Equivalence between radial quadrature and finite series for spherical wave expansion of Bessel beams.

The radial quadrature method was recently proposed for formulating the beam shape coefficients (BSCs) for shaped beams. A new deduction of BSCs using the R-quadrature method is presented in this paper, using the integral of the spherical Bessel functions in the interval ranging from zero to infinity. Based on the scalar description of the Bessel beam, the equivalence between the R-quadrature and the finite series (FS) method is confirmed. The spherical wave expansion of the scalar function allows us to simplify the formulation of the BSCs in the R-quadrature and the FS and to speed up the numerical BSC calculation. As a by-product, FS expansions of the associated Legendre functions are established, which we do not find in the literature.