The possibility of a series expansion for the diffracted fields by a dielectric sphere or circular cylinder is considered by first expanding the Mie scattering coefficients into a geometric series (Debye expansion), and then applying a modified Watson transformation to each term of the series. The diffracted fields can then be obtained by finding poles of the Debye expansion, and evaluating the residues. The locations of the poles due to the Debye expansion as well as the Mie scattering coefficients themselves are discussed. Numerical values of the poles are tabulated. The diffracted field contributions associated with the first five terms of the Debye expansion are presented in the form of a radar scattering cross section. The geometrical optics contribution arising from each term of the Debye expansion is asymptotically evaluated. To compare the magnitude of a sum of the geometrical optics and diffracted field contributions, the radar scattering cross section of the total backscattered field, obtained from the exact Mie theory, is also plotted.