A complete set of scattering coefficients for a multilayered sphere is provided. The set includes coefficients needed to describe the fields within the different regions of the sphere. The analytic expressions are derived directly from the Mie coefficients, using a prescription that relates the coefficients for an r-layered sphere (r is an integer) to those for an (r-1)-layered sphere. This procedure has the advantage that the handling of cumbersome determinants (2rx2r) when the number of layers is (r-1), which normally occur in the expressions, is circumvented. Further, in view of the numerical problems observed in the direct use of such analytic expressions for the homogeneous, single-layer and double-layer sphere cases, the scattering amplitudes for the multi-layered sphere in general are cast in a form amenable to accurate calculations.
"Analytic Expressions, Calculational Forms For Scattering By A Multilayered Sphere", Proc. SPIE 0540, Southwest Conf on Optics '85, (20 November 1985); doi: 10.1117/12.976160; https://doi.org/10.1117/12.976160