Abstract

A new method of visualizing objects with distinct internal dynamics of the constituent scattering particles embedded in a liquid multiple-scattering medium is presented. We report dynamic multiple-light-scattering experiments and a theoretical model, based on diffusing photon-density waves for concentrated colloidal suspensions in Brownian motion, as a background medium into which is inserted a capillary containing (i) the same suspension under flow, or (ii) suspensions of different particle sizes in Brownian motion. These model objects, with purely dynamic but no static scattering contrast, can be visualized by space-resolved measurements of the time autocorrelation function g2(τ) of the scattered light intensity at the sample surface. Maximum contrast occurs at a parameter-dependent finite correlation time τ. The physical origin of this effect is outlined. Our data are in excellent quantitative agreement with the model, with no adjustable parameter.

© 1997 Optical Society of America

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  1. G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
    [CrossRef]
  2. D. A. Weitz, D. J. Pine, P. N. Pusey, and R. J. A. Tough, “Nondiffusive Brownian motion studied by diffusing-wave spectroscopy,” Phys. Rev. Lett. 63, 1747–1750 (1989).
    [CrossRef] [PubMed]
  3. X. L. Wu, D. J. Pine, P. M. Chaikin, J. S. Huang, and D. A. Weitz, “Diffusing-wave spectroscopy in a shear flow,” J. Opt. Soc. Am. B 7, 15–20 (1990).
    [CrossRef]
  4. D. Bicout and G. Maret, “Multiple light scattering in Taylor–Couette flow,” Physica A 210, 87–112 (1994).
    [CrossRef]
  5. W. Leutz and G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B 204, 14–19 (1995).
    [CrossRef]
  6. D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
    [CrossRef] [PubMed]
  7. P. N. den Outer, Th. M. Nieuwenhuizen, and A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
    [CrossRef]
  8. D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
    [CrossRef]
  9. D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
    [CrossRef] [PubMed]
  10. M. Heckmeier and G. Maret, “Visualization of flow in multiple scattering liquids,” Europhys. Lett. 34, 257–262 (1996).
    [CrossRef]
  11. P. M. Chaikin, D. J. Pine, D. A. Weitz, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
    [CrossRef]
  12. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), pp. 175–190.
  13. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  14. M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
    [CrossRef]
  15. D. Bicout, E. Akkermans, and R. Maynard, “Dynamical correlations for multiple light scattering in laminar flow,” J. Phys. (Paris) I 1, 471–491 (1991).
  16. D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, and P. M. Chaikin, “Dynamical correlations of multiply scattered light,” in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, London, 1989), pp. 312–372.
  17. J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), pp. 140–144.
  18. R. C. Haskell, L. V. Swaasand, T. Tsay, T. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  19. B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), p. 40.
  20. The laser beam is incident with an angle of approximately 10° with respect to the backscattering direction. For different setups this angle changes slightly; hence the illuminated regions are not identical, and one obtains somewhat different γ values.
  21. However, we cannot exclude additional small deviations between data and theory that are due to neglect of higher-order scattering of the diffusing waves.

1996 (1)

M. Heckmeier and G. Maret, “Visualization of flow in multiple scattering liquids,” Europhys. Lett. 34, 257–262 (1996).
[CrossRef]

1995 (2)

W. Leutz and G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B 204, 14–19 (1995).
[CrossRef]

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

1994 (3)

R. C. Haskell, L. V. Swaasand, T. Tsay, T. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[CrossRef]

D. Bicout and G. Maret, “Multiple light scattering in Taylor–Couette flow,” Physica A 210, 87–112 (1994).
[CrossRef]

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

1993 (1)

1991 (2)

D. Bicout, E. Akkermans, and R. Maynard, “Dynamical correlations for multiple light scattering in laminar flow,” J. Phys. (Paris) I 1, 471–491 (1991).

D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

1990 (1)

1989 (2)

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

D. A. Weitz, D. J. Pine, P. N. Pusey, and R. J. A. Tough, “Nondiffusive Brownian motion studied by diffusing-wave spectroscopy,” Phys. Rev. Lett. 63, 1747–1750 (1989).
[CrossRef] [PubMed]

1988 (2)

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
[CrossRef]

P. M. Chaikin, D. J. Pine, D. A. Weitz, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef]

1987 (1)

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

Akkermans, E.

D. Bicout, E. Akkermans, and R. Maynard, “Dynamical correlations for multiple light scattering in laminar flow,” J. Phys. (Paris) I 1, 471–491 (1991).

Bicout, D.

D. Bicout and G. Maret, “Multiple light scattering in Taylor–Couette flow,” Physica A 210, 87–112 (1994).
[CrossRef]

D. Bicout, E. Akkermans, and R. Maynard, “Dynamical correlations for multiple light scattering in laminar flow,” J. Phys. (Paris) I 1, 471–491 (1991).

Boas, D. A.

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

Campbell, L. E.

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

Chaikin, P. M.

X. L. Wu, D. J. Pine, P. M. Chaikin, J. S. Huang, and D. A. Weitz, “Diffusing-wave spectroscopy in a shear flow,” J. Opt. Soc. Am. B 7, 15–20 (1990).
[CrossRef]

P. M. Chaikin, D. J. Pine, D. A. Weitz, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef]

Chance, B.

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

Durian, D. J.

D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

Feng, T.

Haskell, R. C.

Heckmeier, M.

M. Heckmeier and G. Maret, “Visualization of flow in multiple scattering liquids,” Europhys. Lett. 34, 257–262 (1996).
[CrossRef]

Herbolzheimer, E.

P. M. Chaikin, D. J. Pine, D. A. Weitz, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef]

Huang, J. S.

Lagendijk, A.

Leutz, W.

W. Leutz and G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B 204, 14–19 (1995).
[CrossRef]

Maret, G.

M. Heckmeier and G. Maret, “Visualization of flow in multiple scattering liquids,” Europhys. Lett. 34, 257–262 (1996).
[CrossRef]

W. Leutz and G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B 204, 14–19 (1995).
[CrossRef]

D. Bicout and G. Maret, “Multiple light scattering in Taylor–Couette flow,” Physica A 210, 87–112 (1994).
[CrossRef]

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

Maynard, R.

D. Bicout, E. Akkermans, and R. Maynard, “Dynamical correlations for multiple light scattering in laminar flow,” J. Phys. (Paris) I 1, 471–491 (1991).

McAdams, M. S.

Nieuwenhuizen, Th. M.

O'Leary, M. A.

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

Outer, P. N. den

Patterson, M. S.

Pine, D. J.

D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

X. L. Wu, D. J. Pine, P. M. Chaikin, J. S. Huang, and D. A. Weitz, “Diffusing-wave spectroscopy in a shear flow,” J. Opt. Soc. Am. B 7, 15–20 (1990).
[CrossRef]

D. A. Weitz, D. J. Pine, P. N. Pusey, and R. J. A. Tough, “Nondiffusive Brownian motion studied by diffusing-wave spectroscopy,” Phys. Rev. Lett. 63, 1747–1750 (1989).
[CrossRef] [PubMed]

P. M. Chaikin, D. J. Pine, D. A. Weitz, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef]

Pusey, P. N.

D. A. Weitz, D. J. Pine, P. N. Pusey, and R. J. A. Tough, “Nondiffusive Brownian motion studied by diffusing-wave spectroscopy,” Phys. Rev. Lett. 63, 1747–1750 (1989).
[CrossRef] [PubMed]

Stephen, M. J.

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
[CrossRef]

Swaasand, L. V.

Tough, R. J. A.

D. A. Weitz, D. J. Pine, P. N. Pusey, and R. J. A. Tough, “Nondiffusive Brownian motion studied by diffusing-wave spectroscopy,” Phys. Rev. Lett. 63, 1747–1750 (1989).
[CrossRef] [PubMed]

Tromberg, B. J.

Tsay, T.

Weitz, D. A.

D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

X. L. Wu, D. J. Pine, P. M. Chaikin, J. S. Huang, and D. A. Weitz, “Diffusing-wave spectroscopy in a shear flow,” J. Opt. Soc. Am. B 7, 15–20 (1990).
[CrossRef]

D. A. Weitz, D. J. Pine, P. N. Pusey, and R. J. A. Tough, “Nondiffusive Brownian motion studied by diffusing-wave spectroscopy,” Phys. Rev. Lett. 63, 1747–1750 (1989).
[CrossRef] [PubMed]

P. M. Chaikin, D. J. Pine, D. A. Weitz, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef]

Wilson, B. C.

Wolf, P. E.

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

Wu, X. L.

Yodh, A. G.

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

Appl. Opt. (1)

Europhys. Lett. (1)

M. Heckmeier and G. Maret, “Visualization of flow in multiple scattering liquids,” Europhys. Lett. 34, 257–262 (1996).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

J. Phys. (Paris) I (1)

D. Bicout, E. Akkermans, and R. Maynard, “Dynamical correlations for multiple light scattering in laminar flow,” J. Phys. (Paris) I 1, 471–491 (1991).

Phys. Rev. B (1)

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
[CrossRef]

Phys. Rev. Lett. (3)

D. A. Weitz, D. J. Pine, P. N. Pusey, and R. J. A. Tough, “Nondiffusive Brownian motion studied by diffusing-wave spectroscopy,” Phys. Rev. Lett. 63, 1747–1750 (1989).
[CrossRef] [PubMed]

P. M. Chaikin, D. J. Pine, D. A. Weitz, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef]

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

Physica A (1)

D. Bicout and G. Maret, “Multiple light scattering in Taylor–Couette flow,” Physica A 210, 87–112 (1994).
[CrossRef]

Physica B (1)

W. Leutz and G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B 204, 14–19 (1995).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

Science (1)

D. J. Durian, D. A. Weitz, and D. J. Pine, “Multiple light-scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

Z. Phys. B (1)

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

Other (6)

B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), p. 40.

The laser beam is incident with an angle of approximately 10° with respect to the backscattering direction. For different setups this angle changes slightly; hence the illuminated regions are not identical, and one obtains somewhat different γ values.

However, we cannot exclude additional small deviations between data and theory that are due to neglect of higher-order scattering of the diffusing waves.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), pp. 175–190.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, and P. M. Chaikin, “Dynamical correlations of multiply scattered light,” in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, London, 1989), pp. 312–372.

J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), pp. 140–144.

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Figures (8)

Fig. 1
Fig. 1

Theoretical time correlation functions for cylindrical (solid curves) and planar (dashed curves) Poiseuille flows embedded in a semi-infinite homogeneous sample. The lower, middle, and upper pairs of curves correspond to depths 3l*, 5l*, and 7l*, respectively. y=0, d=22l*, τ0in=τ0out=2.66×10-4 s, τf=4.6×10-6 s, and γ=2.8. The dotted line corresponds to a homogeneous semi-infinite medium. The inset shows the maximum absolute differences between the correlation functions with flow and with no flow (dotted line) as a function of flow depth (x).

Fig. 2
Fig. 2

Sketch of the light-scattering cell for realization of dynamic heterogeneities inside a turbid liquid. A large cell (5 cm × 4 cm × 2 cm) is completely filled with a concentrated colloidal suspension. The included cylinder consists of an x-ray capillary (optical glass; length: 3 cm, diameter d=1.5 mm, wall thickness: 0.01 mm). A tube connected to the capillary delivers suspension from an elevated tank at flow rates controlled by its height. For the Brownian inclusions the capillary is filled with the corresponding suspensions. x=0: sample surface, y=0: center of the capillary.

Fig. 3
Fig. 3

Experimental time correlation functions for various distances x of the capillary surface from the inner surface of the sample cell [x=2.8l* (○), 4.2l * (⋄), 5.7l * (△), 7.1l * (□); y=0] and Q=0.50 ml/s compared with the case of no flow (+). The inset shows the maximum difference Δg of these correlation functions with respect to the Brownian case as a function of the x position of the capillary. The theoretical predictions are indicated as solid curves.

Fig. 4
Fig. 4

Time correlation functions for backscattering from the suspension (l*=69 µm, D=3.55×10-12 m2/s), without included heterogeneity for different laser-beam sizes. The beam waist is roughly equal to 1 mm (○); the expanded beam diameter is roughly equal to 5 mm (+).

Fig. 5
Fig. 5

Time correlation functions for different flow rates Q. The capillary is placed at x=2.8l* inside the cell and is centered with respect to the incoming laser beam [y=0; Q=0.90 ml/s (○), Q=0.50 ml/s (⋄), Q=0.22 ml/s (□), no flow (△)]. The theoretical predictions are indicated as solid curves.

Fig. 6
Fig. 6

For a fixed x position of 7.1l * inside the cell, Δg is shown as a function of the y position of the flow (Q=0.50 ml/s). The error bars are due to uncertainty in determining Δg; the width of the capillary is indicated by the horizontal line. The solid curve denotes our theoretical prediction.

Fig. 7
Fig. 7

Time correlation functions for different positions of the Brownian heterogeneity [DBin=2.09×10-13 m2/s; x=2.9l* (○), 5.22l * (⋄); y=2.9l*] compared with the homogeneous case without inclusion (+). The inset shows Δg of these correlation functions as a function of the x position of the capillary. The theoretical predictions are indicated as solid curves.

Fig. 8
Fig. 8

Time correlation functions for different Brownian heterogeneities. The capillary is placed at x=3.77l*, y=2.9l*, inside the cell [DBin=2.09×10-13 m2/s (○), DBin=6.08×10-13 m2/s (△), DBin=DBout=3.55×10-12 m2/s (+)]. Inset: For a fixed x position of 3.77l * inside the cell, Δg is shown as a function of the y position for these heterogeneities. The solid curve denotes the theoretical prediction, and the width of the capillary is indicated by the horizontal line.

Equations (11)

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[2-k2(τ)]G1(r, τ)=-S(r)Dp,
G1out(r, τ)-(2/3)l*[n·G1out(r, τ)]=0,rS,
G1in(r, τ)=G1out(r, τ),
Dpin[n·G1in(r, τ)]=Dpout[n·G1out(r, τ)],
rS1.
g10(τ)=exp-γ3τ2τ0out,
g1(τ)=F(ξ1)F(ξ2),
F(ξ)=kinkout exp(koutξ)+tanh(kind)×[kout2 cosh(koutξ)+kin2 sinh(koutξ)]
g1scatt(y, τ)
=-h-x02πl*n=1-θ1θ1 dθscos θsKnkout h-x0cos θs×Kn(kouth2+y2)cos[n(θs-θ)]×koutIn(kouta)In(kina)-kinIn(kout a)In(kina)koutKn(kouta)In(kina)-kinKn(kouta)In(kina).
Δg=max0<τ<|g1(τ)-g10(τ)|,

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