Depth-targeted energy delivery deep inside scattering media

Diffusion makes it difficult to predict and control wave transport through a medium. Overcoming wave diffusion to deliver energy into a target region deep inside a diffusive system is an important challenge for applications, but also represents an interesting fundamental question. It is known that coherently controlling the incident wavefront allows diffraction-limited focusing inside a diffusive system, but in many applications, the targets are significantly larger than a focus and the maximum deliverable energy remains unknown. Here we introduce the 'deposition matrix', which maps an input wavefront to the internal field distribution, and we theoretically predict the ultimate limit on energy enhancement at any depth. Additionally, we find that the maximum obtainable energy enhancement occurs at three-fourths the thickness of the diffusive system, regardless of its scattering strength. We experimentally verify our predictions by measuring the deposition matrix in two-dimensional diffusive waveguides. The experiment gives direct access to the internal field distribution from the third dimension, and we can excite the eigenstates to enhance or suppress the energy within an extended target region. Our analysis reveals that such enhancement or suppression results from both selective transmission-eigenchannel excitation and constructive or destructive interference among these channels. Optimally depositing optical energy into an extended region of a diffusive medium, such as biological tissue, is a challenging task. A matrix that maps the incoming wavefront to the field distribution inside the material can predict the energy enhancement that occurs at a given depth.