Abstract

Previous systems for measuring cross-correlated light scattering by small particles suspended in a liquid with multiple-scattering suppression have illuminated the particles with two laser beams. It is shown that multiple-scattering suppression should also occur in cross correlation for a system that employs a single laser beam and two closely spaced detectors with wide fields of view. The single-scattering, double-scattering, and single–double-scattering cross-term contributions to the intensity cross-correlation function are calculated. It is found that the two cross terms, when added together, are unimportant for both autocorrelation and cross correlation. The amplitude of the double-scattering term can be greatly diminished by judicious detector spacing because the spatial coherence area in the detector plane for double scattering is much smaller than that for single scattering.

© 1997 Optical Society of America

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    [CrossRef]
  7. A. E. Bailey, D. S. Cannell, “Practical method for calculation of multiple light scattering,” Phys. Rev. E 50, 4853–4864 (1994).
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    [CrossRef]
  30. J. A. Lock, “Theory of multiple scattering suppression in cross-correlated light scattering employing a single laser beam,” in Light Scattering and Photon Correlation Spectroscopy, E. R. Pike, J. B. Abbiss, eds., NATO ASI Series (Kluwer, Dordrecht, The Netherlands, 1997).
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    [CrossRef] [PubMed]
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  36. H. M. J. Boots, D. Bedeaux, P. Mazur, “On the theory of multiple scattering. I.,” Physica 79A, 397–419 (1975).
    [CrossRef]
  37. C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Autocorrelation spectroscopy studies of single and multiple scattered light from a critical mixture,” Phys. Rev. A 16, 365–376 (1977).
    [CrossRef]
  38. R. A. Farrell, J. K. Bhattacharjee, “Double-scattering correction for the critical dynamics of a classical fluid,” Phys. Rev. A 19, 348–369 (1979).
    [CrossRef]
  39. D. W. Oxtoby, W. M. Gelbart, “Double-scattering-induced deviations from Ornstein–Zernike behavior near the critical point,” Phys. Rev. A 10, 738–740 (1974).
    [CrossRef]
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    [CrossRef]

1997

1995

1994

Superano, K. Deurloo, P. Stamatelopolous, R. Srivastva, J. C. Thomas, “Light scattering with single-mode fiber collimators,” Appl. Opt. 33, 7200–7205 (1994).
[CrossRef]

A. E. Bailey, D. S. Cannell, “Practical method for calculation of multiple light scattering,” Phys. Rev. E 50, 4853–4864 (1994).
[CrossRef]

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

1993

1992

B. J. Ackerson, R. L. Dougherty, N. M. Reguigui, U. Nobbmann, “Correlation transfer: application of radiative transfer solution methods to photon correlation problems,” J. Thermophys. Heat Transfer 6, 577–588 (1992).
[CrossRef]

1991

H. Wiese, D. Horn, “Single-mode fibers in fiber-optic quasielastic light scattering: a study of the dynamics of concentrated latex dispersions,” J. Chem. Phys. 94, 6429–6443 (1991).
[CrossRef]

K. Schätzel, “Suppression of multiple scattering by photon cross-correlation techniques,” J. Mod. Opt. 38, 1849–1865 (1991).
[CrossRef]

1990

M. Drewel, J. Ahrens, U. Podschus, “Decorrelation of multiple scattering for an arbitrary scattering angle,” J. Opt. Soc. Am. A 7, 206–210 (1990).
[CrossRef]

K. Schätzel, M. Drewel, J. Ahrens, “Suppression of multiple scattering in photon correlation spectroscopy,” J. Phys.: Condensed Matter 2, SA393–SA398 (1990).

1989

F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

H. S. Dhadwal, B. Chu, “A fiber-optic light-scattering spectrometer,” Rev. Sci. Instrum. 50, 845–853 (1989).
[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]

J. G. Shanks, J. V. Sengers, “Double scattering in critically opalescent fluids,” Phys. Rev. A 38, 885–896 (1988).
[CrossRef] [PubMed]

1987

1986

H. J. Mos, C. Pathmamanoharan, J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. II. Experimental,” J. Chem. Phys. 84, 45–49 (1986).
[CrossRef]

1985

J. K. G. Dhont, “Multiple Rayleigh–Gans–Debye scattering in colloidal systems: dynamic light scattering,” Physica 129A, 374–394 (1985).
[CrossRef]

J. K. G. Dhont, C. G. deKruif, A. Vrij, “Light scattering in colloidal suspensions: effects of multiple scattering,” J. Colloid Interface Sci. 105, 539–551 (1985).
[CrossRef]

1983

J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross-correlation. I. Theory,” J. Chem. Phys. 79, 1658–1663 (1983).
[CrossRef]

1981

G. D. J. Phillies, “Suppression of multiple scattering effects in quasielastic light scattering by homodyne cross-correlation techniques,” J. Chem. Phys. 74, 260–262 (1981).
[CrossRef]

G. D. J. Phillies, “Experimental demonstration of multiple-scattering suppression in quasielastic light scattering spectroscopy by homodyne coincidence techniques,” Phys. Rev. A 24, 1939–1943 (1981).
[CrossRef]

1979

R. A. Farrell, J. K. Bhattacharjee, “Double-scattering correction for the critical dynamics of a classical fluid,” Phys. Rev. A 19, 348–369 (1979).
[CrossRef]

1978

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Multiple scattering from a system of Brownian particles,” Phys. Rev. A 17, 2030–2035 (1978).
[CrossRef]

1977

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Autocorrelation spectroscopy studies of single and multiple scattered light from a critical mixture,” Phys. Rev. A 16, 365–376 (1977).
[CrossRef]

1976

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Depolarized correlation function of light double scattered from a system of Brownian particles,” Phys. Rev. A 14, 1520–1532 (1976).
[CrossRef]

1975

H. M. J. Boots, D. Bedeaux, P. Mazur, “On the theory of multiple scattering. I.,” Physica 79A, 397–419 (1975).
[CrossRef]

1974

D. W. Oxtoby, W. M. Gelbart, “Double-scattering-induced deviations from Ornstein–Zernike behavior near the critical point,” Phys. Rev. A 10, 738–740 (1974).
[CrossRef]

1970

N. A. Clark, J. H. Lunacek, G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38, 575–585 (1970).
[CrossRef]

1969

Ackerson, B. J.

U. Nobbmann, S. W. Jones, B. J. Ackerson, “Multiple-scattering suppression: cross correlation with tilted single-mode fibers,” Appl. Opt. 36, 7571–7576 (1997).
[CrossRef]

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

B. J. Ackerson, R. L. Dougherty, N. M. Reguigui, U. Nobbmann, “Correlation transfer: application of radiative transfer solution methods to photon correlation problems,” J. Thermophys. Heat Transfer 6, 577–588 (1992).
[CrossRef]

Ahrens, J.

M. Drewel, J. Ahrens, U. Podschus, “Decorrelation of multiple scattering for an arbitrary scattering angle,” J. Opt. Soc. Am. A 7, 206–210 (1990).
[CrossRef]

K. Schätzel, M. Drewel, J. Ahrens, “Suppression of multiple scattering in photon correlation spectroscopy,” J. Phys.: Condensed Matter 2, SA393–SA398 (1990).

Bailey, A. E.

A. E. Bailey, D. S. Cannell, “Practical method for calculation of multiple light scattering,” Phys. Rev. E 50, 4853–4864 (1994).
[CrossRef]

Bedeaux, D.

H. M. J. Boots, D. Bedeaux, P. Mazur, “On the theory of multiple scattering. I.,” Physica 79A, 397–419 (1975).
[CrossRef]

Benedek, G. B.

N. A. Clark, J. H. Lunacek, G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38, 575–585 (1970).
[CrossRef]

Bhattacharjee, J. K.

R. A. Farrell, J. K. Bhattacharjee, “Double-scattering correction for the critical dynamics of a classical fluid,” Phys. Rev. A 19, 348–369 (1979).
[CrossRef]

Boots, H. M. J.

H. M. J. Boots, D. Bedeaux, P. Mazur, “On the theory of multiple scattering. I.,” Physica 79A, 397–419 (1975).
[CrossRef]

Brown, R. G.

Cannell, D. S.

W. V. Meyer, D. S. Cannell, A. E. Smart, T. W. Taylor, P. Tin “Multiple-scattering suppression by cross correlation,” Appl. Opt. 36, 7551–7558 (1997).
[CrossRef]

A. E. Bailey, D. S. Cannell, “Practical method for calculation of multiple light scattering,” Phys. Rev. E 50, 4853–4864 (1994).
[CrossRef]

W. V. Meyer, D. S. Cannell, A. Smart, T. W. Taylor, P. Tin, “Suppression of multiple scatterring using a single beam cross-correlation method, in Light Scattering and Photon Correlation Spectroscopy, E. R. Pike, J. B. Abbiss, eds., NATO ASI Series (Kluwer, Dordrecht, The Netherlands, 1997).

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]

P. M. Chaikin, T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge U. Press, Cambridge, 1995), Fig. 2.4.1.
[CrossRef]

Chu, B.

H. S. Dhadwal, B. Chu, “A fiber-optic light-scattering spectrometer,” Rev. Sci. Instrum. 50, 845–853 (1989).
[CrossRef]

Clark, N. A.

N. A. Clark, J. H. Lunacek, G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38, 575–585 (1970).
[CrossRef]

de Kruif, C. G.

H. J. Mos, C. Pathmamanoharan, J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. II. Experimental,” J. Chem. Phys. 84, 45–49 (1986).
[CrossRef]

J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross-correlation. I. Theory,” J. Chem. Phys. 79, 1658–1663 (1983).
[CrossRef]

deKruif, C. G.

J. K. G. Dhont, C. G. deKruif, A. Vrij, “Light scattering in colloidal suspensions: effects of multiple scattering,” J. Colloid Interface Sci. 105, 539–551 (1985).
[CrossRef]

Deurloo, K.

Dhadwal, H. S.

H. S. Dhadwal, B. Chu, “A fiber-optic light-scattering spectrometer,” Rev. Sci. Instrum. 50, 845–853 (1989).
[CrossRef]

Dhont, J. K. G.

H. J. Mos, C. Pathmamanoharan, J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. II. Experimental,” J. Chem. Phys. 84, 45–49 (1986).
[CrossRef]

J. K. G. Dhont, “Multiple Rayleigh–Gans–Debye scattering in colloidal systems: dynamic light scattering,” Physica 129A, 374–394 (1985).
[CrossRef]

J. K. G. Dhont, C. G. deKruif, A. Vrij, “Light scattering in colloidal suspensions: effects of multiple scattering,” J. Colloid Interface Sci. 105, 539–551 (1985).
[CrossRef]

J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross-correlation. I. Theory,” J. Chem. Phys. 79, 1658–1663 (1983).
[CrossRef]

Dorri-Nowkoorani, F.

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

B. J. Ackerson, R. L. Dougherty, N. M. Reguigui, U. Nobbmann, “Correlation transfer: application of radiative transfer solution methods to photon correlation problems,” J. Thermophys. Heat Transfer 6, 577–588 (1992).
[CrossRef]

Drewel, M.

M. Drewel, J. Ahrens, U. Podschus, “Decorrelation of multiple scattering for an arbitrary scattering angle,” J. Opt. Soc. Am. A 7, 206–210 (1990).
[CrossRef]

K. Schätzel, M. Drewel, J. Ahrens, “Suppression of multiple scattering in photon correlation spectroscopy,” J. Phys.: Condensed Matter 2, SA393–SA398 (1990).

Egelhaaf, S. U.

Farrell, R. A.

R. A. Farrell, J. K. Bhattacharjee, “Double-scattering correction for the critical dynamics of a classical fluid,” Phys. Rev. A 19, 348–369 (1979).
[CrossRef]

R. W. Hart, R. A. Farrell, “Light scattering in the cornea,” J. Opt. Soc. Am. 59, 766–774 (1969).
[CrossRef] [PubMed]

Gelbart, W. M.

D. W. Oxtoby, W. M. Gelbart, “Double-scattering-induced deviations from Ornstein–Zernike behavior near the critical point,” Phys. Rev. A 10, 738–740 (1974).
[CrossRef]

Gisler, T.

Hart, R. W.

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), p. 532.

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]

Horn, D.

H. Wiese, D. Horn, “Single-mode fibers in fiber-optic quasielastic light scattering: a study of the dynamics of concentrated latex dispersions,” J. Chem. Phys. 94, 6429–6443 (1991).
[CrossRef]

John, S.

F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

Jones, S. W.

Kao, M. H.

Kaplan, P. D.

Lock, J. A.

J. A. Lock, “Theory of multiple scattering suppression in cross-correlated light scattering employing a single laser beam,” in Light Scattering and Photon Correlation Spectroscopy, E. R. Pike, J. B. Abbiss, eds., NATO ASI Series (Kluwer, Dordrecht, The Netherlands, 1997).

Lubensky, T. C.

P. M. Chaikin, T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge U. Press, Cambridge, 1995), Fig. 2.4.1.
[CrossRef]

Lunacek, J. H.

N. A. Clark, J. H. Lunacek, G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38, 575–585 (1970).
[CrossRef]

MacKintosh, F. C.

F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

Mazur, P.

H. M. J. Boots, D. Bedeaux, P. Mazur, “On the theory of multiple scattering. I.,” Physica 79A, 397–419 (1975).
[CrossRef]

Meyer, W. V.

W. V. Meyer, D. S. Cannell, A. E. Smart, T. W. Taylor, P. Tin “Multiple-scattering suppression by cross correlation,” Appl. Opt. 36, 7551–7558 (1997).
[CrossRef]

W. V. Meyer, D. S. Cannell, A. Smart, T. W. Taylor, P. Tin, “Suppression of multiple scatterring using a single beam cross-correlation method, in Light Scattering and Photon Correlation Spectroscopy, E. R. Pike, J. B. Abbiss, eds., NATO ASI Series (Kluwer, Dordrecht, The Netherlands, 1997).

Mockler, R. C.

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Multiple scattering from a system of Brownian particles,” Phys. Rev. A 17, 2030–2035 (1978).
[CrossRef]

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Autocorrelation spectroscopy studies of single and multiple scattered light from a critical mixture,” Phys. Rev. A 16, 365–376 (1977).
[CrossRef]

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Depolarized correlation function of light double scattered from a system of Brownian particles,” Phys. Rev. A 14, 1520–1532 (1976).
[CrossRef]

Mos, H. J.

H. J. Mos, C. Pathmamanoharan, J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. II. Experimental,” J. Chem. Phys. 84, 45–49 (1986).
[CrossRef]

Nobbmann, U.

U. Nobbmann, S. W. Jones, B. J. Ackerson, “Multiple-scattering suppression: cross correlation with tilted single-mode fibers,” Appl. Opt. 36, 7571–7576 (1997).
[CrossRef]

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

B. J. Ackerson, R. L. Dougherty, N. M. Reguigui, U. Nobbmann, “Correlation transfer: application of radiative transfer solution methods to photon correlation problems,” J. Thermophys. Heat Transfer 6, 577–588 (1992).
[CrossRef]

O’Sullivan, W. J.

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Multiple scattering from a system of Brownian particles,” Phys. Rev. A 17, 2030–2035 (1978).
[CrossRef]

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Autocorrelation spectroscopy studies of single and multiple scattered light from a critical mixture,” Phys. Rev. A 16, 365–376 (1977).
[CrossRef]

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Depolarized correlation function of light double scattered from a system of Brownian particles,” Phys. Rev. A 14, 1520–1532 (1976).
[CrossRef]

Oxtoby, D. W.

D. W. Oxtoby, W. M. Gelbart, “Double-scattering-induced deviations from Ornstein–Zernike behavior near the critical point,” Phys. Rev. A 10, 738–740 (1974).
[CrossRef]

Pathmamanoharan, C.

H. J. Mos, C. Pathmamanoharan, J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. II. Experimental,” J. Chem. Phys. 84, 45–49 (1986).
[CrossRef]

Phillies, G. D. J.

G. D. J. Phillies, “Suppression of multiple scattering effects in quasielastic light scattering by homodyne cross-correlation techniques,” J. Chem. Phys. 74, 260–262 (1981).
[CrossRef]

G. D. J. Phillies, “Experimental demonstration of multiple-scattering suppression in quasielastic light scattering spectroscopy by homodyne coincidence techniques,” Phys. Rev. A 24, 1939–1943 (1981).
[CrossRef]

Pine, D. J.

P. D. Kaplan, M. H. Kao, A. G. Yodh, D. J. Pine, “Geometric constraints for the design of diffusing-wave spectroscopy experiments,” Appl. Opt. 32, 3828–3836 (1993).
[CrossRef] [PubMed]

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

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

Podschus, U.

Reguigui, N. M.

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

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Smart, A. E.

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W. V. Meyer, D. S. Cannell, A. E. Smart, T. W. Taylor, P. Tin “Multiple-scattering suppression by cross correlation,” Appl. Opt. 36, 7551–7558 (1997).
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W. V. Meyer, D. S. Cannell, A. E. Smart, T. W. Taylor, P. Tin “Multiple-scattering suppression by cross correlation,” Appl. Opt. 36, 7551–7558 (1997).
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W. V. Meyer, D. S. Cannell, A. Smart, T. W. Taylor, P. Tin, “Suppression of multiple scatterring using a single beam cross-correlation method, in Light Scattering and Photon Correlation Spectroscopy, E. R. Pike, J. B. Abbiss, eds., NATO ASI Series (Kluwer, Dordrecht, The Netherlands, 1997).

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[CrossRef]

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

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

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Depolarized correlation function of light double scattered from a system of Brownian particles,” Phys. Rev. A 14, 1520–1532 (1976).
[CrossRef]

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

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Figures (5)

Fig. 1
Fig. 1

Focused laser beam B that has the wave vector kinc propagates through a scattering cell of volume V, which contains N particles suspended in a liquid. The field of view of detector d is given by the dashed lines. The origin of coordinates O is at the center of the intersection volume of the beam and the detector field of view.

Fig. 2
Fig. 2

Laser beam B is focused by lens L to half-width Ry. It propagates through a test tube of radius Rz, which contains the liquid and the suspended particles. The test tube is enclosed in a coaxial index-matching vat of radius Rv. Wide-field-of-view detectors α and β with completely overlapping fields of view in the scattering cell are a distance Rd from the center of the vat.

Fig. 3
Fig. 3

Four input–output polarization states for scattering in the xz plane: (a) VV, (b) VH, (c) HV, (d) HH.

Fig. 4
Fig. 4

Single scattering by particle j (a) at time tj when the particle is at position rj, (b) at time τ later when the particle has moved by means of diffusion to somewhere within a sphere of radius Δave (τ) centered on rj.

Fig. 5
Fig. 5

Double scattering by particles j and l (a) at times tj and tl when the particles are at the positions rj and rl, (b) at time τ later when the particles have moved by means of diffusion to somewhere within spheres of radii Δave(τ) centered on rj and rl.

Equations (67)

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Eincrj, tj=E0Brjexp-Lbj/2Lscattuˆinc×expikinc·rj-iωtj,
kkinc=2πnLλ.
EscattR, T=E0BrjAMiekR×exp-Lbj+LjR/2Lscatt×expikR-iωT-iκ·rjtj,
kscatt=kRR,
κ=kscatt-kinc, κ=2k sinθ/2,
limka1 AMie=ARayleigh=n2-1n2+2ka3kˆscatt×uˆinc×kˆscatt.
kˆscattsin θûx+cos θûz.
πfARayleighπi=A1P1,
A1=ka3n2-1/n2+2,
P1=VV1VH1HV1HH1.
VV1=1, VH1=HV1=0, HH1=cos θ.
A2=A12,
P2=VV2VH2HV2HH2,
VV2=1-sin2 θjl sin2 ϕjl, VH2=sin2 θjl sin ϕjl cos ϕjl cos θ-sin θjl cos θjl sin ϕjl sin θ, HV2=-sin2 θjl sin ϕjl cos ϕjl, HH2=cos θ-sin2 θjl cos2 ϕjl cos θ+sin θjl cos θjl cos ϕjl sin θ.
kjl=ksin θjl cos ϕjlûx+sin θjl sin ϕjlûy+cos θjlûz.
Ed1Rd, T=E0A1P1kRdexpikRd-iωTj=1N0BrjDrj×exp-Lbj+Ljd/2Lscatt×exp-iκ·rjtj,
rjtj+τ=rjtj+Δjτ
Ed1Rd, T+τ=E0A1P1kRd×expikR-iωT+τS*κ, τ×j=1N0 BrjDrj×exp-Lbj+Ljd/2Lscatt×exp-iκ·rjtj.
BrjtjBrjtj+τ, DrjtjDrjtj+τ.
Sκ, τd3ΔjPΔj, τexpiκ·Δj.
rjl=rjtj-rltl
E2dRd, T=E0A2kRdexpikRd-iωT×j=1N1l=1N2P2BrjDrlkrjl×exp-Lbj+rjl+Lld/2Lscatt×expikinc-kjl·rjtj×expikjl-kscatt·rltl.
E2dRd, T+τ=E0A2kRdexpikRd-iωT+τ×j=1N1l=1N2P2BrjDrlkrjl×exp-Lbj+rjl+Lld/2Lscatt×expikinc-kjl·rjtj×expikjl-kscatt·rltl×S*kscatt-kjl, τS*kjl-kinc, τ.
DαrDβrDr.
cτ=-dTE*T+τ EβT,
cτ=-dTEαT+τEβT,
c1τEα1*τEβ10=E02A12P12k2Rd2expiωτSκα, τ×ρ V0d3rjBrjDrj2×exp-Lbj+Ljd/Lscattexpi·rj+ρ2V0d3rjV0d3rjmB*rjD*rjBrmDrm×grjm-1exp-Lbj+Ljd+Lbm+Lmd/2Lscatt×expi·rjexp-iκβ·rjm+ρ2V0 d3rjBrjDrjexp-Lbj+Ljd/2Lscatt×exp-iκave·rjm2,
=kscattα-kscattβ=κα-κβ, =2k sinδ/2, κave=κα+κβ/2,
c1τE02A12N0k2Rd2expiωτUautoG0P12Sκα, τ.
Uauto1V0V0d3rjBrjDrj2 exp-Lbj+Ljd/Lscatt,
G01UautoV0V0d3rjBrjDrj2×exp-Lbj+Ljd/Lscattexpi·rj,
G0=sinRz cos ξRz cos ξ 2J1Ry sin ξRy sin ξ,
G0=sinδ cos ξ/δcohzδ cos ξ/δcohz 2J1δ sin ξ/δcohyδ sin ξ/δcohy,
δcohy=1kRy
δcohz=1kRz
δcohy=λ2πRy1-nL-1RvnLRd,
δcohz=λ2πnLRz,
c1τEα1τEβ10E02A12N0k2Rd2exp2ikRd-iωτP12×S*κave, τUautoG0*2κave.
c2τE02A22ρ2k2Rd2expiωτV1d3rjBrj2×expi·rjV2d3rjlDrl2k2rjl2P22×Skscattα-kjl, τSkjl-kinc, τ×exp-Lbj+rjl+Lld/Lscattexpi·rjl.
c12τEα1*τEβ20E02A1A2ρ2k2Rd2expiωτSκα, τV0d3rjBrj2×D*rjexpi·rjV2d3rjlDrlkrjlP1P2×exp-2Lbj+Ljd+rjl+Lld/2Lscatt×expikjl-kscattβ·rjl,
c21τEα2*τEβ10E02A1A2ρ2k2Rd2expiωτV0d3rjBrj2×D*rjexpi·rjV2d3rjlDrlkrjlP1P2×Skscattα-kjl, τSkjl-kinc, τ×exp-2Lbj+Ljd+rjl+Lld/2Lscatt×exp-ikjl-kscattα·rjl.
Cτ=-dTIαT+τIβT,
CτEα*τEατEβ*0Eβ0,
Cτ=IαaveIβave+Y*Y+W*W,
Y=Eα1τEβ10+Eα1τEβ20+Eα2τEβ10+Eα2τEβ20,
W=Eα1*τEβ10+Eα1*τEβ20+Eα2*τEβ10+Eα2*τEβ20,
c2τ=E02A12N0k2Rd2expiωτQ2UautoG0S2θ, τ, ,
Q2=3ϕn2-1n2+22ka3kRave,
S2θ, τ, S2VVS2VHS2HVS2HH=14πRave0Ravedrjl0πsin θjldθjl02πdϕjlP22×exp-kscatt-kjl2Dτ×exp-kjl-kinc2Dτexpi·rjl,
S2VVτ=exp-4k2Dτ38Z0b+28Z1b+38Z2b,
S2VHτ=S2HVτ=exp-4k2Dτcos2 γZ0b-6 cos2 γ-4Z1b+5 cos2 γ-4Z2b,
S2HHτ=exp-4k2Dτ198cos4 γ-3 cos2 γ+1×Z0b-148cos4 γ-4 cos2 γ+2×Z1b+38cos4 γ-cos2 γ+1Z2b
b=4k2Dτ cos γ,
Zmb12-11 u2mdu expbu
S2VV=π2Raveexp-4k2DτI0b,
S2VH=S2HV=ORave-3,
S2HH=π2Raveexp-4k2Dτ×cos4 γI0b-2 cos2 γbI1b+3b2I2b,
S2VV=8Rave0Raveduj2uu2,
S2VH=S2HV=1Rave0Raveduj1uu-4j2uu2,
S2HH=1Rave0Raveducos2 θj0u-2j1uu+2 cos2 θ+1j2uu,
c12τ=E02A12N0k2Rd2expiωτQ12UautoG0S12θ, τ,
Q12=12n2+2n2-1ka-6kRave-1Q2,
S12θ, τ=k22πSκα, τ0Raverjlexpikrjldrjl×0πsin θjldθjl02πdϕjlP1P2×exp-ikscattβ·rjl.
c21τ=E02A12N0k2Rd2expiωτQ12UautoG0S21θ, τ,
S21θ, τ=k22π0Rave rjl exp-ikrjldrjl0πsin θjl dθjl×0πdϕjlP1P2exp-kscatt-kjl2Dτ×exp-kjl-kinc2Dτexpikscattα·rjl.
FVVδ=S2VVθ, τ=0, Rave1S2VVθ, τ=0, =0=4π15Rave,
FVVδ4πRy15Rave0.009

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