Abstract

We show that spatially resolved backscattering can be used for simultaneous measurements of static and dynamic properties of highly turbid media. The spatial variation of the backscattered intensity gives access to the transport mean free path. The decay of the temporal intensity-intensity correlation function depends on the point of observation. This property can be used to probe complex dynamics with several time scales. The implementation of the method and the data analysis are tested on concentrated suspensions of polystyrene spheres.

© 2002 Optical Society of America

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  1. D. A. Weitz, D. Pine, “Diffusing-wave spectroscopy,” in Dynamic Light Scattering, W. Brown, ed. (Oxford U. Press, Oxford, 1992), pp. 652–720.
  2. G. Maret, “Diffusive-wave spectroscopy,” Curr. Opin. Coll. Int. Sci. 2, 251–257 (1997).
    [CrossRef]
  3. T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
    [CrossRef] [PubMed]
  4. R. A. J. Groenhuis, H. A. Ferwerda, J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1. Theory,” Appl. Opt. 22, 2456–2462 (1983).
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    [CrossRef] [PubMed]
  6. D. J. Durian, J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source,” J. Opt. Soc. Am. A 16, 837–844 (1999).
    [CrossRef]
  7. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, 1959).
  8. J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
    [CrossRef] [PubMed]
  9. M. U. Vera, D. J. Durian, “The angular distribution of diffusely backscattered light,” Phys. Rev. E 56, 3215–3224 (1996).
    [CrossRef]
  10. D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, “Dynamical correlations of multiply scattered light,” in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed., Vol. 8 of World Scientific Series in Condensed Matter Physics (World Scientific, Singapore, 1990), pp. 312–372.
  11. R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
    [CrossRef]
  12. D. A. Boas, L. E. Campbell, A. G. Yodh, “Scattering and imaging with diffusing temporal field correlation functions,” Phys. Rev. Lett. 75, 1855–1858 (1995).
    [CrossRef] [PubMed]

1999

1997

G. Maret, “Diffusive-wave spectroscopy,” Curr. Opin. Coll. Int. Sci. 2, 251–257 (1997).
[CrossRef]

1996

M. U. Vera, D. J. Durian, “The angular distribution of diffusely backscattered light,” Phys. Rev. E 56, 3215–3224 (1996).
[CrossRef]

1995

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

1994

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

1992

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

1991

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

1983

Ackerson, B. J.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Boas, D. A.

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

Campbell, L. E.

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

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, 1959).

Chaikin, P. M.

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

Dorr-Nowkoorani, F.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Dougherty, R. L.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Durian, D. J.

Farrell, T. J.

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Ferwerda, H. A.

Groenhuis, R. A. J.

Herbolzheimer, E.

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

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, 1959).

Maret, G.

G. Maret, “Diffusive-wave spectroscopy,” Curr. Opin. Coll. Int. Sci. 2, 251–257 (1997).
[CrossRef]

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

Nobbmann, U.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Patterson, M. S.

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Pine, D.

D. A. Weitz, D. Pine, “Diffusing-wave spectroscopy,” in Dynamic Light Scattering, W. Brown, ed. (Oxford U. Press, Oxford, 1992), pp. 652–720.

Pine, D. J.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

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

Reguigui, N. M.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Rudnick, J.

Ten Bosch, J. J.

Vera, M. U.

M. U. Vera, D. J. Durian, “The angular distribution of diffusely backscattered light,” Phys. Rev. E 56, 3215–3224 (1996).
[CrossRef]

Weitz, D. A.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

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

D. A. Weitz, D. Pine, “Diffusing-wave spectroscopy,” in Dynamic Light Scattering, W. Brown, ed. (Oxford U. Press, Oxford, 1992), pp. 652–720.

Wilson, B.

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Wolf, P. E.

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

Yodh, A. G.

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

Zhu, J. X.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Appl. Opt.

Curr. Opin. Coll. Int. Sci.

G. Maret, “Diffusive-wave spectroscopy,” Curr. Opin. Coll. Int. Sci. 2, 251–257 (1997).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Med. Phys.

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Phys. Rev. A

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Phys. Rev. E

M. U. Vera, D. J. Durian, “The angular distribution of diffusely backscattered light,” Phys. Rev. E 56, 3215–3224 (1996).
[CrossRef]

Phys. Rev. Lett.

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

Other

D. A. Weitz, D. Pine, “Diffusing-wave spectroscopy,” in Dynamic Light Scattering, W. Brown, ed. (Oxford U. Press, Oxford, 1992), pp. 652–720.

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

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, 1959).

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Figures (6)

Fig. 1
Fig. 1

Geometry for calculation by the images method of the static and dynamic properties of a semi-infinite homogeneous slab.

Fig. 2
Fig. 2

Experimental setup for spatially resolved DWS in the backscattering geometry. PMT, photomultiplier tube.

Fig. 3
Fig. 3

Variations versus illumination-detection distance of the energy density (experimental count rate) backscattered from suspensions of polystyrene spheres at the following volume fractions: circles, Φ = 9.5 × 10-3; triangles, Φ = 4.75 × 10-3; diamonds, Φ = 2.4 × 10-3; squares, Φ = 9.5 × 10-4.

Fig. 4
Fig. 4

Intensity-intensity correlation functions measured at several distances from the illumination point: from left to right, ρ = 5, ρ = 4, ρ = 3, ρ = 2.5, ρ = 2, ρ = 1.5, and ρ = 1 mm. Solid curves, results of the fitting procedure of the experimental data with Eq. (9).

Fig. 5
Fig. 5

Same curves as in Fig. 3 after proper rescaling on a single master curve. The resultant values of the transport mean free path are the following: circles, l* = 0.23 mm for Φ = 9.5 × 10-3; triangles, l* = 0.46 mm for Φ = 4.75 × 10-3; diamonds, l* = 0.92 mm for Φ = 2.4 × 10-3; and squares, l* = 2.3 mm for Φ = 9.5 × 10-4. Dotted curve, prediction of the Monte Carlo simulations.6 Continuous curve, prediction from the images method. Note: The unit has been suppressed in the final version.

Fig. 6
Fig. 6

Variations of the reduced diffusion coefficient measured for polystyrene spheres as a function of the dimensionless illumination-detection distance.

Equations (11)

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

Ut=DP2U,
Uρ, z, t=Q04πDPt3/2exp-r124DPt-exp-r224DPt,
Pρ, s=Q0163πl*3/2s-5/2zP exp-3zP2+ρ24l*s+zP+2zeexp-3zP+2ze2+ρ24l*s.
Iρ=Q04πzp2ρ2+zp231/2+zp+2ze2ρ2+zp+2ze231/2.
Iρ= Pρ, sexp-slads,
Iρ=Q04πzp1+μρ1ρ13exp-μρ1+zp+2ze1+μρ2ρ23exp-μρ2,
g1ρ, t= Pρ, sexp-k023Δr2tsl*×exp-slads,
sla  k023Δr2tsl*+sla.
g1t=zp1+ρ1xexp-ρ1x/ρ13+zp+2ze1+ρ2xexp-ρ2x/ρ23zp/ρ13+zp+2ze/ρ23.
Iρ/Q0=ρzp+ze2πρ32πρdρ=zp+ze/ρ,
g1t=Q0Iρρ1+ρxexp-ρx×zp+ze2πρ32πρdρ =exp-ρx.

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