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.

Transport and Anderson localization in disordered two-dimensional photonic lattices

Nature volume 446, pages 52–55 (2007)Cite this article

Abstract

One of the most interesting phenomena in solid-state physics is Anderson localization, which predicts that an electron may become immobile when placed in a disordered lattice1. The origin of localization is interference between multiple scatterings of the electron by random defects in the potential, altering the eigenmodes from being extended (Bloch waves) to exponentially localized2. As a result, the material is transformed from a conductor to an insulator. Anderson’s work dates back to 1958, yet strong localization has never been observed in atomic crystals, because localization occurs only if the potential (the periodic lattice and the fluctuations superimposed on it) is time-independent. However, in atomic crystals important deviations from the Anderson model always occur, because of thermally excited phonons and electron–electron interactions. Realizing that Anderson localization is a wave phenomenon relying on interference, these concepts were extended to optics3,4. Indeed, both weak5,6,7,31 and strong8,9,10,11 localization effects were experimentally demonstrated, traditionally by studying the transmission properties of randomly distributed optical scatterers (typically suspensions or powders of dielectric materials). However, in these studies the potential was fully random, rather than being ‘frozen’ fluctuations on a periodic potential, as the Anderson model assumes. Here we report the experimental observation of Anderson localization in a perturbed periodic potential: the transverse localization of light caused by random fluctuations on a two-dimensional photonic lattice. We demonstrate how ballistic transport becomes diffusive in the presence of disorder, and that crossover to Anderson localization occurs at a higher level of disorder. Finally, we study how nonlinearities affect Anderson localization. As Anderson localization is a universal phenomenon, the ideas presented here could also be implemented in other systems (for example, matter waves), thereby making it feasible to explore experimentally long-sought fundamental concepts, and bringing up a variety of intriguing questions related to the interplay between disorder and nonlinearity.

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

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Get just this article for as long as you need it

$39.95

Learn more

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

Figure 1: Transverse localization scheme.
Figure 2: Experimental results for propagation in disordered lattices.
Figure 3: Results of numerical simulations of linear propagation in disordered lattices.
Figure 4: Numerical (top row) and experimental (bottom row) results, showing the effects of nonlinearity on Anderson localization.

References

  1. Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958)

    Article  ADS  CAS  Google Scholar 

  2. Lee, P. A. & Ramakrishnan, T. V. Disordered electronic systems. Rev. Mod. Phys. 57, 287–337 (1985)

    Article  ADS  CAS  Google Scholar 

  3. John, S. Electromagnetic absorption in a disordered medium near a photon mobility edge. Phys. Rev. Lett. 53, 2169–2172 (1984)

    Article  ADS  Google Scholar 

  4. Anderson, P. W. The question of classical localization: a theory of white paint?. Phil. Mag. B 52, 505–509 (1985)

    Article  ADS  CAS  Google Scholar 

  5. Akkermans, E. & Maynard, R. Weak localization of waves. J. Phys. Lett. 46, 1045–1053 (1985)

    Article  Google Scholar 

  6. Van Albada, M. P. & Lagendijk, A. Observation of weak localization of light in a random medium. Phys. Rev. Lett. 55, 2692–2695 (1985)

    Article  ADS  CAS  Google Scholar 

  7. Wolf, P. E. & Maret, G. Weak localization and coherent backscattering of photons in disordered media. Phys. Rev. Lett. 55, 2696–2699 (1985)

    Article  ADS  CAS  Google Scholar 

  8. Wiersma, D. S., Bartolini, P., Lagendijk, A. & Righini, R. Localization of light in a disordered medium. Nature 390, 671–673 (1997)

    Article  ADS  CAS  Google Scholar 

  9. Berry, M. V. & Klein, S. Transparent mirrors: rays, waves and localization. Eur. J. Phys. 18, 222–228 (1997)

    Article  MathSciNet  Google Scholar 

  10. Chabanov, A. A., Stoytchev, M. & Genack, A. Z. Statistical signatures of photon localization. Nature 404, 850–853 (2000)

    Article  ADS  CAS  Google Scholar 

  11. Störzer, M., Gross, P., Aegerter, C. M. & Maret, G. Observation of the critical regime near Anderson localization of light. Phys. Rev. Lett. 96, 063904 (2006)

    Article  ADS  Google Scholar 

  12. Cao, H. et al. Random laser action in semiconductor powder. Phys. Rev. Lett. 82, 2278–2281 (1999)

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  14. Pertsch, T. et al. Nonlinearity and disorder in fiber arrays. Phys. Rev. Lett. 93, 053901 (2004)

    Article  ADS  Google Scholar 

  15. Eisenberg, H. Nonlinear Effects in Waveguide Arrays. PhD thesis (supervisor Y. Silberberg) Weizmann Institute of Science. (2002)

  16. Efremidis, N. K., Sears, S., Christodoulides, D. N., Fleischer, J. W. & Segev, M. Discrete solitons in photorefractive optically induced photonic lattices. Phys. Rev. E 66, 046602 (2002)

    Article  ADS  Google Scholar 

  17. Fleischer, J. W., Carmon, T., Segev, M., Efremidis, N. K. & Christodoulides, D. N. Observation of discrete solitons in optically induced real time waveguide arrays. Phys. Rev. Lett. 90, 023902 (2003)

    Article  ADS  Google Scholar 

  18. Fleischer, J. W., Segev, M., Efremidis, N. K. & Christodoulides, D. N. Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature 422, 147–150 (2003)

    Article  ADS  CAS  Google Scholar 

  19. Neshev, D. N. et al. Observation of discrete vortex solitons in optically induced photonic lattices. Phys. Rev. Lett. 92, 123903 (2004)

    Article  ADS  Google Scholar 

  20. Fleischer, J. W. et al. Observation of vortex-ring “discrete” solitons in 2D photonic lattices. Phys. Rev. Lett. 92, 123904 (2004)

    Article  ADS  Google Scholar 

  21. Cohen, O. et al. Observation of random-phase lattice solitons. Nature 433, 500–503 (2005)

    Article  ADS  CAS  Google Scholar 

  22. Trompeter, H. et al. Bloch oscillations and Zener tunneling in two-dimensional photonic lattices. Phys. Rev. Lett. 96, 053903 (2006)

    Article  ADS  Google Scholar 

  23. Freedman, B. et al. Wave and defect dynamics in nonlinear photonic quasicrystals. Nature 440, 1166–1169 (2006)

    Article  ADS  CAS  Google Scholar 

  24. Bartal, G. et al. Brillouin zone spectroscopy of nonlinear photonic lattices. Phys. Rev. Lett. 94, 163902 (2005)

    Article  ADS  Google Scholar 

  25. Abrahams, E., Anderson, P. W., Licciardello, D. C. & Ramakrishnan, T. V. Scaling theory of localization: Absence of quantum diffusion in two dimensions. Phys. Rev. Lett. 42, 673–676 (1979)

    Article  ADS  Google Scholar 

  26. Mirlin, A. D. Statistics of energy levels and eigenfunctions in disordered systems. Phys. Rep. 326, 259–382 (2000)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  27. Devillard, P. & Souillard, B. Polynomially decaying transmission for the nonlinear Schrödinger equation in a random medium. J. Stat. Phys. 43, 423–439 (1986)

    Article  ADS  Google Scholar 

  28. Doucot, B. & Rammal, R. Invariant-imbedding approach to localization. II. Nonlinear random media. J. Phys. 48, 527–546 (1987)

    Article  Google Scholar 

  29. Li, Q., Soukoulis, C. M., Pnevmatikos & Economou, E. N. Scattering properties of solitons in nonlinear disordered chains. Phys. Rev. B 38, 11888–11891 (1988)

    Article  ADS  CAS  Google Scholar 

  30. McKenna, M. J., Stanley, R. L. & Maynard, J. D. Effects of nonlinearity on Anderson localization. Phys. Rev. Lett. 69, 1807–1810 (1992)

    Article  ADS  CAS  Google Scholar 

  31. Etemad, S., Thompson, R. & Andrejco, M. J. Weak localization of photons: universal fluctuations and ensemble averaging. Phys. Rev. Lett. 57, 575–578 (1986)

    Article  ADS  CAS  Google Scholar 

Download references

Acknowledgements

We are indebted to our colleagues B. Shapiro and E. Akkermans for discussions on Anderson localization. This research was supported by the Israeli Science Foundation, by the German-Israeli DIP Project, and by the Russell Berrie Nanotechnology Institute at the Technion, Israel.

Author information

Authors and Affiliations

  1. Physics Department, Technion—Israel Institute of Technology, Haifa 32000, Israel,

    Tal Schwartz, Guy Bartal, Shmuel Fishman & Mordechai Segev

Authors
  1. Tal Schwartz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guy Bartal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shmuel Fishman
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mordechai Segev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mordechai Segev.

Ethics declarations

Competing interests

Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests.

Supplementary information

Supplementary Information

This file contains Supplementary Discussion, Supplementary Experimental Methods, Supplementary Figure 1 with Legend, Supplementary Numerical Methods and additional references. Supplementary Discussion is shown as Supplementary Information A. Supplementary Information B describes the experimental setup and the generation of a z-invariant disordered lattice and includes Supplementary Figure 1 showing a schematic diagram of the system. Supplementary Information C describes the numerical simulations used and contains a discussion on the estimated values of the mean free path and localization length. (PDF 1074 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Schwartz, T., Bartal, G., Fishman, S. et al. Transport and Anderson localization in disordered two-dimensional photonic lattices. Nature 446, 52–55 (2007). https://doi.org/10.1038/nature05623

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature05623

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

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