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.

  • Article
  • Published:

Transverse localization of transmission eigenchannels

Nature Photonics volume 13, pages 352–358 (2019)Cite this article

Subjects

Abstract

Transmission eigenchannels are building blocks of coherent wave transport in diffusive media, and selective excitation of individual eigenchannels can lead to diverse transport behaviour. An essential yet poorly understood property is the transverse spatial profile of each eigenchannel, which is relevant for the associated energy density and critical for coupling light into and out of it. Here, we discover that the transmission eigenchannels of a disordered slab possess exponentially localized incident and outgoing profiles, even in the diffusive regime far from Anderson localization. Such transverse localization arises from a combination of reciprocity, local coupling of spatial modes and non-local correlations of scattered waves. Experimentally, we observe signatures of such localization even with finite illumination area. The transverse localization of high-transmission channels enhances optical energy densities inside turbid media, which will be important for light–matter interactions and imaging applications.

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: Transverse localization of transmission eigenchannels.
Fig. 2: Bandedness of the real-space transmission matrix.
Fig. 3: Scaling of the asymptotic channel width in diffusive slabs.
Fig. 4: Experimental signatures of transverse localization for high-transmission channels.
Fig. 5: Modification of transmission eigenchannel widths by incomplete control.

Similar content being viewed by others

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

References

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

    Article  ADS  Google Scholar 

  2. Imry, Y. Active transmission channels and universal conductance fluctuations. Europhys. Lett. 1, 249–256 (1986).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  4. Nazarov, Y. V. Limits of universality in disordered conductors. Phys. Rev. Lett. 73, 134–137 (1994).

    Google Scholar 

  5. 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 

  6. Vellekoop, I. M. Feedback-based wavefront shaping. Opt. Express 23, 12189–12206 (2015).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  9. 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 

  10. Kim, M., Choi, W., Yoon, C., Kim, G. H. & Choi, W. Relation between transmission eigenchannels and single-channel optimizing modes in a disordered medium. Opt. Lett. 38, 2994–2996 (2013).

    Article  ADS  Google Scholar 

  11. 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).

    Article  ADS  Google Scholar 

  12. Bosch, J., Goorden, S. A. & Mosk, A. P. Frequency width of open channels in multiple scattering media. Opt. Express 24, 26472–26478 (2016).

    Article  ADS  Google Scholar 

  13. 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 

  14. Sarma, R., Yamilov, A. G., 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).

    Article  ADS  Google Scholar 

  15. Choi, W., Mosk, A. P., Park, Q.-H. & Choi, W. Transmission eigenchannels in a disordered medium. Phys. Rev. B 83, 134207 (2011).

    Article  ADS  Google Scholar 

  16. 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).

    Article  ADS  Google Scholar 

  17. Davy, M., Shi, Z., Park, J., Tian, C. & Genack, A. Z. Universal structure of transmission eigenchannels inside opaque media. Nat. Commun. 6, 6893 (2015).

    Article  ADS  Google Scholar 

  18. Ojambati, O. S., Yılmaz, H., Lagendijk, A., Mosk, A. P. & Vos, W. L. Coupling of energy into the fundamental diffusion mode of a complex nanophotonic medium. New J. Phys. 18, 043032 (2016).

    Article  ADS  Google Scholar 

  19. Koirala, M., Sarma, R., Cao, H. & Yamilov, A. Inverse design of perfectly transmitting eigenchannels in scattering media. Phys. Rev. B 96, 054209 (2017).

    Article  ADS  Google Scholar 

  20. 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).

    Article  Google Scholar 

  21. Pendry, J. B. Quasi-extended electron states in strongly disordered systems. J. Phys. C 20, 733 (1987).

    Article  ADS  Google Scholar 

  22. Bertolotti, J., Gottardo, S., Wiersma, D. S., Ghulinyan, M. & Pavesi, L. Optical necklace states in Anderson localized 1D systems. Phys. Rev. Lett. 94, 113903 (2005).

    Article  ADS  Google Scholar 

  23. Sebbah, P., Hu, B., Klosner, J. M. & Genack, A. Z. Extended quasimodes within nominally localized random waveguides. Phys. Rev. Lett. 96, 183902 (2006).

    Article  ADS  Google Scholar 

  24. Choi, W., Park, Q.-H. & Choi, W. Perfect transmission through Anderson localized systems mediated by a cluster of localized modes. Opt. Express 20, 20721–20729 (2012).

    Article  ADS  Google Scholar 

  25. Peña, 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 

  26. Leseur, O., Pierrat, R., Sáenz, J. J. & Carminati, R. Probing two-dimensional Anderson localization without statistics. Phys. Rev. A 90, 053827 (2014).

    Article  ADS  Google Scholar 

  27. Skipetrov, S. E. & Page, J. H. Red light for Anderson localization. New J. Phys. 18, 021001 (2016).

    Article  ADS  Google Scholar 

  28. De Raedt, H., Lagendijk, A. & de Vries, P. Transverse localization of light. Phys. Rev. Lett. 62, 47–50 (1989).

    Article  ADS  Google Scholar 

  29. Schwartz, T., Bartal, G., Fishman, S. & Segev, M. Transport and Anderson localization in disordered two-dimensional photonic lattices. Nature 446, 52–55 (2007).

    Article  ADS  Google Scholar 

  30. Karbasi, S. et al. Observation of transverse Anderson localization in an optical fiber. Opt. Lett. 37, 2304–2306 (2012).

    Article  ADS  Google Scholar 

  31. Hsieh, P. et al. Photon transport enhanced by transverse Anderson localization in disordered superlattices. Nat. Phys. 11, 268–274 (2015).

    Article  Google Scholar 

  32. Hu, H., Strybulevych, A., Page, J. H., Skipetrov, S. E. & Van Tiggelen, B. A. Localization of ultrasound in a three-dimensional elastic network. Nat. Phys. 4, 945–948 (2008).

    Article  Google Scholar 

  33. Cherroret, N., Skipetrov, S. E. & Van Tiggelen, B. A. Transverse confinement of waves in three-dimensional random media. Phys. Rev. E 82, 056603 (2010).

    Article  ADS  Google Scholar 

  34. Horstmeyer, R., Ruan, H. & Yang, C. Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue. Nat. Photon. 9, 563–571 (2015).

    Article  ADS  Google Scholar 

  35. Kim, M., Choi, W., Choi, Y., Yoon, C. & Choi, W. Transmission matrix of a scattering medium and its applications in biophotonics. Opt. Express 23, 12648–12668 (2015).

    Article  ADS  Google Scholar 

  36. Yu, H. et al. Recent advances in wavefront shaping techniques for biomedical applications. Curr. Appl. Phys. 15, 632–641 (2015).

    Article  ADS  Google Scholar 

  37. Vynck, K., Burresi, M., Riboli, F. & Wiersma, D. S. Photon management in two-dimensional disordered media. Nat. Mater. 11, 1017–1022 (2012).

    Article  ADS  Google Scholar 

  38. Liew, S. F. et al. Coherent control of photocurrent in a strongly scattering photoelectrochemical system. ACS Photon. 3, 449–455 (2016).

    Article  Google Scholar 

  39. Baranger, H. U., DiVincenzo, D. P., Jalabert, R. A. & Stone, A. D. Classical and quantum ballistic-transport anomalies in microjunctions. Phys. Rev. B 44, 10637–10675 (1991).

    Article  ADS  Google Scholar 

  40. Jalas, D. et al. What is—and what is not—an optical isolator. Nat. Photon. 7, 579–582 (2013).

    Article  ADS  Google Scholar 

  41. Freund, I., Rosenbluh, M. & Feng, S. Memory effects in propagation of optical waves through disordered media. Phys. Rev. Lett. 61, 2328–2331 (1988).

    Article  ADS  Google Scholar 

  42. Berkovits, R., Kaveh, M. & Feng, S. Memory effect of waves in disordered systems: a real-space approach. Phys. Rev. B 40, 737–740 (1989).

    Article  ADS  Google Scholar 

  43. Judkewitz, B., Horstmeyer, R., Vellekoop, I. M., Papadopoulos, I. N. & Yang, C. Translation correlations in anisotropically scattering media. Nat. Phys. 11, 684–689 (2015).

    Article  Google Scholar 

  44. Osnabrugge, G., Horstmeyer, R., Papadopoulos, I. N., Judkewitz, B. & Vellekoop, I. M. Generalized optical memory effect. Optica 4, 886–892 (2017).

    Article  Google Scholar 

  45. Casati, G., Molinari, L. & Izrailev, F. Scaling properties of band random matrices. Phys. Rev. Lett. 64, 1851–1854 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  46. Izrailev, F. M. Simple models of quantum chaos: spectrum and eigenfunctions. Phys. Rep. 196, 299–392 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  47. Fyodorov, Y. V. & Mirlin, A. D. Analytical derivation of the scaling law for the inverse participation ratio in quasi-one-dimensional disordered systems. Phys. Rev. Lett. 69, 1093–1096 (1992).

    Article  ADS  Google Scholar 

  48. Stephen, M. J. & Cwilich, G. Intensity correlation functions and fluctuations in light scattered from a random medium. Phys. Rev. Lett. 59, 285–287 (1987).

    Article  ADS  Google Scholar 

  49. Feng, S., Kane, C., Lee, P. A. & Stone, A. D. Correlations and fluctuations of coherent wave transmission through disordered media. Phys. Rev. Lett. 61, 834–837 (1988).

    Article  ADS  Google Scholar 

  50. Mello, P. A., Akkermans, E. & Shapiro, B. Macroscopic approach to correlations in the electronic transmission and reflection from disordered conductors. Phys. Rev. Lett. 61, 459–462 (1988).

    Article  ADS  Google Scholar 

  51. Pnini, R. & Shapiro, B. Fluctuations in transmission of waves through disordered slabs. Phys. Rev. B 39, 6986–6994 (1989).

    Article  ADS  Google Scholar 

  52. Berkovits, R. & Feng, S. Correlations in coherent multiple scattering. Phys. Rep. 238, 135–172 (1994).

    Article  ADS  Google Scholar 

  53. Genack, A. Z., Garcia, N. & Polkosnik, W. Long-range intensity correlation in random media. Phys. Rev. Lett. 65, 2129–2132 (1990).

    Article  ADS  Google Scholar 

  54. Scheffold, F., Härtl, W., Maret, G. & Matijevíc, E. Observation of long-range correlations in temporal intensity fluctuations of light. Phys. Rev. B 56, 10942–10952 (1997).

    Article  ADS  Google Scholar 

  55. Sebbah, P., Hu, B., Genack, A. Z., Pnini, R. & Shapiro, B. Spatial-field correlation: the building block of mesoscopic fluctuations. Phys. Rev. Lett. 88, 123901 (2002).

    Article  ADS  Google Scholar 

  56. Yamilov, A. Relation between channel and spatial mesoscopic correlations in volume-disordered waveguides. Phys. Rev. B 78, 045104 (2008).

    Article  ADS  Google Scholar 

  57. Strudley, T., Zehender, T., Blejean, C., Bakkers, E. P. A. M. & Muskens, O. Mesoscopic light transport by very strong collective multiple scattering in nanowire mats. Nat. Photon. 7, 413–418 (2013).

    Article  ADS  Google Scholar 

  58. Fayard, N., Cazé, A., Pierrat, R. & Carminati, R. Intensity correlations between reflected and transmitted speckle patterns. Phys. Rev. A 92, 033827 (2015).

    Article  ADS  Google Scholar 

  59. Starshynov, I. et al. Non-Gaussian correlations between reflected and transmitted intensity patterns emerging from opaque disordered media. Phys. Rev. X 8, 021041 (2018).

    Google Scholar 

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

  61. 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 

  62. Sheng, P. Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Academic Press, San Diego, CA, 1995).

  63. Kramer, B. & MacKinnon, A. Localization: theory and experiment. Rep. Prog. Phys. 56, 1469–1564 (1993).

    Article  ADS  Google Scholar 

  64. 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 

Download references

Acknowledgements

We thank A. Mosk, A. Genack, B. Shapiro, F. Scheffold, S. Skipetrov, S. Bittner, S. Rotter and T. Kottos for stimulating discussions and useful feedback. We acknowledge financial support by the Office of Naval Research (ONR) under grant no. MURI N00014-13-0649 and by the US–Israel Binational Science Foundation (BSF) under grant no. 2015509, as well as computational resources provided by the Yale High Performance Computing Cluster (Yale HPC).

Author information

Authors and Affiliations

  1. Department of Applied Physics, Yale University, New Haven, CT, USA

    Hasan Yılmaz, Chia Wei Hsu & Hui Cao

  2. Department of Physics, Missouri University of Science & Technology, Rolla, MO, USA

    Alexey Yamilov

Authors
  1. Hasan Yılmaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chia Wei Hsu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexey Yamilov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hui Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

H.Y. performed the experiments and analysed the data. C.W.H. performed the numerical simulations and fabricated the samples. H.Y. analysed the numerical data. C.W.H. helped with experimental data acquisition and contributed to numerical data analysis. H.C. supervised the project. All authors contributed to the interpretation of the results. H.Y. and C.W.H. prepared the manuscript, H.C. edited it and A.Y. provided feedback.

Corresponding author

Correspondence to Hui Cao.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

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

Supplementary information

Supplementary Information

This file contains more information on the work, Supplementary Figures 1–11 and Supplementary Table 1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yılmaz, H., Hsu, C.W., Yamilov, A. et al. Transverse localization of transmission eigenchannels. Nat. Photonics 13, 352–358 (2019). https://doi.org/10.1038/s41566-019-0367-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41566-019-0367-9

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