Abstract

We show in microwave experiments and random matrix calculations that in samples with a large number of channels the statistics of transmission for different incident channels relative to the average transmission is determined by a single parameter, the participation number of the eigenvalues of the transmission matrix, M. Its inverse, M-1, is equal to the variance of relative total transmission of the sample, while the contrast in maximal focusing is equal to M. The distribution of relative total transmission changes from Gaussian to negative exponential over the range in which M-1 changes from 0 to 1. This provides a framework for transmission and imaging in single samples.

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  1. P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B22(8), 3519–3526 (1980).
    [CrossRef]
  2. V. I. Melnikov, “Distribution of resistivity probabilities of a finite, disordered system,” Pis'ma Z. Eksp. Teor. Fiz.32, 244–247 (1980).
  3. A. A. Abrikosov, “The paradox with the static conductivity of a one-dimensional metal,” Solid State Commun.37(12), 997–1000 (1981).
    [CrossRef]
  4. R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, “Observation of h/e Aharonov-Bohm oscillations in normal-metal rings,” Phys. Rev. Lett.54(25), 2696–2699 (1985).
    [CrossRef] [PubMed]
  5. B. L. Altshuler, P. A. Lee, and R. A. Webb, Mesoscopic Phenomena in Solids (Elsevier, Amsterdam, 1991).
  6. E. Abrahams, 50 years of Anderson Localization (World Scientific Publishing Co. Pte. Ltd., 2010).
  7. A. Z. Genack, “Optical transmission in disordered media,” Phys. Rev. Lett.58(20), 2043–2046 (1987).
    [CrossRef] [PubMed]
  8. G. Maret and P. E. Wolf, “Multiple light scattering from disordered media: the effect of Brownian motion of scatterers,” Zeitschrift für Physik B65(4), 409–413 (1987).
    [CrossRef]
  9. P. A. Mello, E. Akkermans, and B. Shapiro, “Macroscopic approach to correlations in the electronic transmission and reflection from disordered conductors,” Phys. Rev. Lett.61(4), 459–462 (1988).
    [CrossRef] [PubMed]
  10. S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett.61(7), 834–837 (1988).
    [CrossRef] [PubMed]
  11. J. de Boer, M. van Rossum, M. van Albada, T. Nieuwenhuizen, and A. Lagendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett.73(19), 2567–2570 (1994).
    [CrossRef] [PubMed]
  12. M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys.71(1), 313–371 (1999).
    [CrossRef]
  13. A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature404(6780), 850–853 (2000).
    [CrossRef] [PubMed]
  14. H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4(12), 945–948 (2008).
    [CrossRef]
  15. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007).
    [CrossRef] [PubMed]
  16. A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson Localization,” Phys. Today62(8), 24–29 (2009).
    [CrossRef]
  17. A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling wave in space and time for imaging and focusing in complex media,” Nat. Photonics6(5), 283–292 (2012).
    [CrossRef]
  18. I. M. Vellekoop and A. P. Mosk, “Universal Optimal Transmission of Light Through Disordered Materials,” Phys. Rev. Lett.101(12), 120601 (2008).
    [CrossRef] [PubMed]
  19. S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
    [CrossRef] [PubMed]
  20. I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (2010).
    [CrossRef]
  21. Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
    [CrossRef] [PubMed]
  22. S. Tripathi, R. Paxman, T. Bifano, and K. C. Toussaint., “Vector transmission matrix for the polarization behavior of light propagation in highly scattering media,” Opt. Express20(14), 16067–16076 (2012).
    [CrossRef] [PubMed]
  23. M. Davy, Z. Shi, and A. Z. Genack, “Focusing through random media: eigenchannel participation number and intensity correlation,” Phys. Rev. B85(3), 035105 (2012).
    [CrossRef]
  24. O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattering of incoherent light,” Nat. Photonics6(8), 549–553 (2012).
    [CrossRef]
  25. E. Abrahams, P. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimension,” Phys. Rev. Lett.42(10), 673–676 (1979).
    [CrossRef]
  26. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  27. Y. Imry and R. Landauer, “Conductance viewed as transmission,” Rev. Mod. Phys.71(2), S306–S312 (1999).
    [CrossRef]
  28. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109(5), 1492–1505 (1958).
    [CrossRef]
  29. O. N. Dorokhov, “On the coexistence of localized and extended electronic states in the metallic phase,” Solid State Commun.51(6), 381–384 (1984).
    [CrossRef]
  30. P. A. Mello, P. Pereyra, and N. Kumar, “Macroscopic approach to multichannel disordered conductors,” Ann. Phys.181(2), 290–317 (1988).
    [CrossRef]
  31. A. D. Stone, P. Mello, K. A. Muttalib, and J.-L. Pichard, “Random Matrix Theory and Maximum Entropy Models for Disordered Conductors,” in Mesoscopic phenomena in solids, B. L. Altshuler, P. A. Lee, and R. A. Webb, eds. (North-Holland, 1991), pp. 369–448.
  32. D. J. Thouless, “Metallic resistance in thin wires,” Phys. Rev. Lett.39(18), 1167–1169 (1977).
    [CrossRef]
  33. Z. Shi and A. Z. Genack, “Transmission eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett.108(4), 043901 (2012).
    [CrossRef] [PubMed]
  34. S. Zhang, Y. Lockerman, and A. Z. Genack, “Mesoscopic speckle,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82(5), 051114 (2010).
    [CrossRef] [PubMed]
  35. Y. Imry, “Active transmission channels and universal conductance fluctuations,” Europhys. Lett.1(5), 249–256 (1986).
    [CrossRef]

2012 (6)

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling wave in space and time for imaging and focusing in complex media,” Nat. Photonics6(5), 283–292 (2012).
[CrossRef]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

M. Davy, Z. Shi, and A. Z. Genack, “Focusing through random media: eigenchannel participation number and intensity correlation,” Phys. Rev. B85(3), 035105 (2012).
[CrossRef]

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattering of incoherent light,” Nat. Photonics6(8), 549–553 (2012).
[CrossRef]

Z. Shi and A. Z. Genack, “Transmission eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett.108(4), 043901 (2012).
[CrossRef] [PubMed]

S. Tripathi, R. Paxman, T. Bifano, and K. C. Toussaint., “Vector transmission matrix for the polarization behavior of light propagation in highly scattering media,” Opt. Express20(14), 16067–16076 (2012).
[CrossRef] [PubMed]

2010 (3)

S. Zhang, Y. Lockerman, and A. Z. Genack, “Mesoscopic speckle,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82(5), 051114 (2010).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (2010).
[CrossRef]

2009 (1)

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson Localization,” Phys. Today62(8), 24–29 (2009).
[CrossRef]

2008 (2)

I. M. Vellekoop and A. P. Mosk, “Universal Optimal Transmission of Light Through Disordered Materials,” Phys. Rev. Lett.101(12), 120601 (2008).
[CrossRef] [PubMed]

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4(12), 945–948 (2008).
[CrossRef]

2007 (1)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007).
[CrossRef] [PubMed]

2000 (1)

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature404(6780), 850–853 (2000).
[CrossRef] [PubMed]

1999 (2)

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys.71(1), 313–371 (1999).
[CrossRef]

Y. Imry and R. Landauer, “Conductance viewed as transmission,” Rev. Mod. Phys.71(2), S306–S312 (1999).
[CrossRef]

1994 (1)

J. de Boer, M. van Rossum, M. van Albada, T. Nieuwenhuizen, and A. Lagendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett.73(19), 2567–2570 (1994).
[CrossRef] [PubMed]

1988 (3)

P. A. Mello, E. Akkermans, and B. Shapiro, “Macroscopic approach to correlations in the electronic transmission and reflection from disordered conductors,” Phys. Rev. Lett.61(4), 459–462 (1988).
[CrossRef] [PubMed]

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett.61(7), 834–837 (1988).
[CrossRef] [PubMed]

P. A. Mello, P. Pereyra, and N. Kumar, “Macroscopic approach to multichannel disordered conductors,” Ann. Phys.181(2), 290–317 (1988).
[CrossRef]

1987 (2)

A. Z. Genack, “Optical transmission in disordered media,” Phys. Rev. Lett.58(20), 2043–2046 (1987).
[CrossRef] [PubMed]

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media: the effect of Brownian motion of scatterers,” Zeitschrift für Physik B65(4), 409–413 (1987).
[CrossRef]

1986 (1)

Y. Imry, “Active transmission channels and universal conductance fluctuations,” Europhys. Lett.1(5), 249–256 (1986).
[CrossRef]

1985 (1)

R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, “Observation of h/e Aharonov-Bohm oscillations in normal-metal rings,” Phys. Rev. Lett.54(25), 2696–2699 (1985).
[CrossRef] [PubMed]

1984 (1)

O. N. Dorokhov, “On the coexistence of localized and extended electronic states in the metallic phase,” Solid State Commun.51(6), 381–384 (1984).
[CrossRef]

1981 (1)

A. A. Abrikosov, “The paradox with the static conductivity of a one-dimensional metal,” Solid State Commun.37(12), 997–1000 (1981).
[CrossRef]

1980 (2)

P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B22(8), 3519–3526 (1980).
[CrossRef]

V. I. Melnikov, “Distribution of resistivity probabilities of a finite, disordered system,” Pis'ma Z. Eksp. Teor. Fiz.32, 244–247 (1980).

1979 (1)

E. Abrahams, P. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimension,” Phys. Rev. Lett.42(10), 673–676 (1979).
[CrossRef]

1977 (1)

D. J. Thouless, “Metallic resistance in thin wires,” Phys. Rev. Lett.39(18), 1167–1169 (1977).
[CrossRef]

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109(5), 1492–1505 (1958).
[CrossRef]

Abrahams, E.

P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B22(8), 3519–3526 (1980).
[CrossRef]

E. Abrahams, P. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimension,” Phys. Rev. Lett.42(10), 673–676 (1979).
[CrossRef]

Abrikosov, A. A.

A. A. Abrikosov, “The paradox with the static conductivity of a one-dimensional metal,” Solid State Commun.37(12), 997–1000 (1981).
[CrossRef]

Akkermans, E.

P. A. Mello, E. Akkermans, and B. Shapiro, “Macroscopic approach to correlations in the electronic transmission and reflection from disordered conductors,” Phys. Rev. Lett.61(4), 459–462 (1988).
[CrossRef] [PubMed]

Anderson, P.

E. Abrahams, P. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimension,” Phys. Rev. Lett.42(10), 673–676 (1979).
[CrossRef]

Anderson, P. W.

P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B22(8), 3519–3526 (1980).
[CrossRef]

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109(5), 1492–1505 (1958).
[CrossRef]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Bifano, T.

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

Chabanov, A. A.

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature404(6780), 850–853 (2000).
[CrossRef] [PubMed]

Choi, W.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Choi, Y.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Dasari, R. R.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Davy, M.

M. Davy, Z. Shi, and A. Z. Genack, “Focusing through random media: eigenchannel participation number and intensity correlation,” Phys. Rev. B85(3), 035105 (2012).
[CrossRef]

de Boer, J.

J. de Boer, M. van Rossum, M. van Albada, T. Nieuwenhuizen, and A. Lagendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett.73(19), 2567–2570 (1994).
[CrossRef] [PubMed]

Dorokhov, O. N.

O. N. Dorokhov, “On the coexistence of localized and extended electronic states in the metallic phase,” Solid State Commun.51(6), 381–384 (1984).
[CrossRef]

Fang-Yen, C.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Feng, S.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett.61(7), 834–837 (1988).
[CrossRef] [PubMed]

Fink, M.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling wave in space and time for imaging and focusing in complex media,” Nat. Photonics6(5), 283–292 (2012).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

Fisher, D. S.

P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B22(8), 3519–3526 (1980).
[CrossRef]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Genack, A. Z.

Z. Shi and A. Z. Genack, “Transmission eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett.108(4), 043901 (2012).
[CrossRef] [PubMed]

M. Davy, Z. Shi, and A. Z. Genack, “Focusing through random media: eigenchannel participation number and intensity correlation,” Phys. Rev. B85(3), 035105 (2012).
[CrossRef]

S. Zhang, Y. Lockerman, and A. Z. Genack, “Mesoscopic speckle,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82(5), 051114 (2010).
[CrossRef] [PubMed]

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature404(6780), 850–853 (2000).
[CrossRef] [PubMed]

A. Z. Genack, “Optical transmission in disordered media,” Phys. Rev. Lett.58(20), 2043–2046 (1987).
[CrossRef] [PubMed]

Gigan, S.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

Hu, H.

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4(12), 945–948 (2008).
[CrossRef]

Imry, Y.

Y. Imry and R. Landauer, “Conductance viewed as transmission,” Rev. Mod. Phys.71(2), S306–S312 (1999).
[CrossRef]

Y. Imry, “Active transmission channels and universal conductance fluctuations,” Europhys. Lett.1(5), 249–256 (1986).
[CrossRef]

Kane, C.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett.61(7), 834–837 (1988).
[CrossRef] [PubMed]

Katz, O.

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattering of incoherent light,” Nat. Photonics6(8), 549–553 (2012).
[CrossRef]

Kim, M.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Kumar, N.

P. A. Mello, P. Pereyra, and N. Kumar, “Macroscopic approach to multichannel disordered conductors,” Ann. Phys.181(2), 290–317 (1988).
[CrossRef]

Lagendijk, A.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling wave in space and time for imaging and focusing in complex media,” Nat. Photonics6(5), 283–292 (2012).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (2010).
[CrossRef]

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson Localization,” Phys. Today62(8), 24–29 (2009).
[CrossRef]

J. de Boer, M. van Rossum, M. van Albada, T. Nieuwenhuizen, and A. Lagendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett.73(19), 2567–2570 (1994).
[CrossRef] [PubMed]

Laibowitz, R. B.

R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, “Observation of h/e Aharonov-Bohm oscillations in normal-metal rings,” Phys. Rev. Lett.54(25), 2696–2699 (1985).
[CrossRef] [PubMed]

Landauer, R.

Y. Imry and R. Landauer, “Conductance viewed as transmission,” Rev. Mod. Phys.71(2), S306–S312 (1999).
[CrossRef]

Lee, K. J.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Lee, P. A.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett.61(7), 834–837 (1988).
[CrossRef] [PubMed]

Lerosey, G.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling wave in space and time for imaging and focusing in complex media,” Nat. Photonics6(5), 283–292 (2012).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

Licciardello, D. C.

E. Abrahams, P. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimension,” Phys. Rev. Lett.42(10), 673–676 (1979).
[CrossRef]

Lockerman, Y.

S. Zhang, Y. Lockerman, and A. Z. Genack, “Mesoscopic speckle,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82(5), 051114 (2010).
[CrossRef] [PubMed]

Maret, G.

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media: the effect of Brownian motion of scatterers,” Zeitschrift für Physik B65(4), 409–413 (1987).
[CrossRef]

Mello, P. A.

P. A. Mello, E. Akkermans, and B. Shapiro, “Macroscopic approach to correlations in the electronic transmission and reflection from disordered conductors,” Phys. Rev. Lett.61(4), 459–462 (1988).
[CrossRef] [PubMed]

P. A. Mello, P. Pereyra, and N. Kumar, “Macroscopic approach to multichannel disordered conductors,” Ann. Phys.181(2), 290–317 (1988).
[CrossRef]

Melnikov, V. I.

V. I. Melnikov, “Distribution of resistivity probabilities of a finite, disordered system,” Pis'ma Z. Eksp. Teor. Fiz.32, 244–247 (1980).

Mosk, A. P.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling wave in space and time for imaging and focusing in complex media,” Nat. Photonics6(5), 283–292 (2012).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (2010).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, “Universal Optimal Transmission of Light Through Disordered Materials,” Phys. Rev. Lett.101(12), 120601 (2008).
[CrossRef] [PubMed]

Nieuwenhuizen, T.

J. de Boer, M. van Rossum, M. van Albada, T. Nieuwenhuizen, and A. Lagendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett.73(19), 2567–2570 (1994).
[CrossRef] [PubMed]

Nieuwenhuizen, T. M.

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys.71(1), 313–371 (1999).
[CrossRef]

Page, J. H.

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4(12), 945–948 (2008).
[CrossRef]

Paxman, R.

Pereyra, P.

P. A. Mello, P. Pereyra, and N. Kumar, “Macroscopic approach to multichannel disordered conductors,” Ann. Phys.181(2), 290–317 (1988).
[CrossRef]

Popoff, S. M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

Ramakrishnan, T. V.

E. Abrahams, P. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimension,” Phys. Rev. Lett.42(10), 673–676 (1979).
[CrossRef]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Segev, M.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Shapiro, B.

P. A. Mello, E. Akkermans, and B. Shapiro, “Macroscopic approach to correlations in the electronic transmission and reflection from disordered conductors,” Phys. Rev. Lett.61(4), 459–462 (1988).
[CrossRef] [PubMed]

Shi, Z.

M. Davy, Z. Shi, and A. Z. Genack, “Focusing through random media: eigenchannel participation number and intensity correlation,” Phys. Rev. B85(3), 035105 (2012).
[CrossRef]

Z. Shi and A. Z. Genack, “Transmission eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett.108(4), 043901 (2012).
[CrossRef] [PubMed]

Silberberg, Y.

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattering of incoherent light,” Nat. Photonics6(8), 549–553 (2012).
[CrossRef]

Skipetrov, S. E.

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4(12), 945–948 (2008).
[CrossRef]

Small, E.

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattering of incoherent light,” Nat. Photonics6(8), 549–553 (2012).
[CrossRef]

Stone, A. D.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett.61(7), 834–837 (1988).
[CrossRef] [PubMed]

Stoytchev, M.

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature404(6780), 850–853 (2000).
[CrossRef] [PubMed]

Strybulevych, A.

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4(12), 945–948 (2008).
[CrossRef]

Thouless, D. J.

P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B22(8), 3519–3526 (1980).
[CrossRef]

D. J. Thouless, “Metallic resistance in thin wires,” Phys. Rev. Lett.39(18), 1167–1169 (1977).
[CrossRef]

Toussaint, K. C.

Tripathi, S.

Umbach, C. P.

R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, “Observation of h/e Aharonov-Bohm oscillations in normal-metal rings,” Phys. Rev. Lett.54(25), 2696–2699 (1985).
[CrossRef] [PubMed]

van Albada, M.

J. de Boer, M. van Rossum, M. van Albada, T. Nieuwenhuizen, and A. Lagendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett.73(19), 2567–2570 (1994).
[CrossRef] [PubMed]

van Rossum, M.

J. de Boer, M. van Rossum, M. van Albada, T. Nieuwenhuizen, and A. Lagendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett.73(19), 2567–2570 (1994).
[CrossRef] [PubMed]

van Rossum, M. C. W.

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys.71(1), 313–371 (1999).
[CrossRef]

van Tiggelen, B.

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson Localization,” Phys. Today62(8), 24–29 (2009).
[CrossRef]

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4(12), 945–948 (2008).
[CrossRef]

Vellekoop, I. M.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (2010).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, “Universal Optimal Transmission of Light Through Disordered Materials,” Phys. Rev. Lett.101(12), 120601 (2008).
[CrossRef] [PubMed]

Washburn, S.

R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, “Observation of h/e Aharonov-Bohm oscillations in normal-metal rings,” Phys. Rev. Lett.54(25), 2696–2699 (1985).
[CrossRef] [PubMed]

Webb, R. A.

R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, “Observation of h/e Aharonov-Bohm oscillations in normal-metal rings,” Phys. Rev. Lett.54(25), 2696–2699 (1985).
[CrossRef] [PubMed]

Wiersma, D. S.

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson Localization,” Phys. Today62(8), 24–29 (2009).
[CrossRef]

Wolf, P. E.

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media: the effect of Brownian motion of scatterers,” Zeitschrift für Physik B65(4), 409–413 (1987).
[CrossRef]

Yang, T. D.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Yoon, C.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Zhang, S.

S. Zhang, Y. Lockerman, and A. Z. Genack, “Mesoscopic speckle,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82(5), 051114 (2010).
[CrossRef] [PubMed]

Ann. Phys. (1)

P. A. Mello, P. Pereyra, and N. Kumar, “Macroscopic approach to multichannel disordered conductors,” Ann. Phys.181(2), 290–317 (1988).
[CrossRef]

Europhys. Lett. (1)

Y. Imry, “Active transmission channels and universal conductance fluctuations,” Europhys. Lett.1(5), 249–256 (1986).
[CrossRef]

Nat. Photonics (3)

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattering of incoherent light,” Nat. Photonics6(8), 549–553 (2012).
[CrossRef]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling wave in space and time for imaging and focusing in complex media,” Nat. Photonics6(5), 283–292 (2012).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (2010).
[CrossRef]

Nat. Phys. (1)

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4(12), 945–948 (2008).
[CrossRef]

Nature (2)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007).
[CrossRef] [PubMed]

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature404(6780), 850–853 (2000).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rev. (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109(5), 1492–1505 (1958).
[CrossRef]

Phys. Rev. B (2)

M. Davy, Z. Shi, and A. Z. Genack, “Focusing through random media: eigenchannel participation number and intensity correlation,” Phys. Rev. B85(3), 035105 (2012).
[CrossRef]

P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B22(8), 3519–3526 (1980).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

S. Zhang, Y. Lockerman, and A. Z. Genack, “Mesoscopic speckle,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82(5), 051114 (2010).
[CrossRef] [PubMed]

Phys. Rev. Lett. (11)

D. J. Thouless, “Metallic resistance in thin wires,” Phys. Rev. Lett.39(18), 1167–1169 (1977).
[CrossRef]

Z. Shi and A. Z. Genack, “Transmission eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett.108(4), 043901 (2012).
[CrossRef] [PubMed]

E. Abrahams, P. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimension,” Phys. Rev. Lett.42(10), 673–676 (1979).
[CrossRef]

R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, “Observation of h/e Aharonov-Bohm oscillations in normal-metal rings,” Phys. Rev. Lett.54(25), 2696–2699 (1985).
[CrossRef] [PubMed]

P. A. Mello, E. Akkermans, and B. Shapiro, “Macroscopic approach to correlations in the electronic transmission and reflection from disordered conductors,” Phys. Rev. Lett.61(4), 459–462 (1988).
[CrossRef] [PubMed]

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett.61(7), 834–837 (1988).
[CrossRef] [PubMed]

J. de Boer, M. van Rossum, M. van Albada, T. Nieuwenhuizen, and A. Lagendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett.73(19), 2567–2570 (1994).
[CrossRef] [PubMed]

A. Z. Genack, “Optical transmission in disordered media,” Phys. Rev. Lett.58(20), 2043–2046 (1987).
[CrossRef] [PubMed]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

I. M. Vellekoop and A. P. Mosk, “Universal Optimal Transmission of Light Through Disordered Materials,” Phys. Rev. Lett.101(12), 120601 (2008).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

Phys. Today (1)

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson Localization,” Phys. Today62(8), 24–29 (2009).
[CrossRef]

Pis'ma Z. Eksp. Teor. Fiz. (1)

V. I. Melnikov, “Distribution of resistivity probabilities of a finite, disordered system,” Pis'ma Z. Eksp. Teor. Fiz.32, 244–247 (1980).

Rev. Mod. Phys. (2)

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys.71(1), 313–371 (1999).
[CrossRef]

Y. Imry and R. Landauer, “Conductance viewed as transmission,” Rev. Mod. Phys.71(2), S306–S312 (1999).
[CrossRef]

Solid State Commun. (2)

O. N. Dorokhov, “On the coexistence of localized and extended electronic states in the metallic phase,” Solid State Commun.51(6), 381–384 (1984).
[CrossRef]

A. A. Abrikosov, “The paradox with the static conductivity of a one-dimensional metal,” Solid State Commun.37(12), 997–1000 (1981).
[CrossRef]

Zeitschrift für Physik B (1)

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media: the effect of Brownian motion of scatterers,” Zeitschrift für Physik B65(4), 409–413 (1987).
[CrossRef]

Other (4)

B. L. Altshuler, P. A. Lee, and R. A. Webb, Mesoscopic Phenomena in Solids (Elsevier, Amsterdam, 1991).

E. Abrahams, 50 years of Anderson Localization (World Scientific Publishing Co. Pte. Ltd., 2010).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

A. D. Stone, P. Mello, K. A. Muttalib, and J.-L. Pichard, “Random Matrix Theory and Maximum Entropy Models for Disordered Conductors,” in Mesoscopic phenomena in solids, B. L. Altshuler, P. A. Lee, and R. A. Webb, eds. (North-Holland, 1991), pp. 369–448.

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Figures (4)

Fig. 1
Fig. 1

Intensity normalized to the peak value in each speckle pattern generated by sources at positions a are represented in the columns with index of detector position b for (a) diffusive and (b) localized waves. (c,d) The transmission eigenvalues are plotted under the corresponding intensity patterns. For localized waves (d), the determination of the third eigenvalue and higher eigenvalues are influenced by the noise level of the measurements. Correlation between speckle patterns for different source positions are clearly seen in (b) due to the small numbers of eigenchannels M contributing appreciably to transmission. (e,f) Distributions of relative intensity P(N2Tba/T) and relative total transmission P(NTa/T) for the two transmission matrices selected in this figure with M−1 = 0.17 (green triangles) and M−1 = 0.99 (red circles).

Fig. 2
Fig. 2

Plot of the var(NTa/T) computed within transmission matrices over a subset of transmission matrices drawn from random ensembles with different values of g with specified value of M−1. The straight line is a plot of var(NTa/T) = M−1. In the inset, the variance of V/M−1 is plotted vs. M−1, where V = var(NTa/T).

Fig. 3
Fig. 3

(a) P(NTa/T) for subsets of transmission matrices with M−1 = 0.17 ± 0.01 drawn from ensembles of samples with L = 61 cm in two frequency ranges in which the wave is diffusive (green circles) and localized (red filled circles). The curve is the theoretical probability distribution of P(NTa/<T>) in which var(NTa/T) is replaced by M−1 in the expression for P(NTa/T) in Ref. 12. (b) P(NTa/T) for M−1 in the range 0.995 ± 0.005 computed for localized waves in samples of two lengths: L = 40 cm (black circles) and L = 61 cm (red filled circles). The straight line represents the exponential distribution, exp(-NTa/T). (c,d) The corresponding intensity distributions P(N2Tba/T) are plotted under (a) and (b).

Fig. 4
Fig. 4

Contrast in maximal focusing vs. eigenchannel participation number M. The open circles and squares represent measurements from transmission matrices N = 30 and 66 channels, respectively. The filled triangles give results for N'×N' matrices with N'=30 for points selected from a larger matrix with size N = 66. Phase conjugation is applied within the reduced matrix to achieve maximal focusing. Equation (3) is represented by the solid red and dashed blue curves for N = 30 and 66, respectively. In the limit of N>>M, the contrast given by Eq. (3) is equal to M, which is shown in long-dashed black line.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

n=1 N ( τ n /T ) 2 N 2 | v na | 4 a + n'n N τ n τ n' / T 2 N 2 | v na | 2 | v n'a | 2 a .
( N T a T ) 2 a = n=1 N τ n 2 + ( n=1 N τ n ) 2 ( n=1 N τ n ) 2 .
var(N T a /T)= n=1 N τ n 2 / ( n=1 N τ n ) 2 M 1 .
C ba,b'a' M = [ T ba T b'a' (T/ N 2 ) 2 ]/ (T/ N 2 ) 2 M .
C ba,b'a' M = δ aa' δ bb' + M 1 ( δ aa' + δ bb' ).
I bβ = | n=1 N τ n u nb u nβ * | 2 T β = n,n'=1 N τ n τ n' u nb u nβ * u n'b * u n'β n=1 N τ n | u nβ | 2 .
I b bβ = 1 N1 b N n,n'=1 N τ n τ n' u nb u nβ * u n'b * u n'β n=1 N τ n | u nβ | 2 1 N1 ( n=1 N τ n | u nβ | 2 ) 2 n=1 N τ n | u nβ | 2 .
I b bβ = 1 N1 n=1 N τ n 2 | u nβ | 2 n=1 N τ n | u nβ | 2 1 N1 n=1 N τ n | u nβ | 2 .
μ= 1 1/M1/N .

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