Inverse scattering of absorptive samples via beam propagation.

Inverse-scattering methods enable label-free, quantitative visualization of a sample's three-dimensional (3D) refractive index (RI), providing intrinsic volumetric contrast. Recent advances have demonstrated successful 3D RI reconstructions in multiple-scattering samples using angle-varying illumination; however, these studies have primarily focused on non-absorptive samples. Here, we extend the multi-slice beam propagation (MSBP) inverse-scattering framework to reconstruct complex-valued RI, encompassing both the sample's conventional RI (real part) and absorptivity (imaginary part). We show that reconstructing complex-valued RI makes the inverse problem ill-posed under angle-varying illumination alone, and that incorporating measurement diversity from both angle-varying illumination and sample defocus enables stable and accurate convergence. Experimental demonstrations were conducted on (1) dyed microsphere samples to characterize the accuracy of the reconstructed RI and absorptivity and (2) diverse scattering and absorptive bio-samples to demonstrate biological utility. These results represent an important step for label-free volumetric imaging in biological tissue, which typically exhibits both scattering and absorption.