Evaluating the beam shape coefficients of a hollow vortex Gaussian beam by using the two-step indirect method based on the scalar translational addition theorem

The two-step indirect method for evaluating the beam shape coefficients (BSCs) of the electromagnetic field of a circular non-vortex Gaussian beam was recently proposed to simplify the analytical work in formulating the BSCs and to speed up the relevant numerical calculation. In this work, the indirect method is extended to a family of circular vortex beams carrying topological charges, which requires an additional recurrence for calculating the translation coefficients. Numerical results of the hollow vortex Gauss beam verify that the proposed method can avoid tedious analytical derivation and improve the efficiency of numerical calculation.