Conditions for a Rotationally Symmetric Spectral Degree of Coherence Produced by Electromagnetic Scattering on an Anisotropic Random Medium

The problem was recently reported that the far-zone electromagnetic momentum of light produced by scattering on a spatially anisotropic random medium can be the same at every azimuthal angle of scattering. Here, we extend the analysis to focus on the possibility of producing a rotationally symmetric spectral degree of coherence (SDOC) generated by scattering by an anisotropic process. The necessary and sufficient conditions for producing such a SDOC in the far zone are derived when a polychromatic electromagnetic plane wave is scattered by an anisotropic Gaussian Schell-model medium. We find that, unlike the generation of a rotationally symmetric momentum flow, it is not enough to simply restrict the structural characteristics of the medium and the incident light source to achieve a SDOC with rotational symmetry. An additional and essential requirement is that the azimuthal angles of scattering corresponding to the two observation points of the SDOC must be constrained to be equal. Only when all these constraints are satisfied simultaneously can a rotationally symmetric electromagnetic SDOC generated by scattering by an anisotropic process be realized. In addition, we find that although the medium parameter conditions for generating a rotationally symmetric SDOC and a rotationally symmetric momentum flow are completely different, it remains possible that the SDOC and the momentum flow produced by a spatially anisotropic medium can still simultaneously exhibit rotational symmetry, provided that the distribution of the correlation function of the scattering potential of the medium is isotropic in the plane perpendicular to the incident direction. Our results not only contribute to a deeper understanding of the far-field distribution of light scattering on an anisotropic scatterer, but also have potential applications in light-field manipulation and in the inverse scattering problem.