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.
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H.C. thanks A. Genack for stimulating discussions. Funding: this work is partly supported by the Office of Naval Research (ONR) under grant no. N00014-20-1-2197, and by the National Science Foundation under grant nos. DMR-1905465, DMR-1905442 and OAC-1919789.
The authors declare no competing interests.
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Nature Physics thanks Alexandre Aubry, Oluwafemi Ojambati and Patrick Sebbah for their contribution to the peer review of this work.
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Bender, N., Yamilov, A., Goetschy, A. et al. Depth-targeted energy delivery deep inside scattering media. Nat. Phys. 18, 309–315 (2022). https://doi.org/10.1038/s41567-021-01475-x
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