Transverse localization of transmission eigenchannels

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.

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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.

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. 1.

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

    ADS  Article  Google Scholar 

  2. 2.

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

    Article  Google Scholar 

  3. 3.

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

    ADS  Article  Google Scholar 

  4. 4.

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

    Google Scholar 

  5. 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).

    ADS  Article  Google Scholar 

  6. 6.

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

    ADS  Article  Google Scholar 

  7. 7.

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

    ADS  Article  Google Scholar 

  8. 8.

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

    ADS  Article  Google Scholar 

  9. 9.

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

    ADS  Article  Google Scholar 

  10. 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).

    ADS  Article  Google Scholar 

  11. 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).

    ADS  Article  Google Scholar 

  12. 12.

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

    ADS  Article  Google Scholar 

  13. 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. 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).

    ADS  Article  Google Scholar 

  15. 15.

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

    ADS  Article  Google Scholar 

  16. 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).

    ADS  Article  Google Scholar 

  17. 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).

    ADS  Article  Google Scholar 

  18. 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).

    ADS  Article  Google Scholar 

  19. 19.

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

    ADS  Article  Google Scholar 

  20. 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. 21.

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

    ADS  Article  Google Scholar 

  22. 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).

    ADS  Article  Google Scholar 

  23. 23.

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

    ADS  Article  Google Scholar 

  24. 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).

    ADS  Article  Google Scholar 

  25. 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).

    ADS  Article  Google Scholar 

  26. 26.

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

    ADS  Article  Google Scholar 

  27. 27.

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

    ADS  Article  Google Scholar 

  28. 28.

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

    ADS  Article  Google Scholar 

  29. 29.

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

    ADS  Article  Google Scholar 

  30. 30.

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

    ADS  Article  Google Scholar 

  31. 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. 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. 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).

    ADS  Article  Google Scholar 

  34. 34.

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

    ADS  Article  Google Scholar 

  35. 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).

    ADS  Article  Google Scholar 

  36. 36.

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

    ADS  Article  Google Scholar 

  37. 37.

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

    ADS  Article  Google Scholar 

  38. 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. 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).

    ADS  Article  Google Scholar 

  40. 40.

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

    ADS  Article  Google Scholar 

  41. 41.

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

    ADS  Article  Google Scholar 

  42. 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).

    ADS  Article  Google Scholar 

  43. 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. 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. 45.

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

    ADS  MathSciNet  Article  Google Scholar 

  46. 46.

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

    ADS  MathSciNet  Article  Google Scholar 

  47. 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).

    ADS  Article  Google Scholar 

  48. 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).

    ADS  Article  Google Scholar 

  49. 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).

    ADS  Article  Google Scholar 

  50. 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).

    ADS  Article  Google Scholar 

  51. 51.

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

    ADS  Article  Google Scholar 

  52. 52.

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

    ADS  Article  Google Scholar 

  53. 53.

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

    ADS  Article  Google Scholar 

  54. 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).

    ADS  Article  Google Scholar 

  55. 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).

    ADS  Article  Google Scholar 

  56. 56.

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

    ADS  Article  Google Scholar 

  57. 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).

    ADS  Article  Google Scholar 

  58. 58.

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

    ADS  Article  Google Scholar 

  59. 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. 60.

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

  61. 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).

    ADS  Article  Google Scholar 

  62. 62.

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

  63. 63.

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

    ADS  Article  Google Scholar 

  64. 64.

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

    ADS  Article  Google Scholar 

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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).

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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.

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Correspondence to Hui Cao.

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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

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