Restoring and tailoring very high dimensional spatial entanglement of a biphoton state transmitted through a scattering medium

We report experimental results where a momentum entangled biphoton state with a Schmidt number of a few thousand is retrieved and manipulated when only one photon of the pair is transmitted through a thin scattering medium. For this purpose, the transmission matrix of the complex medium is first measured with a phase-shifting interferometry measurement method using a spatial light modulator (SLM) illuminated with a laser source. From this matrix, different phase masks are calculated and addressed on the SLM to spatially control the focusing of the laser through the complex medium. These same masks are used to manipulate the phase of the biphoton wave function transmitted by the thin diffuser in order to restore and control in the same way the momentum correlations between the far-field images of twin beams issued from strongly spatial-multi-mode spontaneous parametric down conversion.