Scattering of waves from a random spherical surface—Mie scattering

Abstract

The stochastic theory developed by the authors for the scattering from a random planar surface is extended to the case of a random spherical surface, which is assumed to be a homogeneous random field on the sphere, homogeneous with respect to spherical rotations. Based on the group-theoretical analogies between the two, the formulation of the theory is closely connected to the representation theory of the rotation group. The concept of the ‘‘stochastic’’ spherical harmonics associated with the rotation group and their several formulas are introduced and discussed at the beginning. For the plane wave incident on a random spherical surface, the scattered random wave field can be expanded systematically in terms of the stochastic spherical harmonics in much the same way as the nonrandom case, and several formulas are derived for the coherent scattering amplitude, the coherent and incoherent power flows, and the coherent and incoherent scattering cross sections. The power-flow conservation law is cast into the stochastic version of the optical theorem stating that the total scattering cross section consisting of the coherent and incoherent power flow is equal to the imaginary part of the coherent forward-scattering amplitude. Approximate solutions are obtained for the Mie scattering with a slightly random spherical surface where the single scattering approximation is valid due to the absence of a real resonance, as shown in the previous work on the two-dimensional case. Some numerical calculations are made for the coherent and incoherent scattering cross sections.