Evaluation of beam shape coefficients in T-matrix methods using a finite series technique: on blow-ups using hypergeometric functions and generalized Bessel polynomials

Abstract

In T-matrix methods (generalized Lorenz–Mie theories or extended boundary condition method), beam shape coefficients encoding the shape of the illuminating structured beam have to be evaluated. This may be carried out by using the finite series technique, which, however, generates blow-ups when the partial wave order of the beam shape coefficients increases. Using hypergeometric functions and generalized Bessel polynomials, we demonstrate in the case of on-axis Gaussian beams that these blow-ups are genuine phenomena, not due to a lack of numerical precision, and we establish criteria to evaluate the critical partial wave order that implies blow-ups.