Millisecond vectorial wave-front shaping through scattering media via full transmission-matrix measurement

Optical scattering severely degrades wave-front control and energy delivery in complex media, particularly in biological tissue where polarization mixing further complicates light transport. Although wave-front shaping can compensate for scattering-induced distortions, most existing approaches rely on scalar control and access only partial transmission matrices, which fundamentally limit performance. Here, we present a fast vectorial wave-front shaping framework based on direct measurement of the full vector transmission matrix (VTM). The approach combines compact, reference-free, and single-path optical architecture with an efficient reconstruction algorithm based on the alternating direction method of multipliers, enabling simultaneous retrieval of polarization-coupled scattering responses from intensity-only measurements. Full vectorial wave-front control is realized using a single digital micromirror device with superpixel encoding and Fourier-plane polarization discrimination. We demonstrate millisecond-scale operation through strongly scattering and dynamically decorrelating biological tissue. For N = 64 modes per polarization, full VTM acquisition and focusing are completed within a total time of 46.4 ms (including 24.6 ms for measurement and 21.8 ms for algorithmic reconstruction), corresponding to an average time of approximately 362 µs per mode, enabling stable operation under dynamic scattering conditions.