Abstract

Using time-resolved transmission measurements, we have found indications of Anderson localization of light in bulk three-dimensional systems. The observed deviation from classical diffusion is in good accord with theoretical predictions of localization and cannot be explained by absorption or experimental artifacts such as stratification, fluorescence, or background illumination. Moreover, we show that in our samples the control parameter is given by the mean free path times the wavenumber as required by the Ioffe–Regel criterion. This is in contrast to quasi-one-dimensional systems that were studied with microwaves. There, the control parameter is related to the number of modes inside a waveguide, and deviations from classical diffusion are possible due to a small number of modes.

© 2007 Optical Society of America

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    [Crossref]
  2. P. W. Anderson, "The question of classical localization: a theory of white paint?" Philos. Mag. B 52, 505-509 (1985).
    [Crossref]
  3. S. John, "Electromagnetic absorption in a disordered medium near a photon mobility edge," Phys. Rev. Lett. 53, 2169-2172 (1984).
    [Crossref]
  4. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, "Localization of light in a disordered medium," Nature (London) 390, 671-673 (1997).
    [Crossref]
  5. J. M. Drake and A. Z. Genack, "Observation of nonclassical optical diffusion," Phys. Rev. Lett. 63, 259-262 (1989).
    [Crossref] [PubMed]
  6. F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 206-207 (1999).
    [Crossref]
  7. D. S. Wiersma, J. Gómez Rivas, P. Bartolini, A. Lagendijk, and R. Righini, reply to F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 207 (1999).
    [Crossref]
  8. A. A. Chabanov, M. Stoytchev, and A. Z. Genack, "Statistical signatures of photon localization," Nature (London) 404, 850-853 (2000).
    [Crossref]
  9. M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, "Speed of propagation of classical waves in strongly scattering media," Phys. Rev. Lett. 66, 3132-3135 (1991).
    [Crossref] [PubMed]
  10. G. H. Watson, P. A. Fleury, and S. L. McCall, "Searching for photon localization in the time domain," Phys. Rev. Lett. 58, 945-948 (1987).
    [Crossref] [PubMed]
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    [Crossref]
  12. A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, "Breakdown of diffusion in dynamics of extended waves in mesoscopic media," Phys. Rev. Lett. 90, 203903 (1-4) (2003).
    [Crossref] [PubMed]
  13. P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, "Extended quasimodes within nominally localized random waveguides," Phys. Rev. Lett. 96, 183902 (1-4) (2006).
    [Crossref] [PubMed]
  14. S. K. Cheung, X. Zhang, Z. Q. Zhang, A. A. Chabanov, and A. Z. Genack, "Impact of weak localization in the time domain," Phys. Rev. Lett. 92, 173902 (1-4) (2004).
    [Crossref] [PubMed]
  15. S. E. Skipetrov and B. A. van Tiggelen, "Dynamics of weakly localized waves," Phys. Rev. Lett. 92, 113901 (1-4) (2004).
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    [Crossref] [PubMed]
  18. F. Scheffold, W. Härtl, G. Maret, and E. Matijevic, "Observation of long-range correlations in temporal intensity fluctuations of light," Phys. Rev. B 56, 10942-10952 (1997).
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  19. F. Scheffold and G. Maret, "Universal conductance fluctuations of light," Phys. Rev. Lett. 81, 5800-5803 (1998).
    [Crossref]
  20. S. E. Skipetrov and B. A. van Tiggelen, "Dynamics of Anderson localization in open 3D media," Phys. Rev. Lett. 96, 043902 (1-4) (2006).
    [Crossref] [PubMed]
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  22. R. Berkovitz and M. Kaveh, "Propagation of waves through a slab near the Anderson transition: a local scaling approach," J. Chem. Soc. Dalton Trans. 2, 307-321 (1990).
  23. R. Berkovitz and M. Kaveh, "Backscattering of light near the optical Anderson transition," Phys. Rev. B 36, 9322-9325 (1987).
    [Crossref]
  24. C. M. Aegerter, M. Störzer, and G. Maret, "Experimental determination of critical exponents in Anderson localization of light," Europhys. Lett. 75, 562-568 (2006).
    [Crossref]
  25. P. E Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett. 55, 2696-2699 (1985).
    [Crossref]
  26. M. P. van Albada and A. Lagendijk, "Observation of weak localization of light in a random medium," Phys. Rev. Lett. 55, 2692-2695 (1985).
    [Crossref] [PubMed]
  27. E. Akkermans, P. E. Wolf, and R. Maynard, "Coherent backscattering of light by disordered media: analysis of the peak line shape," Phys. Rev. Lett. 56, 1471-1474 (1986).
    [Crossref] [PubMed]
  28. J. C. M. Garnett, "Colours in metal glasses and in metallic films," Philos. Trans. R. Soc. London, Ser. A 203, 385-420 (1904).
    [Crossref]
  29. J. X. Zhu, D. J. Pine, and D. A. Weitz, "Internal reflection of diffusive light in random media," Phys. Rev. A 44, 3948-3959 (1991).
    [Crossref] [PubMed]
  30. K. Busch and C. M. Soukoulis, "Transport properties of random media: an energy-density CPA approach," Phys. Rev. B 54, 893-899 (1996).
    [Crossref]
  31. P. Gross, "Coherent backscattering close to the transition to strong localization of light," Diploma Thesis, University of Konstanz (2005).
  32. P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. M. Aegerter, "A precise method to determine the angular distribution of backscattered light to high angles," Rev. Sci. Instrum. 78, 033105 (1-6) (2007).
    [Crossref] [PubMed]
  33. M. Störzer, C. M. Aegerter, and G. Maret, "Reduced transport velocity of multiply scattered light due to resonant scattering," Phys. Rev. E 73, 065602(R)1-4 (2006).
    [Crossref]
  34. D. S. Wiersma, "Light in strongly scattering and amplifying random media," Ph.D thesis (University of Amsterdam, 1995).
  35. R. Lenke, R. Tweer, and G. Maret, "Coherent backscattering and localization in a self-attracting random walk model," Eur. Phys. J. B 26, 235-240 (2002).
    [Crossref]
  36. E. Abrahams, P. W. Anderson, D. Licciardello, and T. V. Ramakrishnan, "Scaling theory of localization: absence of quantum diffusion in two dimensions," Phys. Rev. Lett. 42, 673-676 (1979).
    [Crossref]

2007 (1)

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. M. Aegerter, "A precise method to determine the angular distribution of backscattered light to high angles," Rev. Sci. Instrum. 78, 033105 (1-6) (2007).
[Crossref] [PubMed]

2006 (5)

M. Störzer, C. M. Aegerter, and G. Maret, "Reduced transport velocity of multiply scattered light due to resonant scattering," Phys. Rev. E 73, 065602(R)1-4 (2006).
[Crossref]

S. E. Skipetrov and B. A. van Tiggelen, "Dynamics of Anderson localization in open 3D media," Phys. Rev. Lett. 96, 043902 (1-4) (2006).
[Crossref] [PubMed]

C. M. Aegerter, M. Störzer, and G. Maret, "Experimental determination of critical exponents in Anderson localization of light," Europhys. Lett. 75, 562-568 (2006).
[Crossref]

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, "Extended quasimodes within nominally localized random waveguides," Phys. Rev. Lett. 96, 183902 (1-4) (2006).
[Crossref] [PubMed]

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, "Observation of the critical regime near Anderson localization of light," Phys. Rev. Lett. 96, 063904 (1-4) (2006).
[Crossref] [PubMed]

2004 (2)

S. K. Cheung, X. Zhang, Z. Q. Zhang, A. A. Chabanov, and A. Z. Genack, "Impact of weak localization in the time domain," Phys. Rev. Lett. 92, 173902 (1-4) (2004).
[Crossref] [PubMed]

S. E. Skipetrov and B. A. van Tiggelen, "Dynamics of weakly localized waves," Phys. Rev. Lett. 92, 113901 (1-4) (2004).
[Crossref] [PubMed]

2003 (2)

P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, "Time resolved pulse propagation in a strongly scattering material," Phys. Rev. E 68, 016604 (1-9) (2003).
[Crossref]

A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, "Breakdown of diffusion in dynamics of extended waves in mesoscopic media," Phys. Rev. Lett. 90, 203903 (1-4) (2003).
[Crossref] [PubMed]

2002 (1)

R. Lenke, R. Tweer, and G. Maret, "Coherent backscattering and localization in a self-attracting random walk model," Eur. Phys. J. B 26, 235-240 (2002).
[Crossref]

2000 (1)

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, "Statistical signatures of photon localization," Nature (London) 404, 850-853 (2000).
[Crossref]

1999 (2)

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 206-207 (1999).
[Crossref]

D. S. Wiersma, J. Gómez Rivas, P. Bartolini, A. Lagendijk, and R. Righini, reply to F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 207 (1999).
[Crossref]

1998 (1)

F. Scheffold and G. Maret, "Universal conductance fluctuations of light," Phys. Rev. Lett. 81, 5800-5803 (1998).
[Crossref]

1997 (2)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, "Localization of light in a disordered medium," Nature (London) 390, 671-673 (1997).
[Crossref]

F. Scheffold, W. Härtl, G. Maret, and E. Matijevic, "Observation of long-range correlations in temporal intensity fluctuations of light," Phys. Rev. B 56, 10942-10952 (1997).
[Crossref]

1996 (1)

K. Busch and C. M. Soukoulis, "Transport properties of random media: an energy-density CPA approach," Phys. Rev. B 54, 893-899 (1996).
[Crossref]

1991 (2)

J. X. Zhu, D. J. Pine, and D. A. Weitz, "Internal reflection of diffusive light in random media," Phys. Rev. A 44, 3948-3959 (1991).
[Crossref] [PubMed]

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, "Speed of propagation of classical waves in strongly scattering media," Phys. Rev. Lett. 66, 3132-3135 (1991).
[Crossref] [PubMed]

1990 (1)

R. Berkovitz and M. Kaveh, "Propagation of waves through a slab near the Anderson transition: a local scaling approach," J. Chem. Soc. Dalton Trans. 2, 307-321 (1990).

1989 (1)

J. M. Drake and A. Z. Genack, "Observation of nonclassical optical diffusion," Phys. Rev. Lett. 63, 259-262 (1989).
[Crossref] [PubMed]

1987 (2)

G. H. Watson, P. A. Fleury, and S. L. McCall, "Searching for photon localization in the time domain," Phys. Rev. Lett. 58, 945-948 (1987).
[Crossref] [PubMed]

R. Berkovitz and M. Kaveh, "Backscattering of light near the optical Anderson transition," Phys. Rev. B 36, 9322-9325 (1987).
[Crossref]

1986 (1)

E. Akkermans, P. E. Wolf, and R. Maynard, "Coherent backscattering of light by disordered media: analysis of the peak line shape," Phys. Rev. Lett. 56, 1471-1474 (1986).
[Crossref] [PubMed]

1985 (3)

P. E Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett. 55, 2696-2699 (1985).
[Crossref]

M. P. van Albada and A. Lagendijk, "Observation of weak localization of light in a random medium," Phys. Rev. Lett. 55, 2692-2695 (1985).
[Crossref] [PubMed]

P. W. Anderson, "The question of classical localization: a theory of white paint?" Philos. Mag. B 52, 505-509 (1985).
[Crossref]

1984 (1)

S. John, "Electromagnetic absorption in a disordered medium near a photon mobility edge," Phys. Rev. Lett. 53, 2169-2172 (1984).
[Crossref]

1979 (1)

E. Abrahams, P. W. Anderson, D. Licciardello, and T. V. Ramakrishnan, "Scaling theory of localization: absence of quantum diffusion in two dimensions," Phys. Rev. Lett. 42, 673-676 (1979).
[Crossref]

1960 (1)

A. F. Ioffe and A. R. Regel, "Non-crystalline, amorphous and liquid electronic semiconductors," Prog. Semicond. 4, 237-291 (1960).

1958 (1)

P. W. Anderson, "Absence of diffusion in certain random lattices," Phys. Rev. 109, 1492-1505 (1958).
[Crossref]

1904 (1)

J. C. M. Garnett, "Colours in metal glasses and in metallic films," Philos. Trans. R. Soc. London, Ser. A 203, 385-420 (1904).
[Crossref]

Abrahams, E.

E. Abrahams, P. W. Anderson, D. Licciardello, and T. V. Ramakrishnan, "Scaling theory of localization: absence of quantum diffusion in two dimensions," Phys. Rev. Lett. 42, 673-676 (1979).
[Crossref]

Aegerter, C. M.

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. M. Aegerter, "A precise method to determine the angular distribution of backscattered light to high angles," Rev. Sci. Instrum. 78, 033105 (1-6) (2007).
[Crossref] [PubMed]

M. Störzer, C. M. Aegerter, and G. Maret, "Reduced transport velocity of multiply scattered light due to resonant scattering," Phys. Rev. E 73, 065602(R)1-4 (2006).
[Crossref]

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, "Observation of the critical regime near Anderson localization of light," Phys. Rev. Lett. 96, 063904 (1-4) (2006).
[Crossref] [PubMed]

C. M. Aegerter, M. Störzer, and G. Maret, "Experimental determination of critical exponents in Anderson localization of light," Europhys. Lett. 75, 562-568 (2006).
[Crossref]

Akkermans, E.

E. Akkermans, P. E. Wolf, and R. Maynard, "Coherent backscattering of light by disordered media: analysis of the peak line shape," Phys. Rev. Lett. 56, 1471-1474 (1986).
[Crossref] [PubMed]

Anderson, P. W.

P. W. Anderson, "The question of classical localization: a theory of white paint?" Philos. Mag. B 52, 505-509 (1985).
[Crossref]

E. Abrahams, P. W. Anderson, D. Licciardello, and T. V. Ramakrishnan, "Scaling theory of localization: absence of quantum diffusion in two dimensions," Phys. Rev. Lett. 42, 673-676 (1979).
[Crossref]

P. W. Anderson, "Absence of diffusion in certain random lattices," Phys. Rev. 109, 1492-1505 (1958).
[Crossref]

Bartolini, P.

D. S. Wiersma, J. Gómez Rivas, P. Bartolini, A. Lagendijk, and R. Righini, reply to F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 207 (1999).
[Crossref]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, "Localization of light in a disordered medium," Nature (London) 390, 671-673 (1997).
[Crossref]

Berkovitz, R.

R. Berkovitz and M. Kaveh, "Propagation of waves through a slab near the Anderson transition: a local scaling approach," J. Chem. Soc. Dalton Trans. 2, 307-321 (1990).

R. Berkovitz and M. Kaveh, "Backscattering of light near the optical Anderson transition," Phys. Rev. B 36, 9322-9325 (1987).
[Crossref]

Bret, B. P. J.

P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, "Time resolved pulse propagation in a strongly scattering material," Phys. Rev. E 68, 016604 (1-9) (2003).
[Crossref]

Busch, K.

K. Busch and C. M. Soukoulis, "Transport properties of random media: an energy-density CPA approach," Phys. Rev. B 54, 893-899 (1996).
[Crossref]

Chabanov, A. A.

S. K. Cheung, X. Zhang, Z. Q. Zhang, A. A. Chabanov, and A. Z. Genack, "Impact of weak localization in the time domain," Phys. Rev. Lett. 92, 173902 (1-4) (2004).
[Crossref] [PubMed]

A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, "Breakdown of diffusion in dynamics of extended waves in mesoscopic media," Phys. Rev. Lett. 90, 203903 (1-4) (2003).
[Crossref] [PubMed]

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, "Statistical signatures of photon localization," Nature (London) 404, 850-853 (2000).
[Crossref]

Cheung, S. K.

S. K. Cheung, X. Zhang, Z. Q. Zhang, A. A. Chabanov, and A. Z. Genack, "Impact of weak localization in the time domain," Phys. Rev. Lett. 92, 173902 (1-4) (2004).
[Crossref] [PubMed]

Clausen, M.

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. M. Aegerter, "A precise method to determine the angular distribution of backscattered light to high angles," Rev. Sci. Instrum. 78, 033105 (1-6) (2007).
[Crossref] [PubMed]

Drake, J. M.

J. M. Drake and A. Z. Genack, "Observation of nonclassical optical diffusion," Phys. Rev. Lett. 63, 259-262 (1989).
[Crossref] [PubMed]

E Wolf, P.

P. E Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett. 55, 2696-2699 (1985).
[Crossref]

Fiebig, S.

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. M. Aegerter, "A precise method to determine the angular distribution of backscattered light to high angles," Rev. Sci. Instrum. 78, 033105 (1-6) (2007).
[Crossref] [PubMed]

Fleury, P. A.

G. H. Watson, P. A. Fleury, and S. L. McCall, "Searching for photon localization in the time domain," Phys. Rev. Lett. 58, 945-948 (1987).
[Crossref] [PubMed]

Garnett, J. C. M.

J. C. M. Garnett, "Colours in metal glasses and in metallic films," Philos. Trans. R. Soc. London, Ser. A 203, 385-420 (1904).
[Crossref]

Genack, A. Z.

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, "Extended quasimodes within nominally localized random waveguides," Phys. Rev. Lett. 96, 183902 (1-4) (2006).
[Crossref] [PubMed]

S. K. Cheung, X. Zhang, Z. Q. Zhang, A. A. Chabanov, and A. Z. Genack, "Impact of weak localization in the time domain," Phys. Rev. Lett. 92, 173902 (1-4) (2004).
[Crossref] [PubMed]

A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, "Breakdown of diffusion in dynamics of extended waves in mesoscopic media," Phys. Rev. Lett. 90, 203903 (1-4) (2003).
[Crossref] [PubMed]

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, "Statistical signatures of photon localization," Nature (London) 404, 850-853 (2000).
[Crossref]

J. M. Drake and A. Z. Genack, "Observation of nonclassical optical diffusion," Phys. Rev. Lett. 63, 259-262 (1989).
[Crossref] [PubMed]

Gómez Rivas, J.

D. S. Wiersma, J. Gómez Rivas, P. Bartolini, A. Lagendijk, and R. Righini, reply to F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 207 (1999).
[Crossref]

Gross, P.

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. M. Aegerter, "A precise method to determine the angular distribution of backscattered light to high angles," Rev. Sci. Instrum. 78, 033105 (1-6) (2007).
[Crossref] [PubMed]

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, "Observation of the critical regime near Anderson localization of light," Phys. Rev. Lett. 96, 063904 (1-4) (2006).
[Crossref] [PubMed]

P. Gross, "Coherent backscattering close to the transition to strong localization of light," Diploma Thesis, University of Konstanz (2005).

Härtl, W.

F. Scheffold, W. Härtl, G. Maret, and E. Matijevic, "Observation of long-range correlations in temporal intensity fluctuations of light," Phys. Rev. B 56, 10942-10952 (1997).
[Crossref]

Hu, B.

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, "Extended quasimodes within nominally localized random waveguides," Phys. Rev. Lett. 96, 183902 (1-4) (2006).
[Crossref] [PubMed]

Imhof, A.

P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, "Time resolved pulse propagation in a strongly scattering material," Phys. Rev. E 68, 016604 (1-9) (2003).
[Crossref]

Ioffe, A. F.

A. F. Ioffe and A. R. Regel, "Non-crystalline, amorphous and liquid electronic semiconductors," Prog. Semicond. 4, 237-291 (1960).

John, S.

S. John, "Electromagnetic absorption in a disordered medium near a photon mobility edge," Phys. Rev. Lett. 53, 2169-2172 (1984).
[Crossref]

Johnson, P. M.

P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, "Time resolved pulse propagation in a strongly scattering material," Phys. Rev. E 68, 016604 (1-9) (2003).
[Crossref]

Kaveh, M.

R. Berkovitz and M. Kaveh, "Propagation of waves through a slab near the Anderson transition: a local scaling approach," J. Chem. Soc. Dalton Trans. 2, 307-321 (1990).

R. Berkovitz and M. Kaveh, "Backscattering of light near the optical Anderson transition," Phys. Rev. B 36, 9322-9325 (1987).
[Crossref]

Klosner, J. M.

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, "Extended quasimodes within nominally localized random waveguides," Phys. Rev. Lett. 96, 183902 (1-4) (2006).
[Crossref] [PubMed]

Lagendijk, A.

P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, "Time resolved pulse propagation in a strongly scattering material," Phys. Rev. E 68, 016604 (1-9) (2003).
[Crossref]

D. S. Wiersma, J. Gómez Rivas, P. Bartolini, A. Lagendijk, and R. Righini, reply to F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 207 (1999).
[Crossref]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, "Localization of light in a disordered medium," Nature (London) 390, 671-673 (1997).
[Crossref]

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, "Speed of propagation of classical waves in strongly scattering media," Phys. Rev. Lett. 66, 3132-3135 (1991).
[Crossref] [PubMed]

M. P. van Albada and A. Lagendijk, "Observation of weak localization of light in a random medium," Phys. Rev. Lett. 55, 2692-2695 (1985).
[Crossref] [PubMed]

Lenke, R.

R. Lenke, R. Tweer, and G. Maret, "Coherent backscattering and localization in a self-attracting random walk model," Eur. Phys. J. B 26, 235-240 (2002).
[Crossref]

D. S. Wiersma, J. Gómez Rivas, P. Bartolini, A. Lagendijk, and R. Righini, reply to F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 207 (1999).
[Crossref]

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 206-207 (1999).
[Crossref]

Licciardello, D.

E. Abrahams, P. W. Anderson, D. Licciardello, and T. V. Ramakrishnan, "Scaling theory of localization: absence of quantum diffusion in two dimensions," Phys. Rev. Lett. 42, 673-676 (1979).
[Crossref]

Maret, G.

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. M. Aegerter, "A precise method to determine the angular distribution of backscattered light to high angles," Rev. Sci. Instrum. 78, 033105 (1-6) (2007).
[Crossref] [PubMed]

M. Störzer, C. M. Aegerter, and G. Maret, "Reduced transport velocity of multiply scattered light due to resonant scattering," Phys. Rev. E 73, 065602(R)1-4 (2006).
[Crossref]

C. M. Aegerter, M. Störzer, and G. Maret, "Experimental determination of critical exponents in Anderson localization of light," Europhys. Lett. 75, 562-568 (2006).
[Crossref]

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, "Observation of the critical regime near Anderson localization of light," Phys. Rev. Lett. 96, 063904 (1-4) (2006).
[Crossref] [PubMed]

R. Lenke, R. Tweer, and G. Maret, "Coherent backscattering and localization in a self-attracting random walk model," Eur. Phys. J. B 26, 235-240 (2002).
[Crossref]

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 206-207 (1999).
[Crossref]

D. S. Wiersma, J. Gómez Rivas, P. Bartolini, A. Lagendijk, and R. Righini, reply to F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 207 (1999).
[Crossref]

F. Scheffold and G. Maret, "Universal conductance fluctuations of light," Phys. Rev. Lett. 81, 5800-5803 (1998).
[Crossref]

F. Scheffold, W. Härtl, G. Maret, and E. Matijevic, "Observation of long-range correlations in temporal intensity fluctuations of light," Phys. Rev. B 56, 10942-10952 (1997).
[Crossref]

P. E Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett. 55, 2696-2699 (1985).
[Crossref]

Matijevic, E.

F. Scheffold, W. Härtl, G. Maret, and E. Matijevic, "Observation of long-range correlations in temporal intensity fluctuations of light," Phys. Rev. B 56, 10942-10952 (1997).
[Crossref]

Maynard, R.

E. Akkermans, P. E. Wolf, and R. Maynard, "Coherent backscattering of light by disordered media: analysis of the peak line shape," Phys. Rev. Lett. 56, 1471-1474 (1986).
[Crossref] [PubMed]

McCall, S. L.

G. H. Watson, P. A. Fleury, and S. L. McCall, "Searching for photon localization in the time domain," Phys. Rev. Lett. 58, 945-948 (1987).
[Crossref] [PubMed]

Pine, D. J.

J. X. Zhu, D. J. Pine, and D. A. Weitz, "Internal reflection of diffusive light in random media," Phys. Rev. A 44, 3948-3959 (1991).
[Crossref] [PubMed]

Ramakrishnan, T. V.

E. Abrahams, P. W. Anderson, D. Licciardello, and T. V. Ramakrishnan, "Scaling theory of localization: absence of quantum diffusion in two dimensions," Phys. Rev. Lett. 42, 673-676 (1979).
[Crossref]

Regel, A. R.

A. F. Ioffe and A. R. Regel, "Non-crystalline, amorphous and liquid electronic semiconductors," Prog. Semicond. 4, 237-291 (1960).

Righini, R.

D. S. Wiersma, J. Gómez Rivas, P. Bartolini, A. Lagendijk, and R. Righini, reply to F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 207 (1999).
[Crossref]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, "Localization of light in a disordered medium," Nature (London) 390, 671-673 (1997).
[Crossref]

Rivas, J. G.

P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, "Time resolved pulse propagation in a strongly scattering material," Phys. Rev. E 68, 016604 (1-9) (2003).
[Crossref]

Scheffold, F.

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 206-207 (1999).
[Crossref]

D. S. Wiersma, J. Gómez Rivas, P. Bartolini, A. Lagendijk, and R. Righini, reply to F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 207 (1999).
[Crossref]

F. Scheffold and G. Maret, "Universal conductance fluctuations of light," Phys. Rev. Lett. 81, 5800-5803 (1998).
[Crossref]

F. Scheffold, W. Härtl, G. Maret, and E. Matijevic, "Observation of long-range correlations in temporal intensity fluctuations of light," Phys. Rev. B 56, 10942-10952 (1997).
[Crossref]

Sebbah, P.

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, "Extended quasimodes within nominally localized random waveguides," Phys. Rev. Lett. 96, 183902 (1-4) (2006).
[Crossref] [PubMed]

Skipetrov, S. E.

S. E. Skipetrov and B. A. van Tiggelen, "Dynamics of Anderson localization in open 3D media," Phys. Rev. Lett. 96, 043902 (1-4) (2006).
[Crossref] [PubMed]

S. E. Skipetrov and B. A. van Tiggelen, "Dynamics of weakly localized waves," Phys. Rev. Lett. 92, 113901 (1-4) (2004).
[Crossref] [PubMed]

S. E. Skipetrov, sergey.skipetrov@grenoble.cnrs.fr (personal communication, December 2005).

Soukoulis, C. M.

K. Busch and C. M. Soukoulis, "Transport properties of random media: an energy-density CPA approach," Phys. Rev. B 54, 893-899 (1996).
[Crossref]

Störzer, M.

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. M. Aegerter, "A precise method to determine the angular distribution of backscattered light to high angles," Rev. Sci. Instrum. 78, 033105 (1-6) (2007).
[Crossref] [PubMed]

M. Störzer, C. M. Aegerter, and G. Maret, "Reduced transport velocity of multiply scattered light due to resonant scattering," Phys. Rev. E 73, 065602(R)1-4 (2006).
[Crossref]

C. M. Aegerter, M. Störzer, and G. Maret, "Experimental determination of critical exponents in Anderson localization of light," Europhys. Lett. 75, 562-568 (2006).
[Crossref]

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, "Observation of the critical regime near Anderson localization of light," Phys. Rev. Lett. 96, 063904 (1-4) (2006).
[Crossref] [PubMed]

Stoytchev, M.

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, "Statistical signatures of photon localization," Nature (London) 404, 850-853 (2000).
[Crossref]

Tip, A.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, "Speed of propagation of classical waves in strongly scattering media," Phys. Rev. Lett. 66, 3132-3135 (1991).
[Crossref] [PubMed]

Tweer, R.

R. Lenke, R. Tweer, and G. Maret, "Coherent backscattering and localization in a self-attracting random walk model," Eur. Phys. J. B 26, 235-240 (2002).
[Crossref]

D. S. Wiersma, J. Gómez Rivas, P. Bartolini, A. Lagendijk, and R. Righini, reply to F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 207 (1999).
[Crossref]

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 206-207 (1999).
[Crossref]

van Albada, M. P.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, "Speed of propagation of classical waves in strongly scattering media," Phys. Rev. Lett. 66, 3132-3135 (1991).
[Crossref] [PubMed]

M. P. van Albada and A. Lagendijk, "Observation of weak localization of light in a random medium," Phys. Rev. Lett. 55, 2692-2695 (1985).
[Crossref] [PubMed]

van Tiggelen, B. A.

S. E. Skipetrov and B. A. van Tiggelen, "Dynamics of Anderson localization in open 3D media," Phys. Rev. Lett. 96, 043902 (1-4) (2006).
[Crossref] [PubMed]

S. E. Skipetrov and B. A. van Tiggelen, "Dynamics of weakly localized waves," Phys. Rev. Lett. 92, 113901 (1-4) (2004).
[Crossref] [PubMed]

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, "Speed of propagation of classical waves in strongly scattering media," Phys. Rev. Lett. 66, 3132-3135 (1991).
[Crossref] [PubMed]

Watson, G. H.

G. H. Watson, P. A. Fleury, and S. L. McCall, "Searching for photon localization in the time domain," Phys. Rev. Lett. 58, 945-948 (1987).
[Crossref] [PubMed]

Weitz, D. A.

J. X. Zhu, D. J. Pine, and D. A. Weitz, "Internal reflection of diffusive light in random media," Phys. Rev. A 44, 3948-3959 (1991).
[Crossref] [PubMed]

Wiersma, D. S.

D. S. Wiersma, J. Gómez Rivas, P. Bartolini, A. Lagendijk, and R. Righini, reply to F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 207 (1999).
[Crossref]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, "Localization of light in a disordered medium," Nature (London) 390, 671-673 (1997).
[Crossref]

D. S. Wiersma, "Light in strongly scattering and amplifying random media," Ph.D thesis (University of Amsterdam, 1995).

Wolf, P. E.

E. Akkermans, P. E. Wolf, and R. Maynard, "Coherent backscattering of light by disordered media: analysis of the peak line shape," Phys. Rev. Lett. 56, 1471-1474 (1986).
[Crossref] [PubMed]

Zhang, X.

S. K. Cheung, X. Zhang, Z. Q. Zhang, A. A. Chabanov, and A. Z. Genack, "Impact of weak localization in the time domain," Phys. Rev. Lett. 92, 173902 (1-4) (2004).
[Crossref] [PubMed]

Zhang, Z. Q.

S. K. Cheung, X. Zhang, Z. Q. Zhang, A. A. Chabanov, and A. Z. Genack, "Impact of weak localization in the time domain," Phys. Rev. Lett. 92, 173902 (1-4) (2004).
[Crossref] [PubMed]

A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, "Breakdown of diffusion in dynamics of extended waves in mesoscopic media," Phys. Rev. Lett. 90, 203903 (1-4) (2003).
[Crossref] [PubMed]

Zhu, J. X.

J. X. Zhu, D. J. Pine, and D. A. Weitz, "Internal reflection of diffusive light in random media," Phys. Rev. A 44, 3948-3959 (1991).
[Crossref] [PubMed]

Eur. Phys. J. B (1)

R. Lenke, R. Tweer, and G. Maret, "Coherent backscattering and localization in a self-attracting random walk model," Eur. Phys. J. B 26, 235-240 (2002).
[Crossref]

Europhys. Lett. (1)

C. M. Aegerter, M. Störzer, and G. Maret, "Experimental determination of critical exponents in Anderson localization of light," Europhys. Lett. 75, 562-568 (2006).
[Crossref]

J. Chem. Soc. Dalton Trans. (1)

R. Berkovitz and M. Kaveh, "Propagation of waves through a slab near the Anderson transition: a local scaling approach," J. Chem. Soc. Dalton Trans. 2, 307-321 (1990).

Nature (London) (4)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, "Localization of light in a disordered medium," Nature (London) 390, 671-673 (1997).
[Crossref]

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 206-207 (1999).
[Crossref]

D. S. Wiersma, J. Gómez Rivas, P. Bartolini, A. Lagendijk, and R. Righini, reply to F. Scheffold, R. Lenke, R. Tweer, and G. Maret, "Localization or classical diffusion of light?" Nature (London) 398, 207 (1999).
[Crossref]

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, "Statistical signatures of photon localization," Nature (London) 404, 850-853 (2000).
[Crossref]

Philos. Mag. B (1)

P. W. Anderson, "The question of classical localization: a theory of white paint?" Philos. Mag. B 52, 505-509 (1985).
[Crossref]

Philos. Trans. R. Soc. London, Ser. A (1)

J. C. M. Garnett, "Colours in metal glasses and in metallic films," Philos. Trans. R. Soc. London, Ser. A 203, 385-420 (1904).
[Crossref]

Phys. Rev. (1)

P. W. Anderson, "Absence of diffusion in certain random lattices," Phys. Rev. 109, 1492-1505 (1958).
[Crossref]

Phys. Rev. A (1)

J. X. Zhu, D. J. Pine, and D. A. Weitz, "Internal reflection of diffusive light in random media," Phys. Rev. A 44, 3948-3959 (1991).
[Crossref] [PubMed]

Phys. Rev. B (3)

K. Busch and C. M. Soukoulis, "Transport properties of random media: an energy-density CPA approach," Phys. Rev. B 54, 893-899 (1996).
[Crossref]

R. Berkovitz and M. Kaveh, "Backscattering of light near the optical Anderson transition," Phys. Rev. B 36, 9322-9325 (1987).
[Crossref]

F. Scheffold, W. Härtl, G. Maret, and E. Matijevic, "Observation of long-range correlations in temporal intensity fluctuations of light," Phys. Rev. B 56, 10942-10952 (1997).
[Crossref]

Phys. Rev. E (2)

P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, "Time resolved pulse propagation in a strongly scattering material," Phys. Rev. E 68, 016604 (1-9) (2003).
[Crossref]

M. Störzer, C. M. Aegerter, and G. Maret, "Reduced transport velocity of multiply scattered light due to resonant scattering," Phys. Rev. E 73, 065602(R)1-4 (2006).
[Crossref]

Phys. Rev. Lett. (15)

E. Abrahams, P. W. Anderson, D. Licciardello, and T. V. Ramakrishnan, "Scaling theory of localization: absence of quantum diffusion in two dimensions," Phys. Rev. Lett. 42, 673-676 (1979).
[Crossref]

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, "Observation of the critical regime near Anderson localization of light," Phys. Rev. Lett. 96, 063904 (1-4) (2006).
[Crossref] [PubMed]

P. E Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett. 55, 2696-2699 (1985).
[Crossref]

M. P. van Albada and A. Lagendijk, "Observation of weak localization of light in a random medium," Phys. Rev. Lett. 55, 2692-2695 (1985).
[Crossref] [PubMed]

E. Akkermans, P. E. Wolf, and R. Maynard, "Coherent backscattering of light by disordered media: analysis of the peak line shape," Phys. Rev. Lett. 56, 1471-1474 (1986).
[Crossref] [PubMed]

A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, "Breakdown of diffusion in dynamics of extended waves in mesoscopic media," Phys. Rev. Lett. 90, 203903 (1-4) (2003).
[Crossref] [PubMed]

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, "Extended quasimodes within nominally localized random waveguides," Phys. Rev. Lett. 96, 183902 (1-4) (2006).
[Crossref] [PubMed]

S. K. Cheung, X. Zhang, Z. Q. Zhang, A. A. Chabanov, and A. Z. Genack, "Impact of weak localization in the time domain," Phys. Rev. Lett. 92, 173902 (1-4) (2004).
[Crossref] [PubMed]

S. E. Skipetrov and B. A. van Tiggelen, "Dynamics of weakly localized waves," Phys. Rev. Lett. 92, 113901 (1-4) (2004).
[Crossref] [PubMed]

F. Scheffold and G. Maret, "Universal conductance fluctuations of light," Phys. Rev. Lett. 81, 5800-5803 (1998).
[Crossref]

S. E. Skipetrov and B. A. van Tiggelen, "Dynamics of Anderson localization in open 3D media," Phys. Rev. Lett. 96, 043902 (1-4) (2006).
[Crossref] [PubMed]

S. John, "Electromagnetic absorption in a disordered medium near a photon mobility edge," Phys. Rev. Lett. 53, 2169-2172 (1984).
[Crossref]

J. M. Drake and A. Z. Genack, "Observation of nonclassical optical diffusion," Phys. Rev. Lett. 63, 259-262 (1989).
[Crossref] [PubMed]

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, "Speed of propagation of classical waves in strongly scattering media," Phys. Rev. Lett. 66, 3132-3135 (1991).
[Crossref] [PubMed]

G. H. Watson, P. A. Fleury, and S. L. McCall, "Searching for photon localization in the time domain," Phys. Rev. Lett. 58, 945-948 (1987).
[Crossref] [PubMed]

Prog. Semicond. (1)

A. F. Ioffe and A. R. Regel, "Non-crystalline, amorphous and liquid electronic semiconductors," Prog. Semicond. 4, 237-291 (1960).

Rev. Sci. Instrum. (1)

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. M. Aegerter, "A precise method to determine the angular distribution of backscattered light to high angles," Rev. Sci. Instrum. 78, 033105 (1-6) (2007).
[Crossref] [PubMed]

Other (3)

S. E. Skipetrov, sergey.skipetrov@grenoble.cnrs.fr (personal communication, December 2005).

D. S. Wiersma, "Light in strongly scattering and amplifying random media," Ph.D thesis (University of Amsterdam, 1995).

P. Gross, "Coherent backscattering close to the transition to strong localization of light," Diploma Thesis, University of Konstanz (2005).

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Figures (5)

Fig. 1
Fig. 1

Time resolved transmission through a localizing sample. The onset of localization can be seen from the nonexponential tail in the distribution at long times giving rise to deviations from classical diffusion (dotted curve). This tail is not due to fluorescence of the sample, as can be checked by placing a filter behind the sample that suppresses photons of wavelength higher than that of the illuminating laser. Moreover, the tail is well described by a time-dependent diffusion coefficient, decreasing with time as 1 t , as required for localization (dashed curve).

Fig. 2
Fig. 2

The inverse absorption length, L a , does not show a systematic dependence on k l * , in contrast to the localization length. This shows that the deviations from classical diffusion are not associated with absorption. However, the fits to the time-resolved transmission data make an explicit determination of the absorption length possible, such that the exponential decay of the total transmission can be compared with that predicted solely from absorption.

Fig. 3
Fig. 3

Inverse localization length as a function of the control parameter k l * (open symbols) [24]. This physical measure is in good agreement with an empirical measure of the deviation of the data from classical diffusion, which can be obtained from averaging the ratio of the data to a classical fit over a range from t m a x to 3 t m a x (solid symbols) [17].

Fig. 4
Fig. 4

The time dependence of the width of the photon cloud can be described by an exponent, α, which is unity for classical diffusion and zero in the case of localization [23]. Just at the transition, a value of α = 2 3 is predicted from one-parameter scaling theory [36]. From a fit to the time-resolved transmission measurements for different k l * , the transition to localization can be seen to take place at a value of k l * 4 .

Fig. 5
Fig. 5

Static transmission of both a classical [(a), k l * = 25] and a localizing [(b), k l * = 2.5] sample as a function of thickness. While in both cases the decrease of the intensity is exponential and hence cannot be explained by pure classical diffusion (dotted curves), an unknown absorption could still explain the data. For k l * = 25 this is indeed the case, and including the measured absorption length from time-resolved measurements yields the dashed curves, which is in good agreement with the data. For the localizing sample however, this still contradicts the data. Only a description including the measured localization length as well gives an excellent fit to the data without a single adjustable parameter over 12 orders of magnitude. Note that there are no fitting parameters in any of the curves.

Equations (1)

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T ( t ) ( D ( t ) D 0 ) 2 exp [ ( π 2 D ( t ) L 2 + 1 τ a b s ) t ] ,

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