A framework for the finite series method of the generalized Lorenz-Mie theory and its application to freely-propagating Laguerre-Gaussian beams

In face of a revival of interest in the finite series (FS) method due to recent developments upon generalized Lorenz-Mie theories (GLMTs), a more general, understandable, and systematic formulation is proposed. Possibly due to an apparent lack of flexibility in the FS method’s earlier statements, there has been a void in its use since the 1990s. Particularly, the method demands some degree of mathematical labor each time it is applied to a different kind of field profile. Furthermore, the algebraic complexity of its earlier occurrences might also have weighted upon the method’s historical shunning. Dissecting the later works reclaiming the FS, several possibilities for generalization, simplification, and organization were found. Accordingly, with the intent to render the method more approachable and to encourage its use, this work derives an alternative path suitable for both understanding and implementation. Applying the procedure, expressions for the beam shape coefficients of freely propagating Laguerre-Gaussian beams are obtained in closed form in a more straightforward manner when compared to previous formulations – this time not relying on recursions.