Abstract

Dynamic light-scattering spectroscopy is used to study Brownian motion within highly scattering samples. The fluctuations of the light field that is backscattered by a suspension of polystyrene microspheres are measured as power spectra by use of low-coherence interferometry to obtain path-length resolution. The data are modeled as the sum of contributions to the detected light weighted by a Poisson probability for the number of events that each component has experienced. By analyzing the broadening of the power spectra as a function of the path length for various sizes of particles, we determine the contribution of multiple scattering to the detected signal as a function of scattering anisotropy.

© 2001 Optical Society of America

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References

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  1. P. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).
  2. R. Pecora, Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy (Plenum, New York, 1985).
    [CrossRef]
  3. G. Maret, P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
    [CrossRef]
  4. D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
    [CrossRef] [PubMed]
  5. P.-A. Lemieux, M. U. Vera, 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]
  6. D. J. Durian, “Accuracy of diffusing-wave spectroscopy theories,” Phys. Rev. E 51, 3350–3358 (1995).
    [CrossRef]
  7. M. Kempe, A. Z. Genack, W. R. Dorn, P. Dorn, “Ballistic and diffuse light detection in confocal and heterodyne imaging systems,” J. Opt. Soc. Am. A 14, 216–223 (1997).
    [CrossRef]
  8. M. J. Yadlowsky, J. M. Schmitt, R. F. Bonner, “Multiple scattering in optical coherence microscopy,” Appl. Opt. 34, 5699–5707 (1995).
    [CrossRef] [PubMed]
  9. K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: the transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
    [CrossRef]
  10. D. A. Boas, K. K. Bizheva, A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion within highly scattering media,” Opt. Lett. 23, 319–321 (1998).
    [CrossRef]
  11. C. Scott, Introduction to Optics and Optical Imaging (Institute of Electrical and Electronic Engineers, Piscataway, N. J., 1998), p. 310.
  12. A. G. Yodh, P. D. Kaplan, D. J. Pine, “Pulsed diffusing-wave spectroscopy: high resolution through nonlinear optical gating,” Phys. Rev. B 42, 4744–4747 (1990).
    [CrossRef]
  13. R. Bonner, R. Nossal, “Model for laser Doppler measurements of blood flow in tissue,” Appl. Opt. 20, 2097–2107 (1981).
    [CrossRef] [PubMed]
  14. A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
    [CrossRef]
  15. G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
    [CrossRef] [PubMed]
  16. S. Fraden, G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
    [CrossRef] [PubMed]

1999

G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef] [PubMed]

1998

P.-A. Lemieux, M. U. Vera, 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]

K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: the transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
[CrossRef]

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

1997

1995

1992

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

1990

S. Fraden, G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[CrossRef] [PubMed]

A. G. Yodh, P. D. Kaplan, D. J. Pine, “Pulsed diffusing-wave spectroscopy: high resolution through nonlinear optical gating,” Phys. Rev. B 42, 4744–4747 (1990).
[CrossRef]

1988

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

1987

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

1981

Berne, P. J.

P. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Bizheva, K. K.

K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: the transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
[CrossRef]

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

Boas, D. A.

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

K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: the transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
[CrossRef]

Bonner, R.

Bonner, R. F.

M. J. Yadlowsky, J. M. Schmitt, R. F. Bonner, “Multiple scattering in optical coherence microscopy,” Appl. Opt. 34, 5699–5707 (1995).
[CrossRef] [PubMed]

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

Chaikin, P. M.

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

Dorn, P.

Dorn, W. R.

Durian, D. J.

P.-A. Lemieux, M. U. Vera, 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]

D. J. Durian, “Accuracy of diffusing-wave spectroscopy theories,” Phys. Rev. E 51, 3350–3358 (1995).
[CrossRef]

Fraden, S.

S. Fraden, G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[CrossRef] [PubMed]

Gandjbakhche, A. H.

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

Genack, A. Z.

Herbolzheimer, E.

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

Kaplan, P. D.

A. G. Yodh, P. D. Kaplan, D. J. Pine, “Pulsed diffusing-wave spectroscopy: high resolution through nonlinear optical gating,” Phys. Rev. B 42, 4744–4747 (1990).
[CrossRef]

Kempe, M.

Lemieux, P.-A.

P.-A. Lemieux, M. U. Vera, 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.

S. Fraden, G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[CrossRef] [PubMed]

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

Nossal, R.

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

R. Bonner, R. Nossal, “Model for laser Doppler measurements of blood flow in tissue,” Appl. Opt. 20, 2097–2107 (1981).
[CrossRef] [PubMed]

Pecora, R.

R. Pecora, Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy (Plenum, New York, 1985).
[CrossRef]

P. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Pine, D. J.

A. G. Yodh, P. D. Kaplan, D. J. Pine, “Pulsed diffusing-wave spectroscopy: high resolution through nonlinear optical gating,” Phys. Rev. B 42, 4744–4747 (1990).
[CrossRef]

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

Schmitt, J. M.

Scott, C.

C. Scott, Introduction to Optics and Optical Imaging (Institute of Electrical and Electronic Engineers, Piscataway, N. J., 1998), p. 310.

Siegel, A. M.

K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: the transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
[CrossRef]

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

Vera, M. U.

P.-A. Lemieux, M. U. Vera, 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. V.

G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef] [PubMed]

Weitz, D. A.

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

Wolf, P. E.

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

Yadlowsky, M. J.

Yao, G.

G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef] [PubMed]

Yodh, A. G.

A. G. Yodh, P. D. Kaplan, D. J. Pine, “Pulsed diffusing-wave spectroscopy: high resolution through nonlinear optical gating,” Phys. Rev. B 42, 4744–4747 (1990).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

J. Stat. Phys.

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

Opt. Lett.

Phys. Med. Biol.

G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef] [PubMed]

Phys. Rev. B

A. G. Yodh, P. D. Kaplan, D. J. Pine, “Pulsed diffusing-wave spectroscopy: high resolution through nonlinear optical gating,” Phys. Rev. B 42, 4744–4747 (1990).
[CrossRef]

Phys. Rev. E

P.-A. Lemieux, M. U. Vera, 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]

D. J. Durian, “Accuracy of diffusing-wave spectroscopy theories,” Phys. Rev. E 51, 3350–3358 (1995).
[CrossRef]

K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: the transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
[CrossRef]

Phys. Rev. Lett.

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

S. Fraden, G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[CrossRef] [PubMed]

Z. Phys. B

G. Maret, 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

P. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

R. Pecora, Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy (Plenum, New York, 1985).
[CrossRef]

C. Scott, Introduction to Optics and Optical Imaging (Institute of Electrical and Electronic Engineers, Piscataway, N. J., 1998), p. 310.

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Figures (4)

Fig. 1
Fig. 1

Schematic of the experimental setup: BSs, beam splitters; M, mirror; ADC, analog-to-digital converter; SLD, superluminescent diode.

Fig. 2
Fig. 2

Typical frequency spectra for 258-nm microspheres at the photon path lengths indicated. MFPs, mean free paths.

Fig. 3
Fig. 3

(a), (b) Fitted linewidths and (c) amplitudes of the Lorentzian power spectrum measured as a function of the photon path length for 258-nm microspheres. Data (points) are shown compared with (a) single-scattering (dashed line) and diffusion-theory (solid line) predictions and (b), (c) the theoretical predictions (solid curves) presented in this paper.

Fig. 4
Fig. 4

Trends in the empirical parameter α compared with (a) scattering anisotropy and (b) particle diameter. The points represent the measured data; the solid curves represent the theory.

Tables (1)

Tables Icon

Table 1 Properties of Microspheres

Equations (12)

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Sf=1f0A1+f/f02,
πf0=2k2DB
DB=κBT/3πηa,
πf0=k2DBLl1-g,
Fv  exp-v2/vth2.
Fnv  exp-v2/nvth2.
DB  kBT  vth2,
Snf=1nf0A1+f/nf02,
Pnα, L=exp-αL/lαL/lnn!,
STf, α, L=exp-βL/ln=1 Pnα, LSnf.
gτ=0 PSexp-xS/3dS,
Sf=0 PSSf, f0dS,

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