Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

Scattering invariant modes of light in complex media

Nature Photonics volume 15, pages 431–434 (2021)Cite this article

Subjects

An Author Correction to this article was published on 26 April 2021

This article has been updated

Abstract

Random scattering of light in disordered media is an intriguing phenomenon of fundamental relevance to various applications1. Although techniques such as wavefront shaping and transmission matrix measurements2,3 have enabled remarkable progress in advanced imaging concepts4,5,6,7,8,9,10,11, the most successful strategy to obtain clear images through a disordered medium remains the filtering of ballistic light12,13,14. Ballistic photons with a scattering-free propagation are, however, exponentially rare and no known method has been able to increase their proportion. To address these limitations, we introduce and experimentally implement a new set of optical states that we term scattering invariant modes, whose transmitted field pattern is the same, irrespective of whether they scatter through a disordered sample or propagate ballistically through a homogeneous medium. We observe scattering invariant modes that are only weakly attenuated in dense scattering media, and show in simulations that their correlations with the ballistic light can be used to improve imaging inside scattering materials.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Illustration of the concept.
Fig. 2: Experimentally transmitted SIMs and their statistics.
Fig. 3: Experimental realization of a sparse SIM.
Fig. 4: Simulations of imaging using SIMs.

Similar content being viewed by others

Data availability

The data that underlie the plots within this paper and other findings of this study are available from the corresponding authors on reasonable request.

Code availability

The code used to generate simulated data and plots is available from the corresponding authors on reasonable request.

Change history

References

  1. Johnson, C. & Gabriel, D. Laser Light Scattering (Dover Publications, 2015).

  2. 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).

    Article  ADS  Google Scholar 

  3. Rotter, S. & Gigan, S. Light fields in complex media: mesoscopic scattering meets wave control. Rev. Mod. Phys. 89, 015005 (2017).

    Article  ADS  Google Scholar 

  4. Yoon, S. et al. Deep optical imaging within complex scattering media. Nat. Rev. Phys. 2, 141–158 (2020).

    Article  Google Scholar 

  5. Kubby, J., Gigan, S. & Cui, M. (eds) Wavefront Shaping for Biomedical Imaging (Cambridge Univ. Press, 2019).

  6. Bertolotti, J. et al. Non-invasive imaging through opaque scattering layers. Nature 491, 232–234 (2012).

    Article  ADS  Google Scholar 

  7. Katz, O., Heidmann, P., Fink, M. & Gigan, S. Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations. Nat. Photon. 8, 784–790 (2014).

    Article  ADS  Google Scholar 

  8. Kang, S. et al. High-resolution adaptive optical imaging within thick scattering media using closed-loop accumulation of single scattering. Nat. Commun. 8, 2157 (2017).

    Article  ADS  Google Scholar 

  9. Jang, M. et al. Wavefront shaping with disorder-engineered metasurfaces. Nat. Photon. 12, 84–90 (2018).

    Article  ADS  Google Scholar 

  10. Horisaki, R., Mori, Y. & Tanida, J. Incoherent light control through scattering media based on machine learning and its application to multiview stereo displays. Opt. Rev. 26, 709–712 (2019).

    Article  Google Scholar 

  11. Badon, A. et al. Distortion matrix concept for deep optical imaging in scattering media. Sci. Adv. 6, eaay7170 (2020).

    Article  ADS  Google Scholar 

  12. Wang, L. V. & Wu, H. Biomedical Optics: Principles and Imaging (Wiley, 2007).

  13. Ntziachristos, V. Going deeper than microscopy: the optical imaging frontier in biology. Nat. Methods 7, 603–614 (2010).

    Article  Google Scholar 

  14. Drexler, W. & Fujimoto, J. (eds) Optical Coherence Tomography: Technology and Applications (Springer International Publishing, 2015).

  15. Dorokhov, O. On the coexistence of localized and extended electronic states in the metallic phase. Solid State Commun. 51, 381–384 (1984).

    Article  ADS  Google Scholar 

  16. Beenakker, C. W. J. Random-matrix theory of quantum transport. Rev. Mod. Phys. 69, 731–808 (1997).

    Article  ADS  Google Scholar 

  17. Pendry, J., MacKinnon, A. & Pretre, A. Maximal fluctuations—a new phenomenon in disordered systems. Phys. A 168, 400–407 (1990).

    Article  Google Scholar 

  18. Akkermans, E. & Montambaux, G. Mesoscopic Physics of Electrons and Photons (Cambridge Univ. Press, 2007).

  19. Pena, A., Girschik, A., Libisch, F., Rotter, S. & Chabanov, A. A. The single-channel regime of transport through random media. Nat. Commun. 5, 3488 (2014).

    Article  ADS  Google Scholar 

  20. Davy, M., Shi, Z., Wang, J. & Genack, A. Z. Transmission statistics and focusing in single disordered samples. Opt. Express 21, 10367–10375 (2013).

    Article  ADS  Google Scholar 

  21. Miller, D. A. B. Waves, modes, communications and optics. Adv. Opt. Photon. 11, 679–825 (2019).

    Article  Google Scholar 

  22. Vellekoop, I. M. & Mosk, A. P. Universal optimal transmission of light through disordered materials. Phys. Rev. Lett. 101, 120601 (2008).

    Article  ADS  Google Scholar 

  23. 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).

    Article  Google Scholar 

  24. Yu, H. et al. Measuring large optical transmission matrices of disordered media. Phys. Rev. Lett. 111, 153902 (2013).

    Article  ADS  Google Scholar 

  25. Kim, M. et al. Maximal energy transport through disordered media with the implementation of transmission eigenchannels. Nat. Photon. 6, 581–585 (2012).

    Article  ADS  Google Scholar 

  26. Mello, P., Pereyra, P. & Kumar, N. Macroscopic approach to multichannel disordered conductors. Ann. Phys. 181, 290–317 (1988).

    Article  ADS  Google Scholar 

  27. Bosch, J., Goorden, S. A. & Mosk, A. P. Frequency width of open channels in multiple scattering media. Opt. Express 24, 26472 (2016).

    Article  ADS  Google Scholar 

  28. Pai, P., Bosch, J. & Mosk, A. P. Optical transmission matrix measurement sampled on a dense hexagonal lattice. OSA Continuum 3, 637–648 (2020).

    Article  Google Scholar 

  29. Forrester, P. J. & Nagao, T. Eigenvalue statistics of the real Ginibre ensemble. Phys. Rev. Lett. 99, 050603 (2007).

    Article  ADS  Google Scholar 

  30. Goetschy, A. & Stone, A. D. Filtering random matrices: the effect of incomplete channel control in multiple scattering. Phys. Rev. Lett. 111, 063901 (2013).

    Article  ADS  Google Scholar 

  31. Popoff, S. M. 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).

    Article  ADS  Google Scholar 

  32. Cuche, E., Marquet, P. & Depeursinge, C. Spatial filtering for zero-order and twin-image elimination in digital off-axis holography. Appl. Opt. 39, 4070–4075 (2000).

    Article  ADS  Google Scholar 

  33. Schöberl, J. C++11 Implementation of Finite Elements in NGSolve (Institute for Analysis and Scientific Computing, Vienna Univ. Technology, 2014).

Download references

Acknowledgements

We acknowledge helpful discussions with P. Ambichl, D. Bouchet, S. Faez, D. van Oosten, P. Jurrius, D. Killian, C. R. de Kok, F. Salihbegovic and S. Steinhauer. Financial support was provided by the Austrian Science Fund (FWF) under project WAVELAND (grant number P32300 to S.R.) and by the Netherlands Organization for Scientific Research NWO (grant number Vici-68047618 to A.P.M.). The computational results presented in this paper were achieved using the Vienna Scientific Cluster (VSC).

Author information

Authors and Affiliations

  1. Debye Institute for Nanomaterials Science and Center for Extreme Matter and Emergent Phenomena, Utrecht University, Utrecht, the Netherlands

    Pritam Pai, Jeroen Bosch & Allard P. Mosk

  2. Institute for Theoretical Physics, TU Wien, Vienna, Austria

    Matthias Kühmayer & Stefan Rotter

Authors
  1. Pritam Pai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeroen Bosch
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matthias Kühmayer
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Stefan Rotter
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Allard P. Mosk
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The experiments were designed by A.P.M., P.P. and J.B. and implemented by P.P. and J.B. The 2D full-wave numerical simulations and the theoretical analysis were carried out by M.K. and the 3D calculations by A.P.M. S.R. proposed the idea and supervised the theoretical research. All authors analysed the results and contributed to the writing of the manuscript.

Corresponding authors

Correspondence to Stefan Rotter or Allard P. Mosk.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Photonics thanks Roarke Horstmeyer and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–19, discussion and methods.

Supplementary Video 1

Evolution of SIM eigenvalues in the complex plane corresponding to Supplementary Fig. 18 for a scatterer filling fraction of 0.05.

Supplementary Video 2

Evolution of SIM eigenvalues in the complex plane corresponding to Supplementary Fig. 18 for a scatterer filling fraction of 0.10.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pai, P., Bosch, J., Kühmayer, M. et al. Scattering invariant modes of light in complex media. Nat. Photonics 15, 431–434 (2021). https://doi.org/10.1038/s41566-021-00789-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41566-021-00789-9

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing