Coupled amplitude-phase transport in heterogeneous optical media

The transport of intensity equation (TIE) enables deterministic phase retrieval in optical microscopy, yet its application to heterogeneous media exhibiting both refractive-index variations and attenuation remains an open problem. We derive a coupled TIE-TPE (transport of phase equation) framework from the paraxial wave equation with a complex optical potential, in which the refractive index field is decomposed into a spatially uniform mean field and a local fluctuation field through Reynolds decomposition. This formulation yields a nondivergent system of transport equations that simultaneously reconstructs refractive-index fluctuations and attenuation coefficients within the condition | Δ n | < 1 , without requiring linearization or weak-absorption approximations. We establish explicit validity bounds that define the measurable parameter space for a given experimental configuration, emerging from both photon counting statistics and the diffraction limit. Experimental validation using microlens arrays and HeLa cells demonstrates robust recovery of optical properties, including in the transparent-limit regime where attenuation signals approach detection thresholds. We further verify that the attenuation asymmetry parameter remains consistent with zero across three orders of magnitude in optical depth, indicating preservation of optical reciprocity in structurally heterogeneous biological media.