On localized approximations for Laguerre-Gauss beams focused by a lens

When dealing with T-matrix methods for structured beams, e.g. Generalized Lorenz-Mie Theories (GLMTs) or Extended Boundary Condition Method (EBCM), the description of the illuminating beam is encoded by a set of coefficients named Beam Shape Coefficients (BSCs). An efficient method to evaluate the BSCs is by using a localized approximation. In a series of papers, we demonstrated that any existing localized approximation is of limited validity (i) in the case of beams exhibiting a propagation factor of the form exp(ikzcos α), in which α may be called the axicon angle, such as zeroth-order Bessel beams, or (ii) in the case of beams exhibiting a topological charge term, such as Laguerre-Gauss beams propagating freely in space. In the present paper, we consider the case of Laguerre-Gauss beams focused by a lens which exhibit both an axicon angle and a topological charge, and demonstrate that any existing localized approximation should be used with caution as well in this case in contrast with results, available from the literature, in which an integral localized approximation has been used to evaluate the BSCs of such beams.