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• Beam shape coefficients for Laguerre-Gauss beams focused by a lens are discussed. • The method used is the finite series techniques and expressions within this framework are established. • Expressions rely on recurrence equations which are solved. Pursuing a study concerning the evaluation of Beam Shape Coefficients (BSCs) of Laguerre-Gauss beams, after having dealt with the case of beams freely propagating, the present paper establishes finite series expressions to evaluate the BSCs of Laguerre-Gauss beams focused by a lens, which have the known advantage of speeding up computations with respect to the use of numerical quadrature techniques and which may deal with the limitations of localized approximations in the case of beams exhibiting topological charges and/or axicon angles. [ABSTRACT FROM AUTHOR]

• Beam shape coefficients of freely propagating Laguerre–Gauss beams can- not be evaluated speedily enough using quadratures. • Two algorithms to compute freely propagating Laguerre–Gauss finite series beam shape coefficients with similar time complexity are presented. One takes a recursive approach and the other eliminates recursions resulting in an iterative process. • Numerical comparisons between quadrature, localized and finite series computed radial beam shape coefficients are presented as well as their resulting radial electric field profiles. • Deviations from the "intended" field profiles are more pronounced when the beam focus and topological charges are considered. • This work shows that the finite series method can overcome other methods in terms of accuracy or computational cost for freely propagating Laguerre–Gauss beams. In the framework of a series of papers devoted to warnings concerning the evaluation of beam shape coefficients (BSCs) of electromagnetic beams by using a localized approximation, a recent paper provided finite series explicit expressions to evaluate BSCs of paraxial Laguerre–Gauss freely propagating beam. In the present paper, these expressions are numerically implemented and finite series numerical results are compared with results obtained by using quadratures and a localized approximation. These three methods provide a remodeling of the beam, changing the non-Maxwellian paraxial Laguerre–Gauss beam to a beam which perfectly satisfies Maxwell's equations. Comparisons between the three methods are discussed in terms of accuracy and computational time efficiency. [ABSTRACT FROM AUTHOR]