Abstract

Statistics of the phase and intensity of speckles formed with a small number of scattering events (small-N speckles) within multiple-scattering media have been studied. It has been demonstrated that first-order statistics of the intensity fluctuations of small-N speckles nearly obey a Nakagami n distribution in the case considered. The correlation function of the complex amplitude of scattered light is close to a negative exponent. Theoretical results have been experimentally verified using the Shack–Hartmann wavefront analysis technique.

© 2008 Optical Society of America

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  2. P. J. Chandley and H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: The effect of changing the surface height variance,” Opt. Commun. 29, 151-154 (1979).
    [CrossRef]
  3. E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. (Bellingham) 23, 453-461 (1984).
  4. B. Saleh, Photoelectron Statistics with Applications to Spectroscopy and Optical Communication (Springer-Verlag, 1991), pp. 145-149.
  5. S. S. Ulyanov, D. A. Zimnyakov, and V. V. Tuchin, “Fundamentals and applications of dynamic speckles induced by focused laser beam scattering,” Opt. Eng. (Bellingham) 33, 3189-3201 (1994).
    [CrossRef]
  6. S. S. Ulyanov, “Speckled speckles statistics with a small number of scatterers. An implication for blood flow measurements,” J. Biomed. Opt. 3, 227-236 (1998).
    [CrossRef]
  7. M. Kowalczyk and P. Zalicki, “Small-N speckle: Phase-contrast approach,” Proc. SPIE 556, 50-54 (1985).
  8. S. S. Ulyanov, “Dynamics of statistically inhomogeneous speckles: A new type of manifestation of the Doppler effect,” Opt. Lett. 20, 1313-1315 (1995).
    [CrossRef]
  9. S. S. Ulyanov, “New type of manifestation of the Doppler effect: An application to blood and lymph flow measurements,” Opt. Eng. (Bellingham) 34, 2850-2855 (1995).
    [CrossRef]
  10. D. A. Weitz and D. J. Pine, “Diffusing wave spectroscopy,” in Dynamic Light Scattering: The Method and Some Applications, W.Brown, ed. (Claredon, 1993), pp. 652-720.
  11. D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheomer, and P. M. Chaikin, “Dynamical correlations of multiple scattered light,” in Scattering and Localization of Classical Waves in Random Media, Vol. 8 of World Series on Direction in Condensed Matter Physics, P.Sheng, ed. (World Scientific, 1990), pp. 312-372.
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    [CrossRef]
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    [CrossRef] [PubMed]
  15. A. D. Gopal and D. J. Durian, “Nonlinear bubble dynamics in a slowly driven foam,” Phys. Rev. Lett. 75, 2610-2613 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
  17. F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
    [CrossRef]
  18. T. G. Mason, K. Ganesan, J. H. Van-Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282-3285 (1997).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. V. L. Kuzmin, I. V. Meglinsky, and D. Yu. Churmakov, “Stochastic Modelling of Coherent Phenomena in Strongly Inhomogeneous Media,” J. Exp. Theor. Phys. 101, 22-32 (2005).
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    [CrossRef]
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  28. I. S. Gonorovsky, Radiotechnical Circuits and Signals (Sovietskoe Radio, 1977).
  29. J. S. Bendat and A. G. Piersol, Random Data. Analysis and Measurement Procedures (Wiley, 1986).
  30. D. J. Pine, D. A. Weitz, J. X. Zhu, and E. Herbolzheimer, “Diffusing-wave spectroscopy: Dynamic light scattering in the multiple scattering limit,” J. Phys. (France) 51, 2101-2127 (1990).
    [CrossRef]
  31. M. Nakagami, “The m-distribution--a general formula of intensity distribution in rapid fading,” in Statistical Methods on Radio Wave Propagation, W.C.Hoffman, ed. (Pergamon, 1960), pp. 3-36.
  32. M. Yoshikawa and H. Kayano, “Analysis of non-Gaussian speckle by Nakagami m-distribution,” Jpn. J. Appl. Phys., Part 1 26, 974-975 (1987).
    [CrossRef]
  33. A. N. Korolevich and I. V. Meglinsky, “Experimental study of the potential use of diffusing wave spectroscopy to investigate the structural characteristics of blood under multiple scattering,” Bioelectrochemistry 52, 223-227 (2000).
    [CrossRef] [PubMed]
  34. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarsky, Principles of Statistical Radiophysics. Part 2 (Springer, 1989).
  35. W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166-2185 (1990).
    [CrossRef]
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    [PubMed]
  37. A. F. Brooks, T.-L. Kelly, P. J. Veitch, and J. Munch, “Ultra-sensitive wavefront measurement using a Hartmann sensor,” Opt. Express 15, 10370-10375 (2007).
    [CrossRef] [PubMed]
  38. D. A. Boas, K. K. Bizheva, and A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion within highly scattering media,” Opt. Lett. 23, 319-321 (1998).
    [CrossRef]

2007 (1)

2005 (3)

S. Ulyanov, “Diffusing wave spectroscopy with a small number of scattering events: An implication to microflow diagnostics,” Phys. Rev. E 72, 052902 (2005).
[CrossRef]

V. L. Kuzmin, I. V. Meglinsky, and D. Yu. Churmakov, “Stochastic Modelling of Coherent Phenomena in Strongly Inhomogeneous Media,” J. Exp. Theor. Phys. 101, 22-32 (2005).
[CrossRef]

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

2001 (2)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, S573-S577 (2001).
[PubMed]

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[CrossRef]

2000 (1)

A. N. Korolevich and I. V. Meglinsky, “Experimental study of the potential use of diffusing wave spectroscopy to investigate the structural characteristics of blood under multiple scattering,” Bioelectrochemistry 52, 223-227 (2000).
[CrossRef] [PubMed]

1998 (5)

D. A. Boas, K. K. Bizheva, and A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion within highly scattering media,” Opt. Lett. 23, 319-321 (1998).
[CrossRef]

I. V. Meglinsky and S. E. Skipetrov, “Diffusing-wave spectroscopy in randomly inhomogeneous media with spatially localized scatterer flows,” J. Exp. Theor. Phys. 86, 661-665 (1998).
[CrossRef]

P. A. Lemieux, M. U. Vera, and D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: The role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498-4515 (1998).
[CrossRef]

E. M. Furst and A. P. Gast, “Particle dynamics in magnetorheological suspensions using diffusing-wave spectroscopy,” Phys. Rev. E 58, 3372-3376 (1998).
[CrossRef]

S. S. Ulyanov, “Speckled speckles statistics with a small number of scatterers. An implication for blood flow measurements,” J. Biomed. Opt. 3, 227-236 (1998).
[CrossRef]

1997 (3)

N. Menon and D. J. Durian, “Particle motions in a gas-fluidized bed of sand,” Phys. Rev. Lett. 79, 3407-3410 (1997).
[CrossRef]

R. Hohler, S. Cohen-Addad, and H. Hoballah, “Periodic nonlinear bubble motion in aqueous foam under oscillating shear strain,” Phys. Rev. Lett. 79, 1154-1157 (1997).
[CrossRef]

T. G. Mason, K. Ganesan, J. H. Van-Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282-3285 (1997).
[CrossRef]

1995 (3)

A. D. Gopal and D. J. Durian, “Nonlinear bubble dynamics in a slowly driven foam,” Phys. Rev. Lett. 75, 2610-2613 (1995).
[CrossRef] [PubMed]

S. S. Ulyanov, “Dynamics of statistically inhomogeneous speckles: A new type of manifestation of the Doppler effect,” Opt. Lett. 20, 1313-1315 (1995).
[CrossRef]

S. S. Ulyanov, “New type of manifestation of the Doppler effect: An application to blood and lymph flow measurements,” Opt. Eng. (Bellingham) 34, 2850-2855 (1995).
[CrossRef]

1994 (1)

S. S. Ulyanov, D. A. Zimnyakov, and V. V. Tuchin, “Fundamentals and applications of dynamic speckles induced by focused laser beam scattering,” Opt. Eng. (Bellingham) 33, 3189-3201 (1994).
[CrossRef]

1993 (1)

M. H. Kao, A. G. Yodh, and D. J. Pine, “Observation of Brownian motion on the time scale of hydrodynamic interactions,” Phys. Rev. Lett. 70, 242-245 (1993).
[CrossRef] [PubMed]

1990 (2)

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

D. J. Pine, D. A. Weitz, J. X. Zhu, and E. Herbolzheimer, “Diffusing-wave spectroscopy: Dynamic light scattering in the multiple scattering limit,” J. Phys. (France) 51, 2101-2127 (1990).
[CrossRef]

1987 (1)

M. Yoshikawa and H. Kayano, “Analysis of non-Gaussian speckle by Nakagami m-distribution,” Jpn. J. Appl. Phys., Part 1 26, 974-975 (1987).
[CrossRef]

1985 (1)

M. Kowalczyk and P. Zalicki, “Small-N speckle: Phase-contrast approach,” Proc. SPIE 556, 50-54 (1985).

1984 (1)

E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. (Bellingham) 23, 453-461 (1984).

1979 (1)

P. J. Chandley and H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: The effect of changing the surface height variance,” Opt. Commun. 29, 151-154 (1979).
[CrossRef]

Bandyopadhyay, R.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Bendat, J. S.

J. S. Bendat and A. G. Piersol, Random Data. Analysis and Measurement Procedures (Wiley, 1986).

Bizheva, K. K.

Boas, D. A.

Brooks, A. F.

Chaikin, P. M.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheomer, and P. M. Chaikin, “Dynamical correlations of multiple scattered light,” in Scattering and Localization of Classical Waves in Random Media, Vol. 8 of World Series on Direction in Condensed Matter Physics, P.Sheng, ed. (World Scientific, 1990), pp. 312-372.

Chandley, P. J.

P. J. Chandley and H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: The effect of changing the surface height variance,” Opt. Commun. 29, 151-154 (1979).
[CrossRef]

Cheong, W. F.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

Churmakov, D. Yu.

V. L. Kuzmin, I. V. Meglinsky, and D. Yu. Churmakov, “Stochastic Modelling of Coherent Phenomena in Strongly Inhomogeneous Media,” J. Exp. Theor. Phys. 101, 22-32 (2005).
[CrossRef]

Cohen-Addad, S.

R. Hohler, S. Cohen-Addad, and H. Hoballah, “Periodic nonlinear bubble motion in aqueous foam under oscillating shear strain,” Phys. Rev. Lett. 79, 1154-1157 (1997).
[CrossRef]

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics Series (Springer, 1975).

Dixon, P. K.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Durian, D. J.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

P. A. Lemieux, M. U. Vera, and D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: The role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498-4515 (1998).
[CrossRef]

N. Menon and D. J. Durian, “Particle motions in a gas-fluidized bed of sand,” Phys. Rev. Lett. 79, 3407-3410 (1997).
[CrossRef]

A. D. Gopal and D. J. Durian, “Nonlinear bubble dynamics in a slowly driven foam,” Phys. Rev. Lett. 75, 2610-2613 (1995).
[CrossRef] [PubMed]

Escamilla, H. M.

P. J. Chandley and H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: The effect of changing the surface height variance,” Opt. Commun. 29, 151-154 (1979).
[CrossRef]

Furst, E. M.

E. M. Furst and A. P. Gast, “Particle dynamics in magnetorheological suspensions using diffusing-wave spectroscopy,” Phys. Rev. E 58, 3372-3376 (1998).
[CrossRef]

Ganesan, K.

T. G. Mason, K. Ganesan, J. H. Van-Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282-3285 (1997).
[CrossRef]

Gast, A. P.

E. M. Furst and A. P. Gast, “Particle dynamics in magnetorheological suspensions using diffusing-wave spectroscopy,” Phys. Rev. E 58, 3372-3376 (1998).
[CrossRef]

Gittings, A. S.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Gonorovsky, I. S.

I. S. Gonorovsky, Radiotechnical Circuits and Signals (Sovietskoe Radio, 1977).

Goodman, J. W.

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

Gopal, A. D.

A. D. Gopal and D. J. Durian, “Nonlinear bubble dynamics in a slowly driven foam,” Phys. Rev. Lett. 75, 2610-2613 (1995).
[CrossRef] [PubMed]

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, J. X. Zhu, and E. Herbolzheimer, “Diffusing-wave spectroscopy: Dynamic light scattering in the multiple scattering limit,” J. Phys. (France) 51, 2101-2127 (1990).
[CrossRef]

Herbolzheomer, E.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheomer, and P. M. Chaikin, “Dynamical correlations of multiple scattered light,” in Scattering and Localization of Classical Waves in Random Media, Vol. 8 of World Series on Direction in Condensed Matter Physics, P.Sheng, ed. (World Scientific, 1990), pp. 312-372.

Hoballah, H.

R. Hohler, S. Cohen-Addad, and H. Hoballah, “Periodic nonlinear bubble motion in aqueous foam under oscillating shear strain,” Phys. Rev. Lett. 79, 1154-1157 (1997).
[CrossRef]

Hohler, R.

R. Hohler, S. Cohen-Addad, and H. Hoballah, “Periodic nonlinear bubble motion in aqueous foam under oscillating shear strain,” Phys. Rev. Lett. 79, 1154-1157 (1997).
[CrossRef]

Jacques, S. L.

S. L. Jacques and L. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical-Thermal Response of Laser-Irradiated Tissue, A.J.Welch and M.J.van Gemert, eds. (Plenum, 1995), Chap. 4.

Jakeman, E.

E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. (Bellingham) 23, 453-461 (1984).

Kao, M. H.

M. H. Kao, A. G. Yodh, and D. J. Pine, “Observation of Brownian motion on the time scale of hydrodynamic interactions,” Phys. Rev. Lett. 70, 242-245 (1993).
[CrossRef] [PubMed]

Kayano, H.

M. Yoshikawa and H. Kayano, “Analysis of non-Gaussian speckle by Nakagami m-distribution,” Jpn. J. Appl. Phys., Part 1 26, 974-975 (1987).
[CrossRef]

Kelly, T.-L.

Korolevich, A. N.

A. N. Korolevich and I. V. Meglinsky, “Experimental study of the potential use of diffusing wave spectroscopy to investigate the structural characteristics of blood under multiple scattering,” Bioelectrochemistry 52, 223-227 (2000).
[CrossRef] [PubMed]

Kowalczyk, M.

M. Kowalczyk and P. Zalicki, “Small-N speckle: Phase-contrast approach,” Proc. SPIE 556, 50-54 (1985).

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarsky, Principles of Statistical Radiophysics. Part 2 (Springer, 1989).

Kuo, S. C.

T. G. Mason, K. Ganesan, J. H. Van-Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282-3285 (1997).
[CrossRef]

Kuzmin, V. L.

V. L. Kuzmin, I. V. Meglinsky, and D. Yu. Churmakov, “Stochastic Modelling of Coherent Phenomena in Strongly Inhomogeneous Media,” J. Exp. Theor. Phys. 101, 22-32 (2005).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge Univ. Press, 2002).

Lemieux, P. A.

P. A. Lemieux, M. U. Vera, and D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: The role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498-4515 (1998).
[CrossRef]

Maret, G.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheomer, and P. M. Chaikin, “Dynamical correlations of multiple scattered light,” in Scattering and Localization of Classical Waves in Random Media, Vol. 8 of World Series on Direction in Condensed Matter Physics, P.Sheng, ed. (World Scientific, 1990), pp. 312-372.

Mason, T. G.

T. G. Mason, K. Ganesan, J. H. Van-Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282-3285 (1997).
[CrossRef]

Meglinsky, I. V.

V. L. Kuzmin, I. V. Meglinsky, and D. Yu. Churmakov, “Stochastic Modelling of Coherent Phenomena in Strongly Inhomogeneous Media,” J. Exp. Theor. Phys. 101, 22-32 (2005).
[CrossRef]

A. N. Korolevich and I. V. Meglinsky, “Experimental study of the potential use of diffusing wave spectroscopy to investigate the structural characteristics of blood under multiple scattering,” Bioelectrochemistry 52, 223-227 (2000).
[CrossRef] [PubMed]

I. V. Meglinsky and S. E. Skipetrov, “Diffusing-wave spectroscopy in randomly inhomogeneous media with spatially localized scatterer flows,” J. Exp. Theor. Phys. 86, 661-665 (1998).
[CrossRef]

Menon, N.

N. Menon and D. J. Durian, “Particle motions in a gas-fluidized bed of sand,” Phys. Rev. Lett. 79, 3407-3410 (1997).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge Univ. Press, 2002).

Munch, J.

Nakagami, M.

M. Nakagami, “The m-distribution--a general formula of intensity distribution in rapid fading,” in Statistical Methods on Radio Wave Propagation, W.C.Hoffman, ed. (Pergamon, 1960), pp. 3-36.

Piersol, A. G.

J. S. Bendat and A. G. Piersol, Random Data. Analysis and Measurement Procedures (Wiley, 1986).

Pine, D. J.

M. H. Kao, A. G. Yodh, and D. J. Pine, “Observation of Brownian motion on the time scale of hydrodynamic interactions,” Phys. Rev. Lett. 70, 242-245 (1993).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, J. X. Zhu, and E. Herbolzheimer, “Diffusing-wave spectroscopy: Dynamic light scattering in the multiple scattering limit,” J. Phys. (France) 51, 2101-2127 (1990).
[CrossRef]

D. A. Weitz and D. J. Pine, “Diffusing wave spectroscopy,” in Dynamic Light Scattering: The Method and Some Applications, W.Brown, ed. (Claredon, 1993), pp. 652-720.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheomer, and P. M. Chaikin, “Dynamical correlations of multiple scattered light,” in Scattering and Localization of Classical Waves in Random Media, Vol. 8 of World Series on Direction in Condensed Matter Physics, P.Sheng, ed. (World Scientific, 1990), pp. 312-372.

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, S573-S577 (2001).
[PubMed]

Prahl, S. A.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

Romer, S.

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[CrossRef]

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarsky, Principles of Statistical Radiophysics. Part 2 (Springer, 1989).

Saleh, B.

B. Saleh, Photoelectron Statistics with Applications to Spectroscopy and Optical Communication (Springer-Verlag, 1991), pp. 145-149.

Scheffold, F.

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[CrossRef]

Schurtenberger, P.

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[CrossRef]

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, S573-S577 (2001).
[PubMed]

Siegel, A. M.

Skipetrov, S. E.

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[CrossRef]

I. V. Meglinsky and S. E. Skipetrov, “Diffusing-wave spectroscopy in randomly inhomogeneous media with spatially localized scatterer flows,” J. Exp. Theor. Phys. 86, 661-665 (1998).
[CrossRef]

Suh, S. S.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Tatarsky, V. I.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarsky, Principles of Statistical Radiophysics. Part 2 (Springer, 1989).

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge Univ. Press, 2002).

Tuchin, V. V.

S. S. Ulyanov, D. A. Zimnyakov, and V. V. Tuchin, “Fundamentals and applications of dynamic speckles induced by focused laser beam scattering,” Opt. Eng. (Bellingham) 33, 3189-3201 (1994).
[CrossRef]

Ulyanov, S.

S. Ulyanov, “Diffusing wave spectroscopy with a small number of scattering events: An implication to microflow diagnostics,” Phys. Rev. E 72, 052902 (2005).
[CrossRef]

Ulyanov, S. S.

S. S. Ulyanov, “Speckled speckles statistics with a small number of scatterers. An implication for blood flow measurements,” J. Biomed. Opt. 3, 227-236 (1998).
[CrossRef]

S. S. Ulyanov, “Dynamics of statistically inhomogeneous speckles: A new type of manifestation of the Doppler effect,” Opt. Lett. 20, 1313-1315 (1995).
[CrossRef]

S. S. Ulyanov, “New type of manifestation of the Doppler effect: An application to blood and lymph flow measurements,” Opt. Eng. (Bellingham) 34, 2850-2855 (1995).
[CrossRef]

S. S. Ulyanov, D. A. Zimnyakov, and V. V. Tuchin, “Fundamentals and applications of dynamic speckles induced by focused laser beam scattering,” Opt. Eng. (Bellingham) 33, 3189-3201 (1994).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Van-Zanten, J. H.

T. G. Mason, K. Ganesan, J. H. Van-Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282-3285 (1997).
[CrossRef]

Veitch, P. J.

Vera, M. U.

P. A. Lemieux, M. U. Vera, and D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: The role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498-4515 (1998).
[CrossRef]

Wang, L.

S. L. Jacques and L. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical-Thermal Response of Laser-Irradiated Tissue, A.J.Welch and M.J.van Gemert, eds. (Plenum, 1995), Chap. 4.

Weitz, D. A.

D. J. Pine, D. A. Weitz, J. X. Zhu, and E. Herbolzheimer, “Diffusing-wave spectroscopy: Dynamic light scattering in the multiple scattering limit,” J. Phys. (France) 51, 2101-2127 (1990).
[CrossRef]

D. A. Weitz and D. J. Pine, “Diffusing wave spectroscopy,” in Dynamic Light Scattering: The Method and Some Applications, W.Brown, ed. (Claredon, 1993), pp. 652-720.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheomer, and P. M. Chaikin, “Dynamical correlations of multiple scattered light,” in Scattering and Localization of Classical Waves in Random Media, Vol. 8 of World Series on Direction in Condensed Matter Physics, P.Sheng, ed. (World Scientific, 1990), pp. 312-372.

Welch, A. J.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

Wirtz, D.

T. G. Mason, K. Ganesan, J. H. Van-Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282-3285 (1997).
[CrossRef]

Wolf, P. E.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheomer, and P. M. Chaikin, “Dynamical correlations of multiple scattered light,” in Scattering and Localization of Classical Waves in Random Media, Vol. 8 of World Series on Direction in Condensed Matter Physics, P.Sheng, ed. (World Scientific, 1990), pp. 312-372.

Yodh, A. G.

M. H. Kao, A. G. Yodh, and D. J. Pine, “Observation of Brownian motion on the time scale of hydrodynamic interactions,” Phys. Rev. Lett. 70, 242-245 (1993).
[CrossRef] [PubMed]

Yoshikawa, M.

M. Yoshikawa and H. Kayano, “Analysis of non-Gaussian speckle by Nakagami m-distribution,” Jpn. J. Appl. Phys., Part 1 26, 974-975 (1987).
[CrossRef]

Zalicki, P.

M. Kowalczyk and P. Zalicki, “Small-N speckle: Phase-contrast approach,” Proc. SPIE 556, 50-54 (1985).

Zhu, J. X.

D. J. Pine, D. A. Weitz, J. X. Zhu, and E. Herbolzheimer, “Diffusing-wave spectroscopy: Dynamic light scattering in the multiple scattering limit,” J. Phys. (France) 51, 2101-2127 (1990).
[CrossRef]

Zimnyakov, D. A.

S. S. Ulyanov, D. A. Zimnyakov, and V. V. Tuchin, “Fundamentals and applications of dynamic speckles induced by focused laser beam scattering,” Opt. Eng. (Bellingham) 33, 3189-3201 (1994).
[CrossRef]

Bioelectrochemistry (1)

A. N. Korolevich and I. V. Meglinsky, “Experimental study of the potential use of diffusing wave spectroscopy to investigate the structural characteristics of blood under multiple scattering,” Bioelectrochemistry 52, 223-227 (2000).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

J. Biomed. Opt. (1)

S. S. Ulyanov, “Speckled speckles statistics with a small number of scatterers. An implication for blood flow measurements,” J. Biomed. Opt. 3, 227-236 (1998).
[CrossRef]

J. Exp. Theor. Phys. (2)

I. V. Meglinsky and S. E. Skipetrov, “Diffusing-wave spectroscopy in randomly inhomogeneous media with spatially localized scatterer flows,” J. Exp. Theor. Phys. 86, 661-665 (1998).
[CrossRef]

V. L. Kuzmin, I. V. Meglinsky, and D. Yu. Churmakov, “Stochastic Modelling of Coherent Phenomena in Strongly Inhomogeneous Media,” J. Exp. Theor. Phys. 101, 22-32 (2005).
[CrossRef]

J. Phys. (France) (1)

D. J. Pine, D. A. Weitz, J. X. Zhu, and E. Herbolzheimer, “Diffusing-wave spectroscopy: Dynamic light scattering in the multiple scattering limit,” J. Phys. (France) 51, 2101-2127 (1990).
[CrossRef]

J. Refract. Surg. (1)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, S573-S577 (2001).
[PubMed]

Jpn. J. Appl. Phys., Part 1 (1)

M. Yoshikawa and H. Kayano, “Analysis of non-Gaussian speckle by Nakagami m-distribution,” Jpn. J. Appl. Phys., Part 1 26, 974-975 (1987).
[CrossRef]

Opt. Commun. (1)

P. J. Chandley and H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: The effect of changing the surface height variance,” Opt. Commun. 29, 151-154 (1979).
[CrossRef]

Opt. Eng. (Bellingham) (3)

E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. (Bellingham) 23, 453-461 (1984).

S. S. Ulyanov, D. A. Zimnyakov, and V. V. Tuchin, “Fundamentals and applications of dynamic speckles induced by focused laser beam scattering,” Opt. Eng. (Bellingham) 33, 3189-3201 (1994).
[CrossRef]

S. S. Ulyanov, “New type of manifestation of the Doppler effect: An application to blood and lymph flow measurements,” Opt. Eng. (Bellingham) 34, 2850-2855 (1995).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. E (4)

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[CrossRef]

E. M. Furst and A. P. Gast, “Particle dynamics in magnetorheological suspensions using diffusing-wave spectroscopy,” Phys. Rev. E 58, 3372-3376 (1998).
[CrossRef]

P. A. Lemieux, M. U. Vera, and D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: The role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498-4515 (1998).
[CrossRef]

S. Ulyanov, “Diffusing wave spectroscopy with a small number of scattering events: An implication to microflow diagnostics,” Phys. Rev. E 72, 052902 (2005).
[CrossRef]

Phys. Rev. Lett. (5)

M. H. Kao, A. G. Yodh, and D. J. Pine, “Observation of Brownian motion on the time scale of hydrodynamic interactions,” Phys. Rev. Lett. 70, 242-245 (1993).
[CrossRef] [PubMed]

A. D. Gopal and D. J. Durian, “Nonlinear bubble dynamics in a slowly driven foam,” Phys. Rev. Lett. 75, 2610-2613 (1995).
[CrossRef] [PubMed]

R. Hohler, S. Cohen-Addad, and H. Hoballah, “Periodic nonlinear bubble motion in aqueous foam under oscillating shear strain,” Phys. Rev. Lett. 79, 1154-1157 (1997).
[CrossRef]

T. G. Mason, K. Ganesan, J. H. Van-Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282-3285 (1997).
[CrossRef]

N. Menon and D. J. Durian, “Particle motions in a gas-fluidized bed of sand,” Phys. Rev. Lett. 79, 3407-3410 (1997).
[CrossRef]

Proc. SPIE (1)

M. Kowalczyk and P. Zalicki, “Small-N speckle: Phase-contrast approach,” Proc. SPIE 556, 50-54 (1985).

Rev. Sci. Instrum. (1)

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Other (12)

S. L. Jacques and L. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical-Thermal Response of Laser-Irradiated Tissue, A.J.Welch and M.J.van Gemert, eds. (Plenum, 1995), Chap. 4.

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

I. S. Gonorovsky, Radiotechnical Circuits and Signals (Sovietskoe Radio, 1977).

J. S. Bendat and A. G. Piersol, Random Data. Analysis and Measurement Procedures (Wiley, 1986).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge Univ. Press, 2002).

D. A. Weitz and D. J. Pine, “Diffusing wave spectroscopy,” in Dynamic Light Scattering: The Method and Some Applications, W.Brown, ed. (Claredon, 1993), pp. 652-720.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheomer, and P. M. Chaikin, “Dynamical correlations of multiple scattered light,” in Scattering and Localization of Classical Waves in Random Media, Vol. 8 of World Series on Direction in Condensed Matter Physics, P.Sheng, ed. (World Scientific, 1990), pp. 312-372.

B. Saleh, Photoelectron Statistics with Applications to Spectroscopy and Optical Communication (Springer-Verlag, 1991), pp. 145-149.

M. Nakagami, “The m-distribution--a general formula of intensity distribution in rapid fading,” in Statistical Methods on Radio Wave Propagation, W.C.Hoffman, ed. (Pergamon, 1960), pp. 3-36.

J. C. Dainty, Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics Series (Springer, 1975).

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarsky, Principles of Statistical Radiophysics. Part 2 (Springer, 1989).

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Figures (7)

Fig. 1
Fig. 1

Probability density functions, expressed by Eq. (24) (circles) and Eq. (25) (boxes), in the case of a small number of scattering events, n = 7 . The relative difference between the rigorous formula, expressing first-order statistics, and the Nakagami n distribution is less than 7%.

Fig. 2
Fig. 2

Optical scheme for observation of the temporal fluctuations of the phase and intensities of multiple-scattered light: a, collimated laser beam; b, scattering object; c (gray line), object plane; d, micro-objective; e (gray line), image plane of micro-objective; f, pinhole; g, lens; h (gray line), lens focal plane; i, complementary metal-oxide semiconductor (CMOS) camera.

Fig. 3
Fig. 3

Speckle pattern formed in the image plane of the micro-objective.

Fig. 4
Fig. 4

Illustration of the wavefront analysis. The position of the light spot in the focal plane of the lenslet is shown.

Fig. 5
Fig. 5

Typical realizations of the temporal fluctuations of phase (a) and normalized intensity (b) of dynamic multiple-scattered speckles with a small number of scattering events.

Fig. 6
Fig. 6

Temporal autocorrelation function of the amplitude of scattered light. Solid curve, theoretical curve; dots, experimental data.

Fig. 7
Fig. 7

Temporal autocorrelation function of the phase of scattered light. Solid curve, theoretical curve; dots, experimental data.

Equations (36)

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e 2 e 0 λ s 1 exp ( 2 π i s 1 λ ) ,
E 2 1 S 1 exp ( 2 π i S 1 ) ,
E 3 1 S 1 S 2 exp ( 2 π i S 1 ) exp ( 2 π i S 2 ) ,
E n 1 i = 1 n S i exp ( 2 π i i = 1 n S i ) .
p ( S ) = M s exp ( M s S ) ,
p ( Z ) = p ( S ) p ( Z S ) d S S .
p ( Z ) = M S 2 0 exp ( M s S ) exp ( M s Z S ) d S S .
p ( Z ) = 2 M S 2 K 0 ( 2 M s Z ) .
p ( ξ ) = 4 M S 4 K 0 ( 2 M s Z ) K 0 ( 2 M s ξ Z ) d Z Z .
p ( ξ ) 0 exp ( 2 M s Z ) exp ( 2 M s ξ Z ) d Z Z .
p ( ξ ) K o ( 4 M s ξ 1 4 ) .
p ( n ) K o ( n M s n 1 n ) ,
p ( E n ) = p ( n ) d E n d n .
d E n d n = 1 n 2
p ( E n ) const ( n ) 1 E n 2 K 0 ( n M s ( E n ) 1 n ) ,
const ( n ) = 1 0 1 E 2 K 0 ( n M s E 1 n ) d E .
const ( n ) = n ( n 1 ) M S n 2 ( n 2 ) ( Γ [ n 2 ] ) 2 .
p ( E n ) n ( n 1 ) M S n 2 ( n 2 ) ( Γ [ n 2 ] ) 2 1 E 2 K 0 ( n M s ( E ) 1 n ) .
P ( E ) = s p ( E S ) p ( S ) exp ( 2 π i S ) ,
p ( S ) = n p ( S n ) P ( n ) ,
p ( S n ) = M S 2 S ( n 1 ) ( n 1 ) ! exp ( M S S ) .
P ( n ) = n n exp ( n ) n ! .
p ( S ) = n p ( S n ) P ( n ) = [ exp ( M s S ) exp ( n ) 1 S ] n ( M S S n ) n n ! ( n 1 ) ! [ exp ( M s S ) exp ( n ) 1 S ] [ I 1 ( M s S n ) + 1 ] ,
S = 0 S p ( S ) d S = 1 M s exp ( n ) n n ( n 1 ) ! = n M s .
p ( E ) = n n ( n 2 ) ( M S ) n + 1 2 ( n 2 ) ( Γ [ n 2 ] ) 2 1 E 2 K 0 ( n M s ( E ) 1 n ) [ exp ( ( 1 + 2 π i M s ) n ) exp ( n ) ] [ I 1 ( n n ) + 1 ] .
p ( E ) n n E n 2 Γ ( n ) E n 2 exp ( n E E ) ,
U ( t ) = E ( t ) exp ( i Ψ ( t ) ) ,
Γ ( τ ) = U ( t ) U ( t + τ ) * .
Γ ( τ ) = E ( t ) E ( t + τ ) * exp ( i { Ψ ( t ) Ψ ( t + τ ) } ) .
Γ ( τ ) factor ( n ) exp ( i { Ψ ( t ) Ψ ( t + τ ) } ) ,
factor ( n ) 0 E 2 p ( E ) d E
exp ( i { Ψ ( t ) Ψ ( t + τ ) } ) = exp ( 0.5 { Ψ ( t ) Ψ ( t + τ ) } 2 ) ,
Γ ( τ ) factor ( n ) exp ( σ Ψ 2 [ K Ψ ( τ ) 1 ] ) ,
sin ( ς ) λ l v ,
g 1 sin ( ς ) 2 1 [ λ M o 2 ] 2 = 0.8 ,
N sc = Δ μ s ( 1 g ) 0.6 ,

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