Optical turbulence retrieval of heterogeneous media

The transport of intensity equation (TIE) has revolutionized phase retrieval in optical microscopy, yet its application to complex media with absorption/scattering remains challenging. Here, we present a coupled TIE-TPE (transport of phase equation) framework derived directly from the paraxial wave equation with complex optical potential. By decomposing the refractive index field into a spatially uniform mean field and local fluctuation field, our approach enables simultaneous reconstruction of refractive-index fluctuations and attenuation coefficients without linearization assumptions. We establish reconstruction validity bounds that define the measurable parameter region where reconstruction remains physically consistent. Experimental demonstration with microlens arrays and HeLa cells shows robust recovery of optical properties even in the transparent-limit regime where attenuation signals approach detection thresholds. Furthermore, we provide the first experimental verification of attenuation symmetry -- a fundamental property of wave propagation that characterizes reciprocity in light-matter interactions.