion here is based on an exact expansion that was obtained for a vector plane wave in terms of spherical vector wave functions. Since the simpler vector plane wave representation that previously enabled an exact solution is replaced here by the more complicated narrow Gaussian beam representation, an approximation of the associated Legendre function will be applied in order to obtain an approximate expansion of a narrow Gaussian beam in terms of spherical vector wave functions. Although the expansion is taken about the source point, field points close to the beam axis located at distances near and far from the source are found to have good accuracy.