Deep imaging inside scattering media through virtual spatiotemporal wavefront shaping

The multiple scattering of light makes materials opaque and obstructs imaging. Optimized wavefronts can overcome scattering to focus but typically require restrictive guidestars and only work within an isoplanatic patch. Focusing by lenses and wavefront shaping by spatial light modulators also limit the imaging volume and update speed. Here, we introduce scattering matrix tomography (SMT): use the measured scattering matrix of the sample to construct its volumetric image by scanning a confocal spatiotemporal focus with input and output wavefront correction for every isoplanatic patch, dispersion compensation, and index-mismatch correction--all performed digitally during post-processing without a physical guidestar. The digital focusing offers a large depth of field without constraint by the focal plane's Rayleigh range, and the digital wavefront correction enables image optimization with fast updates unrestricted by the speed of the hardware. We demonstrate SMT with sub-micron diffraction-limited lateral resolution and one-micron bandwidth-limited axial resolution at one millimeter beneath ex vivo mouse brain tissue and inside a dense colloid, where conventional imaging methods fail due to the overwhelming multiple scattering. SMT translates deep-tissue imaging into a computational reconstruction and optimization problem. It is noninvasive and label-free, with prospective applications in medical diagnosis, biological science, colloidal physics, and device inspection.