Angular spectrum decomposition method and quadrature method in the generalized Lorenz–Mie theory for evaluating the beam shape coefficients of TEM0l∗ doughnut beam

The evaluation of beam shape coefficients (BSCs) which encodes the description of the illuminating beam is an essential issue in light scattering theories of spherical particles, such as the generalized Lorenz–Mie theory (GLMT). Different methods have been developed in this context, including the traditional (quadratures, finites series, localized approximations), to be complemented by use of the angular spectrum decomposition (ASD). The present paper is devoted to a comprehensive study of the relationship between the traditional quadrature method and the ASD method to the evaluation of BSCs. We shall establish that evaluation of BSCs using the quadrature method can be modified into expressions in the spectral space, leading to the same results as those obtained using the ASD. These BSCs are afterwards approximated under the paraxial conditions, leading to the same results as these obtained by using the localized approximations. The TEM0l∗ doughnut beam is taken as an example of the shaped beam and the theoretical analysis is confirmed by numerical calculations.