On the accuracy of approximate descriptions of discrete superpositions of Bessel beams in the generalized Lorenz-Mie theory

This paper aims to investigate one of the most important schemes, viz. the localized approximation (LA), for describing arbitrary-shaped beams in the generalized Lorenz-Mie theory. Our focus is on a specific class of non-diffracting beams called discrete frozen waves, which are constructed from superpositions of Bessel beams and can be designed to provide virtually any longitudinal intensity pattern of interest. Recently, the LA was applied to frozen waves allowing, for the first time, for the analysis of light scattering problems from spherical scatterers and the subsequent determination of the physical/optical quantities in optical trapping. Since the LA cannot be rigorously applied to Bessel beams, it is of interest to determine whether those results and predictions previously established in the literature are reliable or not. To do so, we rely on exact descriptions of frozen waves and establish the limits to the validity of such an approximation scheme. It is revealed that, although serious doubts can be raised against its use, specially due to cumulative errors, the LA is much more robust than previously thought, and it may serve well to both paraxial and non-paraxial Frozen Waves, under certain circumstances.