Abstract

We measure the distribution of intensity of microwave radiation transmitted through absorbing random waveguides of lengths L up to localization length ξ. For large intensity values the distribution is given by a negative stretched exponential to the 1/2 power, in agreement with predictions by Nieuwenhuizen and van Rossum[Phys. Rev. Lett. 74, 2674 (1995)] for diffusing waves in nonabsorbing samples, as opposed to a negative exponential given by Rayleigh statistics. The intensity distribution is well described by a transform derived by Kogan and Kaveh [Phys.  Rev.  B 52, R3813 (1995)] of the measured distribution of total transmission.

© 1999 Optical Society of America

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  1. J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984), pp. 9–75.
  2. Rayleigh, Proc. London Math. Soc. 3, 267 (1871).
  3. B. Shapiro, Phys. Rev. Lett. 57, 2168 (1986).
    [CrossRef] [PubMed]
  4. P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. (Paris) 49, 63 (1988).
    [CrossRef]
  5. N. Garcia and A. Z. Genack, Phys. Rev. Lett. 63, 1678 (1989).
    [CrossRef] [PubMed]
  6. G. Maret and E. Wolf, Z. Phys. B 65, 409 (1987).
    [CrossRef]
  7. A. Z. Genack, Phys. Rev. Lett. 58, 2043 (1987).
    [CrossRef] [PubMed]
  8. M. J. Stephen and G. Cwilich, Phys. Rev. Lett. 59, 285 (1987).
    [CrossRef] [PubMed]
  9. S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 61, 834 (1988).
    [CrossRef] [PubMed]
  10. P. A. Mello, E. Akkermans, and B. Shapiro, Phys. Rev. Lett. 61, 459 (1988).
    [CrossRef] [PubMed]
  11. A. Z. Genack, N. Garcia, and W. Polkosnik, Phys. Rev. Lett. 65, 2129 (1990).
    [CrossRef] [PubMed]
  12. M. P. van Albada, J. F. de Boer, and A. Lagendijk, Phys. Rev. Lett. 64, 2787 (1990).
    [CrossRef] [PubMed]
  13. P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).
    [CrossRef] [PubMed]
  14. B. L. Altshuler and D. E. Khmelnitskii, JETP Lett. 42, 359 (1985).
  15. Th. M. Nieuwenhuizen and M. C. W. van Rossum, Phys. Rev. Lett. 74, 2674 (1995).
    [CrossRef] [PubMed]
  16. E. Kogan and M. Kaveh, Phys. Rev. B 52, R3813 (1995).
    [CrossRef]
  17. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
    [CrossRef]
  18. D. J. Thouless, Phys. Rev. Lett. 39, 1167 (1977).
    [CrossRef]
  19. N. Garcia, A. Z. Genack, R. Pnini, and B. Shapiro, Phys. Lett. A 176, 458 (1993).
    [CrossRef]
  20. A. Z. Genack and N. Garcia, Europhys. Lett. 21, 753 (1993).
    [CrossRef]
  21. N. Shnerb and M. Kaveh, Phys. Rev. B 43, 1279 (1991).
    [CrossRef]
  22. E. Kogan, M. Kaveh, R. Baumgartner, and R. Berkovits, Phys. Rev. B 48, 9404 (1993).
    [CrossRef]
  23. A. D. Stone, P. A. Mello, K. A. Muttalib, and J. L. Pichard, in Mesoscopic Phenomena in Solids, B. L. Altshuler, P. A. Lee, and R. A. Webb, eds. (Elsevier, New York, 1991), pp. 369–448C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).
    [CrossRef]
  24. A. Z. Genack, in The Scattering and Localization of Classical Waves, P. Sheng, ed. (World Scientific, Singapore, 1990), p. 264.
  25. M. Stoytchev and A. Z. Genack, Phys. Rev. Lett. 79, 309 (1997).
    [CrossRef]
  26. P. W. Brouwer, Phys. Rev. B 57, 10526 (1998).
    [CrossRef]
  27. A. Z. Genack, J. H. Li, N. Garcia, and A. A. Lisyansky, in Photonic Band Gaps and Localization, C. M. Soukoulis, ed. (Plenum, New York, 1993), pp. 23–55.
    [CrossRef]

1998 (1)

P. W. Brouwer, Phys. Rev. B 57, 10526 (1998).
[CrossRef]

1997 (1)

M. Stoytchev and A. Z. Genack, Phys. Rev. Lett. 79, 309 (1997).
[CrossRef]

1995 (2)

Th. M. Nieuwenhuizen and M. C. W. van Rossum, Phys. Rev. Lett. 74, 2674 (1995).
[CrossRef] [PubMed]

E. Kogan and M. Kaveh, Phys. Rev. B 52, R3813 (1995).
[CrossRef]

1993 (3)

N. Garcia, A. Z. Genack, R. Pnini, and B. Shapiro, Phys. Lett. A 176, 458 (1993).
[CrossRef]

A. Z. Genack and N. Garcia, Europhys. Lett. 21, 753 (1993).
[CrossRef]

E. Kogan, M. Kaveh, R. Baumgartner, and R. Berkovits, Phys. Rev. B 48, 9404 (1993).
[CrossRef]

1991 (1)

N. Shnerb and M. Kaveh, Phys. Rev. B 43, 1279 (1991).
[CrossRef]

1990 (2)

A. Z. Genack, N. Garcia, and W. Polkosnik, Phys. Rev. Lett. 65, 2129 (1990).
[CrossRef] [PubMed]

M. P. van Albada, J. F. de Boer, and A. Lagendijk, Phys. Rev. Lett. 64, 2787 (1990).
[CrossRef] [PubMed]

1989 (1)

N. Garcia and A. Z. Genack, Phys. Rev. Lett. 63, 1678 (1989).
[CrossRef] [PubMed]

1988 (3)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 61, 834 (1988).
[CrossRef] [PubMed]

P. A. Mello, E. Akkermans, and B. Shapiro, Phys. Rev. Lett. 61, 459 (1988).
[CrossRef] [PubMed]

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. (Paris) 49, 63 (1988).
[CrossRef]

1987 (3)

G. Maret and E. Wolf, Z. Phys. B 65, 409 (1987).
[CrossRef]

A. Z. Genack, Phys. Rev. Lett. 58, 2043 (1987).
[CrossRef] [PubMed]

M. J. Stephen and G. Cwilich, Phys. Rev. Lett. 59, 285 (1987).
[CrossRef] [PubMed]

1986 (1)

B. Shapiro, Phys. Rev. Lett. 57, 2168 (1986).
[CrossRef] [PubMed]

1985 (2)

P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).
[CrossRef] [PubMed]

B. L. Altshuler and D. E. Khmelnitskii, JETP Lett. 42, 359 (1985).

1979 (1)

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
[CrossRef]

1977 (1)

D. J. Thouless, Phys. Rev. Lett. 39, 1167 (1977).
[CrossRef]

1871 (1)

Rayleigh, Proc. London Math. Soc. 3, 267 (1871).

Abrahams, E.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
[CrossRef]

Akkermans, E.

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. (Paris) 49, 63 (1988).
[CrossRef]

P. A. Mello, E. Akkermans, and B. Shapiro, Phys. Rev. Lett. 61, 459 (1988).
[CrossRef] [PubMed]

Altshuler, B. L.

B. L. Altshuler and D. E. Khmelnitskii, JETP Lett. 42, 359 (1985).

Anderson, P. W.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
[CrossRef]

Baumgartner, R.

E. Kogan, M. Kaveh, R. Baumgartner, and R. Berkovits, Phys. Rev. B 48, 9404 (1993).
[CrossRef]

Berkovits, R.

E. Kogan, M. Kaveh, R. Baumgartner, and R. Berkovits, Phys. Rev. B 48, 9404 (1993).
[CrossRef]

Brouwer, P. W.

P. W. Brouwer, Phys. Rev. B 57, 10526 (1998).
[CrossRef]

Cwilich, G.

M. J. Stephen and G. Cwilich, Phys. Rev. Lett. 59, 285 (1987).
[CrossRef] [PubMed]

de Boer, J. F.

M. P. van Albada, J. F. de Boer, and A. Lagendijk, Phys. Rev. Lett. 64, 2787 (1990).
[CrossRef] [PubMed]

Feng, S.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 61, 834 (1988).
[CrossRef] [PubMed]

Garcia, N.

A. Z. Genack and N. Garcia, Europhys. Lett. 21, 753 (1993).
[CrossRef]

N. Garcia, A. Z. Genack, R. Pnini, and B. Shapiro, Phys. Lett. A 176, 458 (1993).
[CrossRef]

A. Z. Genack, N. Garcia, and W. Polkosnik, Phys. Rev. Lett. 65, 2129 (1990).
[CrossRef] [PubMed]

N. Garcia and A. Z. Genack, Phys. Rev. Lett. 63, 1678 (1989).
[CrossRef] [PubMed]

A. Z. Genack, J. H. Li, N. Garcia, and A. A. Lisyansky, in Photonic Band Gaps and Localization, C. M. Soukoulis, ed. (Plenum, New York, 1993), pp. 23–55.
[CrossRef]

Genack, A. Z.

M. Stoytchev and A. Z. Genack, Phys. Rev. Lett. 79, 309 (1997).
[CrossRef]

N. Garcia, A. Z. Genack, R. Pnini, and B. Shapiro, Phys. Lett. A 176, 458 (1993).
[CrossRef]

A. Z. Genack and N. Garcia, Europhys. Lett. 21, 753 (1993).
[CrossRef]

A. Z. Genack, N. Garcia, and W. Polkosnik, Phys. Rev. Lett. 65, 2129 (1990).
[CrossRef] [PubMed]

N. Garcia and A. Z. Genack, Phys. Rev. Lett. 63, 1678 (1989).
[CrossRef] [PubMed]

A. Z. Genack, Phys. Rev. Lett. 58, 2043 (1987).
[CrossRef] [PubMed]

A. Z. Genack, J. H. Li, N. Garcia, and A. A. Lisyansky, in Photonic Band Gaps and Localization, C. M. Soukoulis, ed. (Plenum, New York, 1993), pp. 23–55.
[CrossRef]

A. Z. Genack, in The Scattering and Localization of Classical Waves, P. Sheng, ed. (World Scientific, Singapore, 1990), p. 264.

Goodman, J. W.

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984), pp. 9–75.

Kane, C.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 61, 834 (1988).
[CrossRef] [PubMed]

Kaveh, M.

E. Kogan and M. Kaveh, Phys. Rev. B 52, R3813 (1995).
[CrossRef]

E. Kogan, M. Kaveh, R. Baumgartner, and R. Berkovits, Phys. Rev. B 48, 9404 (1993).
[CrossRef]

N. Shnerb and M. Kaveh, Phys. Rev. B 43, 1279 (1991).
[CrossRef]

Khmelnitskii, D. E.

B. L. Altshuler and D. E. Khmelnitskii, JETP Lett. 42, 359 (1985).

Kogan, E.

E. Kogan and M. Kaveh, Phys. Rev. B 52, R3813 (1995).
[CrossRef]

E. Kogan, M. Kaveh, R. Baumgartner, and R. Berkovits, Phys. Rev. B 48, 9404 (1993).
[CrossRef]

Lagendijk, A.

M. P. van Albada, J. F. de Boer, and A. Lagendijk, Phys. Rev. Lett. 64, 2787 (1990).
[CrossRef] [PubMed]

Lee, P. A.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 61, 834 (1988).
[CrossRef] [PubMed]

P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).
[CrossRef] [PubMed]

Li, J. H.

A. Z. Genack, J. H. Li, N. Garcia, and A. A. Lisyansky, in Photonic Band Gaps and Localization, C. M. Soukoulis, ed. (Plenum, New York, 1993), pp. 23–55.
[CrossRef]

Licciardello, D. C.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
[CrossRef]

Lisyansky, A. A.

A. Z. Genack, J. H. Li, N. Garcia, and A. A. Lisyansky, in Photonic Band Gaps and Localization, C. M. Soukoulis, ed. (Plenum, New York, 1993), pp. 23–55.
[CrossRef]

Maret, G.

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. (Paris) 49, 63 (1988).
[CrossRef]

G. Maret and E. Wolf, Z. Phys. B 65, 409 (1987).
[CrossRef]

Maynard, R.

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. (Paris) 49, 63 (1988).
[CrossRef]

Mello, P. A.

P. A. Mello, E. Akkermans, and B. Shapiro, Phys. Rev. Lett. 61, 459 (1988).
[CrossRef] [PubMed]

A. D. Stone, P. A. Mello, K. A. Muttalib, and J. L. Pichard, in Mesoscopic Phenomena in Solids, B. L. Altshuler, P. A. Lee, and R. A. Webb, eds. (Elsevier, New York, 1991), pp. 369–448C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).
[CrossRef]

Muttalib, K. A.

A. D. Stone, P. A. Mello, K. A. Muttalib, and J. L. Pichard, in Mesoscopic Phenomena in Solids, B. L. Altshuler, P. A. Lee, and R. A. Webb, eds. (Elsevier, New York, 1991), pp. 369–448C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).
[CrossRef]

Nieuwenhuizen, Th. M.

Th. M. Nieuwenhuizen and M. C. W. van Rossum, Phys. Rev. Lett. 74, 2674 (1995).
[CrossRef] [PubMed]

Pichard, J. L.

A. D. Stone, P. A. Mello, K. A. Muttalib, and J. L. Pichard, in Mesoscopic Phenomena in Solids, B. L. Altshuler, P. A. Lee, and R. A. Webb, eds. (Elsevier, New York, 1991), pp. 369–448C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).
[CrossRef]

Pnini, R.

N. Garcia, A. Z. Genack, R. Pnini, and B. Shapiro, Phys. Lett. A 176, 458 (1993).
[CrossRef]

Polkosnik, W.

A. Z. Genack, N. Garcia, and W. Polkosnik, Phys. Rev. Lett. 65, 2129 (1990).
[CrossRef] [PubMed]

Ramakrishnan, T. V.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
[CrossRef]

Rayleigh,

Rayleigh, Proc. London Math. Soc. 3, 267 (1871).

Shapiro, B.

N. Garcia, A. Z. Genack, R. Pnini, and B. Shapiro, Phys. Lett. A 176, 458 (1993).
[CrossRef]

P. A. Mello, E. Akkermans, and B. Shapiro, Phys. Rev. Lett. 61, 459 (1988).
[CrossRef] [PubMed]

B. Shapiro, Phys. Rev. Lett. 57, 2168 (1986).
[CrossRef] [PubMed]

Shnerb, N.

N. Shnerb and M. Kaveh, Phys. Rev. B 43, 1279 (1991).
[CrossRef]

Stephen, M. J.

M. J. Stephen and G. Cwilich, Phys. Rev. Lett. 59, 285 (1987).
[CrossRef] [PubMed]

Stone, A. D.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 61, 834 (1988).
[CrossRef] [PubMed]

P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).
[CrossRef] [PubMed]

A. D. Stone, P. A. Mello, K. A. Muttalib, and J. L. Pichard, in Mesoscopic Phenomena in Solids, B. L. Altshuler, P. A. Lee, and R. A. Webb, eds. (Elsevier, New York, 1991), pp. 369–448C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).
[CrossRef]

Stoytchev, M.

M. Stoytchev and A. Z. Genack, Phys. Rev. Lett. 79, 309 (1997).
[CrossRef]

Thouless, D. J.

D. J. Thouless, Phys. Rev. Lett. 39, 1167 (1977).
[CrossRef]

van Albada, M. P.

M. P. van Albada, J. F. de Boer, and A. Lagendijk, Phys. Rev. Lett. 64, 2787 (1990).
[CrossRef] [PubMed]

van Rossum, M. C. W.

Th. M. Nieuwenhuizen and M. C. W. van Rossum, Phys. Rev. Lett. 74, 2674 (1995).
[CrossRef] [PubMed]

Wolf, E.

G. Maret and E. Wolf, Z. Phys. B 65, 409 (1987).
[CrossRef]

Wolf, P. E.

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. (Paris) 49, 63 (1988).
[CrossRef]

Europhys. Lett. (1)

A. Z. Genack and N. Garcia, Europhys. Lett. 21, 753 (1993).
[CrossRef]

J. Phys. (Paris) (1)

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. (Paris) 49, 63 (1988).
[CrossRef]

JETP Lett. (1)

B. L. Altshuler and D. E. Khmelnitskii, JETP Lett. 42, 359 (1985).

Phys. Lett. A (1)

N. Garcia, A. Z. Genack, R. Pnini, and B. Shapiro, Phys. Lett. A 176, 458 (1993).
[CrossRef]

Phys. Rev. B (4)

P. W. Brouwer, Phys. Rev. B 57, 10526 (1998).
[CrossRef]

E. Kogan and M. Kaveh, Phys. Rev. B 52, R3813 (1995).
[CrossRef]

N. Shnerb and M. Kaveh, Phys. Rev. B 43, 1279 (1991).
[CrossRef]

E. Kogan, M. Kaveh, R. Baumgartner, and R. Berkovits, Phys. Rev. B 48, 9404 (1993).
[CrossRef]

Phys. Rev. Lett. (13)

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
[CrossRef]

D. J. Thouless, Phys. Rev. Lett. 39, 1167 (1977).
[CrossRef]

Th. M. Nieuwenhuizen and M. C. W. van Rossum, Phys. Rev. Lett. 74, 2674 (1995).
[CrossRef] [PubMed]

N. Garcia and A. Z. Genack, Phys. Rev. Lett. 63, 1678 (1989).
[CrossRef] [PubMed]

A. Z. Genack, Phys. Rev. Lett. 58, 2043 (1987).
[CrossRef] [PubMed]

M. J. Stephen and G. Cwilich, Phys. Rev. Lett. 59, 285 (1987).
[CrossRef] [PubMed]

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 61, 834 (1988).
[CrossRef] [PubMed]

P. A. Mello, E. Akkermans, and B. Shapiro, Phys. Rev. Lett. 61, 459 (1988).
[CrossRef] [PubMed]

A. Z. Genack, N. Garcia, and W. Polkosnik, Phys. Rev. Lett. 65, 2129 (1990).
[CrossRef] [PubMed]

M. P. van Albada, J. F. de Boer, and A. Lagendijk, Phys. Rev. Lett. 64, 2787 (1990).
[CrossRef] [PubMed]

P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).
[CrossRef] [PubMed]

B. Shapiro, Phys. Rev. Lett. 57, 2168 (1986).
[CrossRef] [PubMed]

M. Stoytchev and A. Z. Genack, Phys. Rev. Lett. 79, 309 (1997).
[CrossRef]

Proc. London Math. Soc. (1)

Rayleigh, Proc. London Math. Soc. 3, 267 (1871).

Z. Phys. B (1)

G. Maret and E. Wolf, Z. Phys. B 65, 409 (1987).
[CrossRef]

Other (4)

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984), pp. 9–75.

A. D. Stone, P. A. Mello, K. A. Muttalib, and J. L. Pichard, in Mesoscopic Phenomena in Solids, B. L. Altshuler, P. A. Lee, and R. A. Webb, eds. (Elsevier, New York, 1991), pp. 369–448C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).
[CrossRef]

A. Z. Genack, in The Scattering and Localization of Classical Waves, P. Sheng, ed. (World Scientific, Singapore, 1990), p. 264.

A. Z. Genack, J. H. Li, N. Garcia, and A. A. Lisyansky, in Photonic Band Gaps and Localization, C. M. Soukoulis, ed. (Plenum, New York, 1993), pp. 23–55.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Intensity spectrum for a single sample configuration for the sample with L=520 cm ξ and d=5 cm.

Fig. 2
Fig. 2

Measured intensity distributions for three samples: (a) L=67 cm, d=7.5 cm, L/ξ1/15; (b) L=200 cm, d=5 cm, L/ξ2/5; (c) L=520 cm, d=5 cm, L/ξ1. The Rayleigh distribution is represented by the dotted line.

Fig. 3
Fig. 3

Comparison of experiment and theory for sample (b) from Fig.  2. The smooth solid curve represents the transform of the total transmission distribution measured in Ref.  25. The dotted curve gives the fit to the tail with relation  (1). The value of the fitting parameter of 3.20±0.12 obtained from the tail of Psab is close to the value of g=3.06±0.07 obtained from total transmission measurements for the same sample.25

Equations (2)

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Psabexp-2gsab.
Psab=0dsasaPsaexp-sabsa.

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