  Studying the internal structure of complex samples with light is an important
task, but a difficult challenge due to light scattering. While the complex
optical distortions induced by multiple scattering can be effectively undone
with the knowledge of the medium's scattering-matrix, this matrix is generally
unknown, and cannot be measured with high resolution without the presence of
fluorescent or absorbing probes at all points of interest. To overcome these
limitations, we introduce here the concept of the acousto-optic transmission
matrix (AOTM). Taking advantage of the near scattering-free propagation of
ultrasound in complex samples, we noninvasively measure an
ultrasonically-encoded, spatially-resolved, optical scattering-matrix. We
demonstrate that a singular value decomposition analysis of the AOTM, acquired
using a single or multiple ultrasonic beams, allows controlled optical focusing
beyond the acoustic diffraction limit in scattering media. Our approach
provides a generalized framework for analyzing acousto-optical experiments, and
for noninvasive, high-resolution study of complex media.
