Dyad of equivalence theorems of partially coherent beams on scattering from different types of media.

Although the equivalence theorem (ET) concerning partially coherent beams on scattering has recently emerged as an advancement in potential scattering theory, the current analysis remains confined to scatterers belonging to the same category, i.e., continuous scatterers. Here, we generalize the ET to a more general and practical situation where partially coherent beams are scattered by different types of media, i.e., the continuous medium and the discrete particulate medium. We find that the normalized far-zone spectral densities produced by Gaussian Schell-model (GSM) beams on scattering from different types of media can be identical if the structural characteristics of both the incident beam and the scattering medium obey certain constraints. Based on this, we identify two ETs for the normalized far-zone spectral density: medium-driven equivalence and source-medium joint equivalence, which are referred to here collectively as a dyad. As an example, we analyze the scattering of GSM beams by a Gaussian-correlated quasihomogeneous continuous medium and by a system of deterministic Gaussian soft-edge particles distributed with a GSM position correlation profile, and derive the explicit analytical constraints for the structural parameters of both the incident beam and the scattering medium to achieve identical normalized spectral densities. These findings reveal structural ambiguities in inverse scattering, offering insights for improving reconstruction accuracy in optical remote sensing, laser plasma diagnostics, and biomedical imaging.