Pupil phase series: a fast, accurate, and energy-conserving model for forward and inverse light scattering in thick biological samples

We present the pupil phase series (PPS), a fast and accurate forward scattering algorithm for simulating and inverting multiple light scattering in large biological samples. PPS achieves high-angle scattering accuracy and energy conservation simultaneously by introducing a spatially varying phase modulation in the pupil plane. By expanding the scattering term into a Taylor series, PPS achieves high precision while maintaining computational efficiency. We integrate PPS into a quasi-Newton inverse solver to reconstruct the three-dimensional refractive index of a 180 μm-thick human organoid. Compared to linear reconstruction, our method recovers subcellular features—such as nuclei and vesicular structures—deep within the sample volume. PPS offers a scalable and interpretable alternative to conventional solvers, paving the way for high-throughput, label-free imaging of optically thick biological tissues.