Abstract

The interaction between coherent waves and random media is a complicated, deterministic process that is usually examined upon ensemble averaging. The result of one realization of the interaction process depends on the specific disorder present in an experimentally controllable interaction volume. We show that this randomness can be quantified and structural information not apparent in the ensemble average can be obtained. We use the information entropy as a viable measure of randomness and we demonstrate that its rate of change provides means for discriminating between media with identical mean characteristics.

© 2009 Optical Society of America

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References

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  1. P. A. Lee and A. D. Stone, "Universal Conductance Fluctuations in Metals," Phys. Rev. Lett. 55(15), 1622-1625 (1985).
    [CrossRef]
  2. 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]
  3. S. Etemad, R. Thompson, and M. J. Andrejco, "Weak localization of photons: universal fluctuations and ensemble averagign," Phys. Rev. Lett. 57(5), 575-578 (1986).
    [CrossRef]
  4. M. Kaveh, M. Rosenbluh, I. Edrei, and I. Freund, "Weak Localization and Light Scattering from Disordered Solids," Phys. Rev. Lett. 57(16), 2049-2052 (1986).
    [CrossRef]
  5. B. Shapiro, "Large Intensity Fluctuations for Wave Propagation in Random Media," Phys. Rev. Lett. 57(17), 2168-2171 (1986).
    [CrossRef]
  6. M. J. Stephen and G. Cwilich, "Intensity correlation functions and fluctuations in light scattered from a random medium," Phys. Rev. Lett. 59(3), 285-287 (1987).
    [CrossRef]
  7. J. W. Goodman, Speckle Phenomena in Optics, 1st ed. (Roberts & Co., Englewood, 2007).
  8. A. Ishimaru, Wave Propagation and Scattering in Random Media, vol. 1 (Academic, New York, 1971).
  9. C. E. Shannon, "A Mathematical Theory of Communication," Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).
  10. G. Popescu and A. Dogariu, "Scattering of low coherence radiation and applications," Eur. Phys. J. Appl. Phys. 32(2), 73-93 (2005).
    [CrossRef]
  11. G. Popescu and A. Dogariu, "Optical path-length spectroscopy of wave propagation in random media," Opt. Lett. 24(7), 442-444 (1999).
    [CrossRef]
  12. A. H. Gandjbakhche and G. H. Weiss, "Random walk and diffusion-like model of photon migration in turbid media," Prog. in Opt. 34, 333-402 (1995).
    [CrossRef]
  13. A. Apostol, D. Haefner, and A. Dogariu, "Near-field characterization of effective optical interfaces," Phys. Rev. E 74(6), 066603-6 (2006).
    [CrossRef]
  14. E. Hartveit and M. L. Veruki, "Studying properties of neurotransmitter receptors by non-stationary noise analysis of spontaneous postsynaptic currents and agonist-evoked responses in outside-out patches," Nature Protocols 2(2), 434-448 (2007).
    [CrossRef]

2007

E. Hartveit and M. L. Veruki, "Studying properties of neurotransmitter receptors by non-stationary noise analysis of spontaneous postsynaptic currents and agonist-evoked responses in outside-out patches," Nature Protocols 2(2), 434-448 (2007).
[CrossRef]

2006

A. Apostol, D. Haefner, and A. Dogariu, "Near-field characterization of effective optical interfaces," Phys. Rev. E 74(6), 066603-6 (2006).
[CrossRef]

2005

G. Popescu and A. Dogariu, "Scattering of low coherence radiation and applications," Eur. Phys. J. Appl. Phys. 32(2), 73-93 (2005).
[CrossRef]

1999

1995

A. H. Gandjbakhche and G. H. Weiss, "Random walk and diffusion-like model of photon migration in turbid media," Prog. in Opt. 34, 333-402 (1995).
[CrossRef]

1988

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]

1987

M. J. Stephen and G. Cwilich, "Intensity correlation functions and fluctuations in light scattered from a random medium," Phys. Rev. Lett. 59(3), 285-287 (1987).
[CrossRef]

1986

S. Etemad, R. Thompson, and M. J. Andrejco, "Weak localization of photons: universal fluctuations and ensemble averagign," Phys. Rev. Lett. 57(5), 575-578 (1986).
[CrossRef]

M. Kaveh, M. Rosenbluh, I. Edrei, and I. Freund, "Weak Localization and Light Scattering from Disordered Solids," Phys. Rev. Lett. 57(16), 2049-2052 (1986).
[CrossRef]

B. Shapiro, "Large Intensity Fluctuations for Wave Propagation in Random Media," Phys. Rev. Lett. 57(17), 2168-2171 (1986).
[CrossRef]

1985

P. A. Lee and A. D. Stone, "Universal Conductance Fluctuations in Metals," Phys. Rev. Lett. 55(15), 1622-1625 (1985).
[CrossRef]

1948

C. E. Shannon, "A Mathematical Theory of Communication," Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).

Andrejco, M. J.

S. Etemad, R. Thompson, and M. J. Andrejco, "Weak localization of photons: universal fluctuations and ensemble averagign," Phys. Rev. Lett. 57(5), 575-578 (1986).
[CrossRef]

Apostol, A.

A. Apostol, D. Haefner, and A. Dogariu, "Near-field characterization of effective optical interfaces," Phys. Rev. E 74(6), 066603-6 (2006).
[CrossRef]

Cwilich, G.

M. J. Stephen and G. Cwilich, "Intensity correlation functions and fluctuations in light scattered from a random medium," Phys. Rev. Lett. 59(3), 285-287 (1987).
[CrossRef]

Dogariu, A.

A. Apostol, D. Haefner, and A. Dogariu, "Near-field characterization of effective optical interfaces," Phys. Rev. E 74(6), 066603-6 (2006).
[CrossRef]

G. Popescu and A. Dogariu, "Scattering of low coherence radiation and applications," Eur. Phys. J. Appl. Phys. 32(2), 73-93 (2005).
[CrossRef]

G. Popescu and A. Dogariu, "Optical path-length spectroscopy of wave propagation in random media," Opt. Lett. 24(7), 442-444 (1999).
[CrossRef]

Edrei, I.

M. Kaveh, M. Rosenbluh, I. Edrei, and I. Freund, "Weak Localization and Light Scattering from Disordered Solids," Phys. Rev. Lett. 57(16), 2049-2052 (1986).
[CrossRef]

Etemad, S.

S. Etemad, R. Thompson, and M. J. Andrejco, "Weak localization of photons: universal fluctuations and ensemble averagign," Phys. Rev. Lett. 57(5), 575-578 (1986).
[CrossRef]

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]

Freund, I.

M. Kaveh, M. Rosenbluh, I. Edrei, and I. Freund, "Weak Localization and Light Scattering from Disordered Solids," Phys. Rev. Lett. 57(16), 2049-2052 (1986).
[CrossRef]

Gandjbakhche, A. H.

A. H. Gandjbakhche and G. H. Weiss, "Random walk and diffusion-like model of photon migration in turbid media," Prog. in Opt. 34, 333-402 (1995).
[CrossRef]

Haefner, D.

A. Apostol, D. Haefner, and A. Dogariu, "Near-field characterization of effective optical interfaces," Phys. Rev. E 74(6), 066603-6 (2006).
[CrossRef]

Hartveit, E.

E. Hartveit and M. L. Veruki, "Studying properties of neurotransmitter receptors by non-stationary noise analysis of spontaneous postsynaptic currents and agonist-evoked responses in outside-out patches," Nature Protocols 2(2), 434-448 (2007).
[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]

Kaveh, M.

M. Kaveh, M. Rosenbluh, I. Edrei, and I. Freund, "Weak Localization and Light Scattering from Disordered Solids," Phys. Rev. Lett. 57(16), 2049-2052 (1986).
[CrossRef]

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]

P. A. Lee and A. D. Stone, "Universal Conductance Fluctuations in Metals," Phys. Rev. Lett. 55(15), 1622-1625 (1985).
[CrossRef]

Popescu, G.

G. Popescu and A. Dogariu, "Scattering of low coherence radiation and applications," Eur. Phys. J. Appl. Phys. 32(2), 73-93 (2005).
[CrossRef]

G. Popescu and A. Dogariu, "Optical path-length spectroscopy of wave propagation in random media," Opt. Lett. 24(7), 442-444 (1999).
[CrossRef]

Rosenbluh, M.

M. Kaveh, M. Rosenbluh, I. Edrei, and I. Freund, "Weak Localization and Light Scattering from Disordered Solids," Phys. Rev. Lett. 57(16), 2049-2052 (1986).
[CrossRef]

Shannon, C. E.

C. E. Shannon, "A Mathematical Theory of Communication," Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).

Shapiro, B.

B. Shapiro, "Large Intensity Fluctuations for Wave Propagation in Random Media," Phys. Rev. Lett. 57(17), 2168-2171 (1986).
[CrossRef]

Stephen, M. J.

M. J. Stephen and G. Cwilich, "Intensity correlation functions and fluctuations in light scattered from a random medium," Phys. Rev. Lett. 59(3), 285-287 (1987).
[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]

P. A. Lee and A. D. Stone, "Universal Conductance Fluctuations in Metals," Phys. Rev. Lett. 55(15), 1622-1625 (1985).
[CrossRef]

Thompson, R.

S. Etemad, R. Thompson, and M. J. Andrejco, "Weak localization of photons: universal fluctuations and ensemble averagign," Phys. Rev. Lett. 57(5), 575-578 (1986).
[CrossRef]

Veruki, M. L.

E. Hartveit and M. L. Veruki, "Studying properties of neurotransmitter receptors by non-stationary noise analysis of spontaneous postsynaptic currents and agonist-evoked responses in outside-out patches," Nature Protocols 2(2), 434-448 (2007).
[CrossRef]

Weiss, G. H.

A. H. Gandjbakhche and G. H. Weiss, "Random walk and diffusion-like model of photon migration in turbid media," Prog. in Opt. 34, 333-402 (1995).
[CrossRef]

Bell Syst. Tech. J.

C. E. Shannon, "A Mathematical Theory of Communication," Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).

Eur. Phys. J. Appl. Phys.

G. Popescu and A. Dogariu, "Scattering of low coherence radiation and applications," Eur. Phys. J. Appl. Phys. 32(2), 73-93 (2005).
[CrossRef]

Nature Protocols

E. Hartveit and M. L. Veruki, "Studying properties of neurotransmitter receptors by non-stationary noise analysis of spontaneous postsynaptic currents and agonist-evoked responses in outside-out patches," Nature Protocols 2(2), 434-448 (2007).
[CrossRef]

Opt. Lett.

Phys. Rev. E

A. Apostol, D. Haefner, and A. Dogariu, "Near-field characterization of effective optical interfaces," Phys. Rev. E 74(6), 066603-6 (2006).
[CrossRef]

Phys. Rev. Lett.

P. A. Lee and A. D. Stone, "Universal Conductance Fluctuations in Metals," Phys. Rev. Lett. 55(15), 1622-1625 (1985).
[CrossRef]

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]

S. Etemad, R. Thompson, and M. J. Andrejco, "Weak localization of photons: universal fluctuations and ensemble averagign," Phys. Rev. Lett. 57(5), 575-578 (1986).
[CrossRef]

M. Kaveh, M. Rosenbluh, I. Edrei, and I. Freund, "Weak Localization and Light Scattering from Disordered Solids," Phys. Rev. Lett. 57(16), 2049-2052 (1986).
[CrossRef]

B. Shapiro, "Large Intensity Fluctuations for Wave Propagation in Random Media," Phys. Rev. Lett. 57(17), 2168-2171 (1986).
[CrossRef]

M. J. Stephen and G. Cwilich, "Intensity correlation functions and fluctuations in light scattered from a random medium," Phys. Rev. Lett. 59(3), 285-287 (1987).
[CrossRef]

Prog. in Opt.

A. H. Gandjbakhche and G. H. Weiss, "Random walk and diffusion-like model of photon migration in turbid media," Prog. in Opt. 34, 333-402 (1995).
[CrossRef]

Other

J. W. Goodman, Speckle Phenomena in Optics, 1st ed. (Roberts & Co., Englewood, 2007).

A. Ishimaru, Wave Propagation and Scattering in Random Media, vol. 1 (Academic, New York, 1971).

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Figures (4)

Fig. 1.
Fig. 1.

Sketch of path-length distributions for two media with identical mean properties (same D). The two media consist of scatterers of different cross-sections σ and different number densities NV and are examined over the same range of path-lengths s. The medium with smaller number density provides fewer possible paths of given length s resulting in larger fluctuations of pα (s).

Fig. 2.
Fig. 2.

The averaged backscattered intensities for medium A (blue solid line) and medium B (red dashed line). The insets show typical micrographs of the materials examined.

Fig. 3.
Fig. 3.

Typical mean square fluctuations δ2 α (s,ξα) of path-length distributions for media A and B shown in Fig.2.

Fig. 4.
Fig. 4.

Average normalized entropy hα (Δ) for medium A (blue circles) and medium B (red boxes) for increasing volumes of interaction in a bistatic configuration as depicted in the inset.

Equations (4)

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

pα(s)=f (s,D)[1+δα(s,ξα)] .
Hα(s1,s2)=s1s2δα2(s,ξα)s1s2δα2(s,ξα)dslog(δα2(s,ξα)s1s2δα2(s,ξα)ds)ds.
hα(s1,s2)=Hα(s1,s2)log(1S).
p(s,Δ)~exp(ze2+Δ24sD)D32s52,

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