Second modified localized approximation for use in generalized Lorenz-Mie theory and other theories revisited

Arbitrary electromagnetic shaped beams may be described by using expansions over a set of basis functions, with expansion coefficients containing subcoefficients named "beam shape coefficients" (BSCs). When BSCs cannot be obtained in closed form, and/or when the beam description does not exactly satisfy Maxwell's equations, the most efficient method to evaluate the BSCs is to rely on localized approximations. One of them, named the second modified localized approximation, has been presented in a way that may be found ambiguous in some cases. The aim of the present paper is to remove any ambiguity on the use of the second modified localized approximation.