  Multiple scattering of waves in disordered media is a nightmare whether it be
for detection or imaging purposes. The best approach so far to get rid of
multiple scattering is optical coherence tomography. It basically combines
confocal microscopy and coherence time-gating to discriminate ballistic photons
from a predominant multiple scattering background. Nevertheless, the imaging
depth range remains limited to 1 mm at best in human soft tissues. Here we
propose a matrix approach of optical imaging to push back this fundamental
limit. By combining a matrix discrimination of ballistic waves and iterative
time-reversal, we show both theoretically and experimentally an extension of
the imaging-depth limit by at least a factor two compared to optical coherence
tomography. In particular, the reported experiment demonstrates imaging through
a strongly scattering layer from which only one reflected photon over 1000
billion is ballistic. This approach opens a new route towards ultra-deep tissue
imaging.
