Abstract

Dynamic light-scattering techniques provide noninvasive probes of diverse media, such as colloidal suspensions, granular materials, or foams. In homodyne photon correlation spectroscopy, the dynamical properties of the medium are extracted from the intensity autocorrelation g(2)(τ) of the scattered light by means of the Siegert relation g(2)(τ)=1+|E(0)E*(τ)|2/EE*2. This approach is unfortunately limited to systems where the electric field is a Gaussian random variable and thus breaks down when the scattering sites are few or correlated. We propose to extend the traditional analysis by introducing intensity correlation functions g(n) of higher order, which allow us both to detect non-Gaussian scattering processes and to extract information not available in g(2) alone. The g(n) are experimentally measured by a combination of a commercial correlator and a custom digital delay line. Experimental results for g(3) and g(4) are presented for both Gaussian and non-Gaussian light-scattering processes and compared with theoretical predictions.

© 1999 Optical Society of America

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    [CrossRef] [PubMed]

1998 (3)

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

I. Flammer, G. Bucher, J. Ricka, “Diffusive wave illumination: light-scattering study of colloidal dynamics in opaque media,” J. Opt. Soc. Am. A 15, 2066–2077 (1998).
[CrossRef]

A. Krall, D. Weitz, “Internal dynamics and elasticity of fractal colloidal gels,” Phys. Rev. Lett. 80, 778–781 (1998).
[CrossRef]

1997 (2)

N. Menon, D. Durian, “Diffusing-wave spectroscopy of dynamics in a three-dimensional granular flow,” Science 275, 1920–1922 (1997).
[CrossRef] [PubMed]

G. Maret, “Diffusing-wave spectroscopy,” Curr. Opin. Colloid Interface Sci. 2, 251–257 (1997).
[CrossRef]

1995 (4)

B. Frisken, F. Ferri, D. Cannell, “Critical behavior in the presence of a disordered environment,” Phys. Rev. E 51, 5922–5943 (1995).
[CrossRef]

S. Brauer, G. Stephenson, M. Sutton, R. Bruning, E. Dufresne, S. Mochrie, G. Grubel, J. Als-Nielsen, D. Abernathy, “X-ray intensity fluctuation spectroscopy observation of critical dynamics in Fe3Al,” Phys. Rev. Lett. 74, 2010–2013 (1995).
[CrossRef] [PubMed]

S. Dierker, R. Pindak, R. Fleming, I. Robinson, L. Berman, “X-ray photon correlation spectroscopy study of Brownian motion of gold colloids in glycerol,” Phys. Rev. Lett. 75, 449–452 (1995).
[CrossRef] [PubMed]

T. G. Mason, D. A. Weitz, “Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250–1253 (1995).
[CrossRef] [PubMed]

1994 (1)

1992 (1)

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-l. Wu, P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[CrossRef] [PubMed]

1991 (1)

T. Bellini, M. A. Glaser, N. A. Clark, V. Degiorgio, “Effects of finite laser coherence in quasielastic multiple scattering,” Phys. Rev. A 44, 5215–5223 (1991).
[CrossRef] [PubMed]

1989 (1)

P. N. Pusey, W. van Megen, “Dynamic light scattering by non-ergodic media,” Physica A 157, 705–741 (1989).
[CrossRef]

1986 (1)

M. T. Bishop, K. H. Langley, F. E. Karasz, “Diffusion of a flexible polymer in a random porous material,” Phys. Rev. Lett. 57, 1741–1744 (1986).
[CrossRef] [PubMed]

1983 (1)

1979 (1)

P. N. Pusey, “Number fluctuations of interacting particles,” J. Phys. A 12, 1805–1818 (1979).
[CrossRef]

1976 (1)

M. Corti, V. Degiorgio, “Intrinsic third-order correlations in laser light near threshold,” Phys. Rev. A 14, 1475–1478 (1976).
[CrossRef]

1974 (1)

M. Corti, A. D. Agostini, V. Degiorgio, “Fast digital correlator for weak optical signals,” Rev. Sci. Instrum. 45, 888–893 (1974).
[CrossRef]

1973 (2)

S. Chopra, L. Mandel, “Higher-order correlation properties of a laser beam,” Phys. Rev. Lett. 30, 60–63 (1973).
[CrossRef]

C. D. Cantrel, M. Lax, W. A. Smith, “Third- and higher-order intensity correlation in laser light,” Phys. Rev. A 7, 175–181 (1973).
[CrossRef]

1972 (1)

1971 (1)

1970 (2)

H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” Prog. Opt. 8, 133–200 (1970).
[CrossRef]

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

1965 (1)

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Abernathy, D.

S. Brauer, G. Stephenson, M. Sutton, R. Bruning, E. Dufresne, S. Mochrie, G. Grubel, J. Als-Nielsen, D. Abernathy, “X-ray intensity fluctuation spectroscopy observation of critical dynamics in Fe3Al,” Phys. Rev. Lett. 74, 2010–2013 (1995).
[CrossRef] [PubMed]

Agostini, A. D.

M. Corti, A. D. Agostini, V. Degiorgio, “Fast digital correlator for weak optical signals,” Rev. Sci. Instrum. 45, 888–893 (1974).
[CrossRef]

Als-Nielsen, J.

S. Brauer, G. Stephenson, M. Sutton, R. Bruning, E. Dufresne, S. Mochrie, G. Grubel, J. Als-Nielsen, D. Abernathy, “X-ray intensity fluctuation spectroscopy observation of critical dynamics in Fe3Al,” Phys. Rev. Lett. 74, 2010–2013 (1995).
[CrossRef] [PubMed]

Balakrishnan, A. V.

A. V. Balakrishnan, Introduction to Random Processes in Engineering (Wiley, New York, 1995).

Bellini, T.

T. Bellini, M. A. Glaser, N. A. Clark, V. Degiorgio, “Effects of finite laser coherence in quasielastic multiple scattering,” Phys. Rev. A 44, 5215–5223 (1991).
[CrossRef] [PubMed]

Berman, L.

S. Dierker, R. Pindak, R. Fleming, I. Robinson, L. Berman, “X-ray photon correlation spectroscopy study of Brownian motion of gold colloids in glycerol,” Phys. Rev. Lett. 75, 449–452 (1995).
[CrossRef] [PubMed]

Bertolotti, M.

B. Crosignani, P. DiPorto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

Bishop, M. T.

M. T. Bishop, K. H. Langley, F. E. Karasz, “Diffusion of a flexible polymer in a random porous material,” Phys. Rev. Lett. 57, 1741–1744 (1986).
[CrossRef] [PubMed]

Bluemel, V.

Bohren, C.

C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Brauer, S.

S. Brauer, G. Stephenson, M. Sutton, R. Bruning, E. Dufresne, S. Mochrie, G. Grubel, J. Als-Nielsen, D. Abernathy, “X-ray intensity fluctuation spectroscopy observation of critical dynamics in Fe3Al,” Phys. Rev. Lett. 74, 2010–2013 (1995).
[CrossRef] [PubMed]

Bruning, R.

S. Brauer, G. Stephenson, M. Sutton, R. Bruning, E. Dufresne, S. Mochrie, G. Grubel, J. Als-Nielsen, D. Abernathy, “X-ray intensity fluctuation spectroscopy observation of critical dynamics in Fe3Al,” Phys. Rev. Lett. 74, 2010–2013 (1995).
[CrossRef] [PubMed]

Bucher, G.

Cannell, D.

B. Frisken, F. Ferri, D. Cannell, “Critical behavior in the presence of a disordered environment,” Phys. Rev. E 51, 5922–5943 (1995).
[CrossRef]

Cantrel, C. D.

C. D. Cantrel, M. Lax, W. A. Smith, “Third- and higher-order intensity correlation in laser light,” Phys. Rev. A 7, 175–181 (1973).
[CrossRef]

Chaikin, P. M.

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-l. Wu, P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[CrossRef] [PubMed]

Chopra, S.

S. Chopra, L. Mandel, “Higher-order correlation properties of a laser beam,” Phys. Rev. Lett. 30, 60–63 (1973).
[CrossRef]

Chu, B.

B. Chu, Laser Light Scattering, Basic Principles and Practice (Academic, New York, 1991).

Clark, N. A.

T. Bellini, M. A. Glaser, N. A. Clark, V. Degiorgio, “Effects of finite laser coherence in quasielastic multiple scattering,” Phys. Rev. A 44, 5215–5223 (1991).
[CrossRef] [PubMed]

Corti, M.

M. Corti, V. Degiorgio, “Intrinsic third-order correlations in laser light near threshold,” Phys. Rev. A 14, 1475–1478 (1976).
[CrossRef]

M. Corti, A. D. Agostini, V. Degiorgio, “Fast digital correlator for weak optical signals,” Rev. Sci. Instrum. 45, 888–893 (1974).
[CrossRef]

Crosignani, B.

B. Crosignani, P. DiPorto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

Cummins, H. Z.

H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” Prog. Opt. 8, 133–200 (1970).
[CrossRef]

Degiorgio, V.

T. Bellini, M. A. Glaser, N. A. Clark, V. Degiorgio, “Effects of finite laser coherence in quasielastic multiple scattering,” Phys. Rev. A 44, 5215–5223 (1991).
[CrossRef] [PubMed]

M. Corti, V. Degiorgio, “Intrinsic third-order correlations in laser light near threshold,” Phys. Rev. A 14, 1475–1478 (1976).
[CrossRef]

M. Corti, A. D. Agostini, V. Degiorgio, “Fast digital correlator for weak optical signals,” Rev. Sci. Instrum. 45, 888–893 (1974).
[CrossRef]

Dierker, S.

S. Dierker, R. Pindak, R. Fleming, I. Robinson, L. Berman, “X-ray photon correlation spectroscopy study of Brownian motion of gold colloids in glycerol,” Phys. Rev. Lett. 75, 449–452 (1995).
[CrossRef] [PubMed]

DiPorto, P.

B. Crosignani, P. DiPorto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

Dufresne, E.

S. Brauer, G. Stephenson, M. Sutton, R. Bruning, E. Dufresne, S. Mochrie, G. Grubel, J. Als-Nielsen, D. Abernathy, “X-ray intensity fluctuation spectroscopy observation of critical dynamics in Fe3Al,” Phys. Rev. Lett. 74, 2010–2013 (1995).
[CrossRef] [PubMed]

Durian, D.

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

N. Menon, D. Durian, “Diffusing-wave spectroscopy of dynamics in a three-dimensional granular flow,” Science 275, 1920–1922 (1997).
[CrossRef] [PubMed]

Estes, L. E.

Ferri, F.

B. Frisken, F. Ferri, D. Cannell, “Critical behavior in the presence of a disordered environment,” Phys. Rev. E 51, 5922–5943 (1995).
[CrossRef]

Flammer, I.

Fleming, R.

S. Dierker, R. Pindak, R. Fleming, I. Robinson, L. Berman, “X-ray photon correlation spectroscopy study of Brownian motion of gold colloids in glycerol,” Phys. Rev. Lett. 75, 449–452 (1995).
[CrossRef] [PubMed]

Frisken, B.

B. Frisken, F. Ferri, D. Cannell, “Critical behavior in the presence of a disordered environment,” Phys. Rev. E 51, 5922–5943 (1995).
[CrossRef]

Glaser, M. A.

T. Bellini, M. A. Glaser, N. A. Clark, V. Degiorgio, “Effects of finite laser coherence in quasielastic multiple scattering,” Phys. Rev. A 44, 5215–5223 (1991).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984).

Grubel, G.

S. Brauer, G. Stephenson, M. Sutton, R. Bruning, E. Dufresne, S. Mochrie, G. Grubel, J. Als-Nielsen, D. Abernathy, “X-ray intensity fluctuation spectroscopy observation of critical dynamics in Fe3Al,” Phys. Rev. Lett. 74, 2010–2013 (1995).
[CrossRef] [PubMed]

Harris, M.

Hill, C. A.

Hill, W.

P. Horowitz, W. Hill, The Art of Electronics, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1989).

Horowitz, P.

P. Horowitz, W. Hill, The Art of Electronics, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1989).

Huffman, D.

C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jakeman, E.

E. Jakeman, “Photon correlation,” in Photon Correlation and Light-Beating Spectroscopy, Vol. B3 of NATO Advanced Study Institutes Series, H. Cummins, E. Pike, eds. (Plenum, New York, 1974), pp. 75–149.

Karasz, F. E.

M. T. Bishop, K. H. Langley, F. E. Karasz, “Diffusion of a flexible polymer in a random porous material,” Phys. Rev. Lett. 57, 1741–1744 (1986).
[CrossRef] [PubMed]

Krall, A.

A. Krall, D. Weitz, “Internal dynamics and elasticity of fractal colloidal gels,” Phys. Rev. Lett. 80, 778–781 (1998).
[CrossRef]

Kurz, T.

W. Lauterborn, T. Kurz, M. Wiesenfeldt, Coherent Optics: Fundamentals and Applications (Springer, Berlin, 1995).

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Langley, K. H.

M. T. Bishop, K. H. Langley, F. E. Karasz, “Diffusion of a flexible polymer in a random porous material,” Phys. Rev. Lett. 57, 1741–1744 (1986).
[CrossRef] [PubMed]

Lauterborn, W.

W. Lauterborn, T. Kurz, M. Wiesenfeldt, Coherent Optics: Fundamentals and Applications (Springer, Berlin, 1995).

Lax, M.

C. D. Cantrel, M. Lax, W. A. Smith, “Third- and higher-order intensity correlation in laser light,” Phys. Rev. A 7, 175–181 (1973).
[CrossRef]

Lemieux, P.-A.

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

Lohmann, A.

Mandel, L.

S. Chopra, L. Mandel, “Higher-order correlation properties of a laser beam,” Phys. Rev. Lett. 30, 60–63 (1973).
[CrossRef]

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Maret, G.

G. Maret, “Diffusing-wave spectroscopy,” Curr. Opin. Colloid Interface Sci. 2, 251–257 (1997).
[CrossRef]

Mason, T. G.

T. G. Mason, D. A. Weitz, “Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250–1253 (1995).
[CrossRef] [PubMed]

Menon, N.

N. Menon, D. Durian, “Diffusing-wave spectroscopy of dynamics in a three-dimensional granular flow,” Science 275, 1920–1922 (1997).
[CrossRef] [PubMed]

Milner, S. T.

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-l. Wu, P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[CrossRef] [PubMed]

Mochrie, S.

S. Brauer, G. Stephenson, M. Sutton, R. Bruning, E. Dufresne, S. Mochrie, G. Grubel, J. Als-Nielsen, D. Abernathy, “X-ray intensity fluctuation spectroscopy observation of critical dynamics in Fe3Al,” Phys. Rev. Lett. 74, 2010–2013 (1995).
[CrossRef] [PubMed]

Narducci, L. M.

Pearson, G. N.

Pecora, R.

R. Pecora, Dynamic Light Scattering: Application of Photon Correlation Spectroscopy (Plenum, New York, 1985).

Pindak, R.

S. Dierker, R. Pindak, R. Fleming, I. Robinson, L. Berman, “X-ray photon correlation spectroscopy study of Brownian motion of gold colloids in glycerol,” Phys. Rev. Lett. 75, 449–452 (1995).
[CrossRef] [PubMed]

Pine, D.

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

Pine, D. J.

J.-Z. Xue, D. J. Pine, S. T. Milner, X.-l. Wu, P. M. Chaikin, “Nonergodicity and light scattering from polymer gels,” Phys. Rev. A 46, 6550–6563 (1992).
[CrossRef] [PubMed]

Pusey, P. N.

P. N. Pusey, W. van Megen, “Dynamic light scattering by non-ergodic media,” Physica A 157, 705–741 (1989).
[CrossRef]

P. N. Pusey, “Number fluctuations of interacting particles,” J. Phys. A 12, 1805–1818 (1979).
[CrossRef]

P. N. Pusey, “Statistical properties of scattered radiation,” in Photon Correlation Spectroscopy and Velocimetry, Vol. B23 of NATO Advanced Study Institutes Series, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1977), pp. 45–141.

P. N. Pusey, Department of Physics and Astronomy, University of Edinburgh, James Clerk Maxwell Bldg., Mayfield Road, Edinburgh EH9 3JZ, UK. E-mail, p.n.pusey@ed.ac.uk (personal communication, 1998).

Ricka, J.

Robinson, I.

S. Dierker, R. Pindak, R. Fleming, I. Robinson, L. Berman, “X-ray photon correlation spectroscopy study of Brownian motion of gold colloids in glycerol,” Phys. Rev. Lett. 75, 449–452 (1995).
[CrossRef] [PubMed]

Smith, W. A.

C. D. Cantrel, M. Lax, W. A. Smith, “Third- and higher-order intensity correlation in laser light,” Phys. Rev. A 7, 175–181 (1973).
[CrossRef]

Stephenson, G.

S. Brauer, G. Stephenson, M. Sutton, R. Bruning, E. Dufresne, S. Mochrie, G. Grubel, J. Als-Nielsen, D. Abernathy, “X-ray intensity fluctuation spectroscopy observation of critical dynamics in Fe3Al,” Phys. Rev. Lett. 74, 2010–2013 (1995).
[CrossRef] [PubMed]

Sutton, M.

S. Brauer, G. Stephenson, M. Sutton, R. Bruning, E. Dufresne, S. Mochrie, G. Grubel, J. Als-Nielsen, D. Abernathy, “X-ray intensity fluctuation spectroscopy observation of critical dynamics in Fe3Al,” Phys. Rev. Lett. 74, 2010–2013 (1995).
[CrossRef] [PubMed]

Swinney, H. L.

H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” Prog. Opt. 8, 133–200 (1970).
[CrossRef]

Tuft, R. A.

van de Hulst, H.

H. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

van Megen, W.

P. N. Pusey, W. van Megen, “Dynamic light scattering by non-ergodic media,” Physica A 157, 705–741 (1989).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Dependence of higher-order light intensity moments I3/I3 and I4/I4 on the second moment I2/I2. Data were obtained by using a dilute suspension of polystyrene spheres in the multiple-scattering limit, a system designed to exhibit Gaussian scattering characteristics. Using various collection optics, we varied the parameter β1, i.e., the measured value of the second moment I2/I2=1+β1. The two sets of predictions are for a Gaussian scattering process [Eqs. (16) and (17)] and contain no fitting parameters; the pinhole detection predictions take into account spatial averaging over an integer number of speckles, while the single-speckle predictions solely include field polarization effects. Data for both backscattered and transmitted light appear identical, indicating that coherence effects are negligible here.

Fig. 2
Fig. 2

Experimental method for measuring intensity correlations of second, third, and fourth order. A single commercial correlator is used successively in cross-correlation and dual autocorrelation mode. The delay and multiply operations are performed by a custom digital delay line.

Fig. 3
Fig. 3

Systematic procedure used to determine whether the scattered field is a Gaussian variable and thus whether the Siegert relation holds. No assumptions are made on the nature of the scattering process.

Fig. 4
Fig. 4

Second- and fourth-order intensity correlation functions, g(2) and g(4), respectively, for a Gaussian scattering process. The multiple datasets correspond to various values of the delay T, i.e., various time slices of g(4). The predictions are for a Gaussian scattering process and are generated from g(2) according to Eqs. (13a) and (13c) and the procedure described in Fig. 3. The excellent agreement is a strong indication that the process is Gaussian and that the Siegert relation can safely be used to extract the dynamics of the system. The resurgence of correlation at τ=T is simply a characteristic feature of the g(4)(T, τ, τ+T) slice of the more general g(4)(τ1, τ2, τ3). Data were obtained by using a dilute suspension of polystyrene spheres in the multiple-scattering limit.

Fig. 5
Fig. 5

Second- and fourth-order intensity correlation functions for a fluctuating source intensity. The fluctuations were modeled by switching the laser tube current to a random value every 2 s, as labeled. For stronger fluctuations the intensity distribution P(I) is widened, and thus the intercept of g(2)(τ), i.e., the second moment I2/I2, is increased. The second decay of g(2), which is due to the intensity switching, might be mistaken for a signal that is due to scattering dynamics. This effect can be detected by considering the agreement between the measured and Gaussian-predicted fourth-order intensity correlation functions g(4). The Gaussian contributions are obtained by dividing out the intensity-switching contribution, according to Eqs. (23). Data were obtained by using a dilute suspension of polystyrene spheres in the multiple-scattering limit.

Fig. 6
Fig. 6

Second- and fourth-order intensity correlation functions for two light sources with different coherence lengths. The reduced coherence length of the diode laser pointer decreases the contrast of the speckle pattern and thus lowers the intercept of g(2), i.e., the second moment I2/I2. The decay rate of |γ(τ)| is also decreased, an effect that would lead to the wrong interpretation of the dynamics of the scattering sites. The reduced coherence can, however, be detected by considering the agreement between the measured and Gaussian-predicted fourth-order intensity correlation functions. Data were obtained by using a dilute suspension of polystyrene spheres in the multiple-scattering limit.

Fig. 7
Fig. 7

Second- and fourth-order intensity correlation functions in the case of an additional static (heterodyne) component in the scattered field. The contrast of the speckle pattern is decreased, and thus the intercept of g(2) is lowered. The decay rate of |γ(τ)| is also increased, an effect that would lead to the wrong interpretation of the dynamics of the scattering sites if the Siegert relation were used. This can, however, be detected by considering the agreement between the measured and the Gaussian-predicted g(4). Also shown for comparison is the same system without the presence of the static component. Data were obtained by using a dilute suspension of polystyrene spheres in the multiple-scattering limit.

Fig. 8
Fig. 8

Second- and fourth-order intensity correlation functions for a system exhibiting number fluctuations. A rotating polycarbonate disk with 100-μm surface roughness was used. As the size of the illuminated region is decreased, the number of effective scattering sites diminishes to the point that the central limit theorem fails, and the scattered field is no longer Gaussian. For a small number of scattering sites, lighthouse effects are observed. Therefore the width of the intensity distribution increases, and, accordingly, the intercept of g(2) is enhanced. For a large spot, there is a strong agreement between the measured and the Gaussian-predicted g(4). This agreement quickly disappears as the spot size is reduced, and the central limit theorem fails. Data were obtained by using a rotating polycarbonate ground disk.

Fig. 9
Fig. 9

Second- and fourth-order intensity correlation functions for a system exhibiting intermittent dynamics. The system consisted of an auger rotating intermittently through a sand column. The first decay of g(2) is due to the rapid motion of the grains of sand while the auger is turning, while the second decay of g(2) is a sign of the intermittent nature of the dynamics. The strongly correlated nature of the dynamics leads to a failure of the Siegert relation, as seen in the disagreement between the experimental and the Gaussian-predicted g(4). The periodic structure apparent in the baseline of g(4) is a characteristic feature of periodic intermittency (the oscillation appears to decay at longer times because of the logarithmic spacing of the correlator channels). This information, absent in g(2), helps one to characterize the system. Also shown is the case of continuous auger rotation, where, as expected, the light-scattering process appears Gaussian.

Equations (32)

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g(2)(τ)=1+|E(0)E*(τ)|2|EE*|21+|γ(τ)|2.
γ(τ)=|γ(τ)|exp(iω0τ).
I(t)E(t)E*(t).
E(t)=jNEj(t).
γ(τ)=E(0)E*(τ)EE*,
I(0)I(τ)=E(0)E*(0)E(τ)E*(τ).
I(0)I(τ)=E(0)E*(0)E(τ)E*(τ)+E(0)E(τ)E*(0)E*(τ)+E(0)E*(τ)E*(0)E(τ)=I2+|E(0)E(τ)|2+|E(0)E*(τ)|2.
g(n)(τ1,,τn-1)I(0)I(τ1)I(τn-1)/In.
g(3)(τ1, τ2)=1+|γ01|2+|γ02|2+|γ12|2+2 Re(γ01γ12γ20),
g(4)(τ1, τ2, τ3)=1+|γ01|2+|γ02|2+|γ03|2+|γ12|2+|γ13|2+|γ23|2+|γ01|2|γ23|2+|γ02|2|γ13|2+|γ03|2|γ12|2+2 Re(γ01γ12γ20+γ01γ13γ30+γ02γ23γ30+γ12γ23γ31)+2 Re(γ01γ12γ23γ30+γ02γ23γ31γ10+γ02γ21γ13γ30),
In/In=limτi0 g(n)=n!,
dλr/l,
I(t)=j=1NIj(t),
Ij(0)Ik(τ)=Ij2[1+|γ(τ)|2],j=kIjIk,jk.
g(2)(τ1)=jkIjIk+j=kIj(0)Ik(τ1)j,kIjIk=1+β1|γ01|2,
g(3)(τ1, τ2)=1+β1(|γ01|2+|γ02|2+|γ12|2)+2β2 Re(γ01γ12γ20),
g(4)(τ1, τ2, τ3)=1+β1(|γ01|2+|γ02|2+|γ03|2+|γ12|2+|γ13|2+|γ23|2)+β12(|γ01|2|γ23|2+|γ02|2|γ13|2+|γ03|2|γ12|2)+2β2 Re(γ01γ12γ20+γ01γ13γ30+γ02γ23γ30+γ12γ23γ31)+2β3 Re(γ01γ12γ23γ30+γ02γ23γ31γ10+γ02γ21γ13γ30).
βi=j=1NIji+1/j=1NIji+1,
βi=βi,
In/In=1[1+β][1+2β][1+(n-1)β].
βi=1+pi+1(1+p)i+1.
I(0)I(T1)I(Tn-2)I(τ)I(0)I(T1)I(Tn-2)I.
g^(3)(T, τ)=I(0)I(T)I(τ)I(0)I(T)I,
g^(4)(T, τ)=I(0)I(T)I(τ)I(τ+T)I(0)I(T)2
Ia(0)Ib(τ)IaIb=na(0)nb(τ)nanb,
g(3)(T, τ)=(τsRI)(τsRII)(τsRI)3 g^(3)(T, τ)=RIIτsRI2 g^(3)(T, τ),
g(4)(T, τ, τ+T)=(τsRII)(τsRII)(τsRI)4 g^(4)(T, τ)=RII2τs2RI4 g^(4)(T, τ),
g(2)(τ)=α(0)α(τ)Ic(0)Ic(τ)/I2,
g(4)(T, τ, τ+T)=α(0)α(T)α(τ)α(τ+T)×Ic(0)Ic(T)Ic(τ)Ic(τ+T)/I4,
Eh(t)=E(t)+Es exp(-iω0t),
gh(2)(τ)=1+β(If+Is)2 [If2|γ(τ)|2+2IfIs|γ(τ)|],
gh(3)(τ1, τ2)=1+β(If+Is)3 {If3(|γ01|2+|γ02|2+|γ12|2+2β|γ01γ12γ20|)+IsIf2[|γ01|2+|γ12|2+|γ02|2+2(β|γ01γ02|+|γ01γ12|+|γ02γ12|)]+2(IsIf2+Is2If)(|γ01|+|γ02|+|γ12|)},

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