Abstract
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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Data availability
Source data are available for this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
References
Mosk, A. P., Lagendijk, A., Lerosey, G. & Fink, M. Controlling waves in space and time for imaging and focusing in complex media. Nat. Photon. 6, 283–292 (2012).
Rotter, S. & Gigan, S. Light fields in complex media: mesoscopic scattering meets wave control. Rev. Mod. Phys. 89, 015005 (2017).
Yu, H. et al. Recent advances in wavefront shaping techniques for biomedical applications. Curr. Appl. Phys. 15, 632–641 (2015).
Yoon, S. et al. Deep optical imaging within complex scattering media. Nat. Rev. Phys. 2, 141–158 (2020).
Yoon, J. et al. Optogenetic control of cell signaling pathway through scattering skull using wavefront shaping. Sci. Rep. 5, 13289 (2015).
Ruan, H. et al. Deep tissue optical focusing and optogenetic modulation with time-reversed ultrasonically encoded light. Sci. Adv. 3, eaao5520 (2017).
Pernot, M. et al. In vivo transcranial brain surgery with an ultrasonic time reversal mirror. J. Neurosurg. 106, 1061–1066 (2007).
Liew, S. F. et al. Coherent control of photocurrent in a strongly scattering photoelectrochemical system. ACS Photon. 3, 449–455 (2016).
Vellekoop, I. M. & Mosk, A. Focusing coherent light through opaque strongly scattering media. Opt. Lett. 32, 2309–2311 (2007).
Yaqoob, Z., Psaltis, D., Feld, M. S. & Yang, C. Optical phase conjugation for turbidity suppression in biological samples. Nat. Photon. 2, 110–115 (2008).
Vellekoop, I. M., Van Putten, E., Lagendijk, A. & Mosk, A. Demixing light paths inside disordered metamaterials. Opt. Express 16, 67–80 (2008).
Xu, X., Liu, H. & Wang, L. V. Time-reversed ultrasonically encoded optical focusing into scattering media. Nat. Photon. 5, 154–157 (2011).
Judkewitz, B., Wang, Y. M., Horstmeyer, R., Mathy, A. & Yang, C. Speckle-scale focusing in the diffusive regime with time reversal of variance-encoded light (TROVE). Nat. Photon. 7, 300–305 (2013).
Horstmeyer, R., Ruan, H. & Yang, C. Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue. Nat. Photon. 9, 563–571 (2015).
Vellekoop, I. M. Feedback-based wavefront shaping. Opt. Express 23, 12189–12206 (2015).
Fink, M. et al. Time-reversed acoustics. Rep. Prog. Phys. 63, 1933 (2000).
Vellekoop, I. M. & Mosk, A. P. Universal optimal transmission of light through disordered materials. Phys. Rev. Lett. 101, 120601 (2008).
Hsu, C. W., Liew, S. F., Goetschy, A., Cao, H. & Stone, A. D. Correlation-enhanced control of wave focusing in disordered media. Nat. Phys. 13, 497–502 (2017).
Popoff, S. et al. Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media. Phys. Rev. Lett. 104, 100601 (2010).
Choi, W., Mosk, A. P., Park, Q.-H. & Choi, W. Transmission eigenchannels in a disordered medium. Phys. Rev. B 83, 134207 (2011).
Kim, M. et al. Maximal energy transport through disordered media with the implementation of transmission eigenchannels. Nat. Photon. 6, 581–585 (2012).
Yu, H. et al. Measuring large optical transmission matrices of disordered media. Phys. Rev. Lett. 111, 153902 (2013).
Popoff, S. M., Goetschy, A., Liew, S. F., Stone, A. D. & Cao, H. Coherent control of total transmission of light through disordered media. Phys. Rev. Lett. 112, 133903 (2014).
Gérardin, B., Laurent, J., Derode, A., Prada, C. & Aubry, A. Full transmission and reflection of waves propagating through a maze of disorder. Phys. Rev. Lett. 113, 173901 (2014).
Davy, M., Shi, Z., Park, J., Tian, C. & Genack, A. Z. Universal structure of transmission eigenchannels inside opaque media. Nat. Commun. 6, 6893 (2015).
Yılmaz, H., Hsu, C. W., Yamilov, A. & Cao, H. Transverse localization of transmission eigenchannels. Nat. Photon. 13, 352–358 (2019).
Bender, N., Yamilov, A., Yılmaz, H. & Cao, H. Fluctuations and correlations of transmission eigenchannels in diffusive media. Phys. Rev. Lett. 125, 165901 (2020).
Hsu, C. W., Goetschy, A., Bromberg, Y., Stone, A. D. & Cao, H. Broadband coherent enhancement of transmission and absorption in disordered media. Phys. Rev. Lett. 115, 223901 (2015).
Cheng, X. & Genack, A. Z. Focusing and energy deposition inside random media. Opt. Lett. 39, 6324–6327 (2014).
Chaigne, T. et al. Controlling light in scattering media non-invasively using the photoacoustic transmission matrix. Nat. Photon. 8, 58–64 (2014).
Ambichl, P. et al. Focusing inside disordered media with the generalized Wigner-Smith operator. Phys. Rev. Lett. 119, 033903 (2017).
Horodynski, M. et al. Optimal wave fields for micromanipulation in complex scattering environments. Nat. Photon. 14, 149–153 (2020).
Jeong, S. et al. Focusing of light energy inside a scattering medium by controlling the time-gated multiple light scattering. Nat. Photon. 12, 277–283 (2018).
Badon, A. et al. Smart optical coherence tomography for ultra-deep imaging through highly scattering media. Sci. Adv. 2, e1600370 (2016).
Katz, O., Ramaz, F., Gigan, S. & Fink, M. Controlling light in complex media beyond the acoustic diffraction-limit using the acousto-optic transmission matrix. Nat. Commun. 10, 717 (2019).
Durand, M., Popoff, S. M., Carminati, R. & Goetschy, A. Optimizing light storage in scattering media with the dwell-time operator. Phys. Rev. Lett. 123, 243901 (2019).
Lambert, W., Cobus, L. A., Frappart, T., Fink, M. & Aubry, A. Distortion matrix approach for ultrasound imaging of random scattering media. Proc. Natl Acad. Sci. USA 117, 14645–14656 (2020).
Badon, A. et al. Distortion matrix concept for deep optical imaging in scattering media. Sci. Adv. 6, eaay7170 (2020).
Bouchet, D., Rotter, S. & Mosk, A. P. Maximum information states for coherent scattering measurements. Nat. Phys. 17, 564–568 (2021).
Groth, C. W., Wimmer, M., Akhmerov, A. R. & Waintal, X. Kwant: a software package for quantum transport. New J. Phys. 16, 063065 (2014).
Beenakker, C. W. Random-matrix theory of quantum transport. Rev. Mod. Phys. 69, 731–808 (1997).
Marchenko, V. A. & Pastur, L. A. Distribution of eigenvalues for some sets of random matrices. Math. USSR Sb. 1, 457 (1967).
Goetschy, A. & Stone, A. D. Filtering random matrices: the effect of incomplete channel control in multiple scattering. Phys. Rev. Lett. 111, 063901 (2013).
van Rossum, M. C. W. & Nieuwenhuizen, T. M. Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion. Rev. Mod. Phys. 71, 313 (1999).
Pnini, R. & Shapiro, B. Fluctuations in transmission of waves through disordered slabs. Phys. Rev. B 39, 6986–6994 (1989).
Sarma, R., Yamilov, A., Neupane, P., Shapiro, B. & Cao, H. Probing long-range intensity correlations inside disordered photonic nanostructures. Phys. Rev. B 90, 014203 (2014).
Yamilov, A. G. et al. Position-dependent diffusion of light in disordered waveguides. Phys. Rev. Lett. 112, 023904 (2014).
Sarma, R., Yamilov, A., Petrenko, S., Bromberg, Y. & Cao, H. Control of energy density inside a disordered medium by coupling to open or closed channels. Phys. Rev. Lett. 117, 086803 (2016).
Ojambati, O. S., Mosk, A. P., Vellekoop, I. M., Lagendijk, A. & Vos, W. L. Mapping the energy density of shaped waves in scattering media onto a complete set of diffusion modes. Opt. Express 24, 18525–18540 (2016).
Koirala, M., Sarma, R., Cao, H. & Yamilov, A. Inverse design of perfectly transmitting eigenchannels in scattering media. Phys. Rev. B 96, 054209 (2017).
Hong, P., Ojambati, O. S., Lagendijk, A., Mosk, A. P. & Vos, W. L. Three-dimensional spatially resolved optical energy density enhanced by wavefront shaping. Optica 5, 844–849 (2018).
Fang, P. et al. Universality of eigenchannel structures in dimensional crossover. Phys. Rev. B 99, 094202 (2019).
Uppu, R., Adhikary, M., Harteveld, C. A. M. & Vos, W. L. Spatially shaping waves to penetrate deep inside a forbidden gap. Phys. Rev. Lett. 126, 177402 (2021).
Acknowledgements
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.
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N.B. conducted the experiments and analysed the data. A.Y. performed the numerical simulations. A.G. developed the analytical model. H.Y. participated in the experimental study. C.W.H. contributed to the theoretical analysis. H.C. initiated the project and supervised the research. All the authors contributed to manuscript preparation.
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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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DOI: https://doi.org/10.1038/s41567-021-01475-x
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