Abstract

Two single-mode fibers collect light with the same scattered wave vector from two spatially separated regions in a sample. These regions are illuminated by a single coherent laser beam, so that the collected signals interfere when combined by means of a fiber-optic coupler, before they are directed to a photomultiplier tube. The fibers and the coupler are polarization preserving to guarantee a high signal-to-noise ratio. The measured intensity fluctuations are used to determine the velocity difference δv(L) for spatial separations L in the sample. Specifically, an intensity autocorrelation function is calculated theoretically for rigid body rotation and is tested experimentally. Experimental results span two orders of magnitude in L and agree with theoretical predictions with an error of less than 5%. This new technique will be very useful in the study of turbulent flow and particle settling dynamics.

© 1998 Optical Society of America

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References

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  1. B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).
  2. B. J. Ackerson, N. A. Clark, “Dynamic light scattering at low rates of shear,” J. Phys. (Paris) 42, 929–936 (1981).
    [CrossRef]
  3. G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in a laminar flow by homodyne light-scattering spectroscopy,” J. Fluid Mech. 100, 555–575 (1980).
    [CrossRef]
  4. P. Tong, W. I. Goldburg, C. K. Chan, A. Sirivat, “Turbulent transition by photon correlation spectroscopy,” Phys. Rev. A 37, 2125–2133 (1988).
    [CrossRef] [PubMed]
  5. H. K. Pak, W. I. Goldburg, A. Sirivat, “Measuring the probability distribution of the relative velocities in grid-generated turbulence,” Phys. Rev. Lett. 68, 938–941 (1992); “An experimental study of weak turbulence,” Fluid Dyn. Res. 8, 19–31 (1991).
    [CrossRef] [PubMed]
  6. P. Tong, Y. Shen, “Relative velocity fluctuations in turbulent Rayleigh–Bénard convection,” Phys. Rev. Lett. 69, 2066–2069 (1992).
    [CrossRef]
  7. H. Kellay, X.-L. Wu, W. I. Goldburg, “Experiments with turbulent soap films,” Phys. Rev. Lett. 74, 3975–3978 (1995).
    [CrossRef] [PubMed]
  8. P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
    [CrossRef]
  9. S. Hanson, “The laser gradient anemometer,” in Photon Correlation Techniques in Fluid Mechanics, E. O. Schulz-DuBois, ed. (Springer-Verlag, Berlin, 1983), pp. 212–220.
  10. C. Keveloh, W. Staude, “The application of cross-correlation measurements to the determination of velocity gradients in fluid flow,” in Photon Correlation Techniques in Fluid Mechanics, E. O. Schulz-DuBois, ed. (Springer-Verlag, Berlin, 1983), pp. 221–224.
  11. G. I. Taylor, “The spectrum of turbulence,” Proc. R. Soc. London, Ser. A 164, 476–490 (1938).
    [CrossRef]
  12. K. J. Måløy, W. I. Goldburg, H. K. Pak, “Spatial coherence of homodyne light scattering from particles in a convective field,” Phys. Rev. A 46, 3288–3291 (1992).
    [CrossRef]
  13. T. Narayanan, C. Cheung, P. Tong, W. I. Goldburg, X.-L. Wu, “Measurement of the velocity difference by photon correlation spectroscopy: an improved scheme,” Appl. Opt. 36, 7639–7644 (1997).
    [CrossRef]
  14. See, e.g., P. Tong, W. I. Goldburg, “Relative velocity fluctuations in turbulent flows at moderate Reynolds number II. Model calculation,” Phys. Fluids 31, 3253–3259 (1988).
    [CrossRef]
  15. H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 8, pp. 133–200.
  16. U. Nobbmann, S. W. Jones, B. J. Ackerson, “Multiple scattering suppression: cross correlation with tilted single-mode fibers,” Appl. Opt. 36, 7571–7582 (1997).
    [CrossRef]

1997

1995

H. Kellay, X.-L. Wu, W. I. Goldburg, “Experiments with turbulent soap films,” Phys. Rev. Lett. 74, 3975–3978 (1995).
[CrossRef] [PubMed]

1992

K. J. Måløy, W. I. Goldburg, H. K. Pak, “Spatial coherence of homodyne light scattering from particles in a convective field,” Phys. Rev. A 46, 3288–3291 (1992).
[CrossRef]

H. K. Pak, W. I. Goldburg, A. Sirivat, “Measuring the probability distribution of the relative velocities in grid-generated turbulence,” Phys. Rev. Lett. 68, 938–941 (1992); “An experimental study of weak turbulence,” Fluid Dyn. Res. 8, 19–31 (1991).
[CrossRef] [PubMed]

P. Tong, Y. Shen, “Relative velocity fluctuations in turbulent Rayleigh–Bénard convection,” Phys. Rev. Lett. 69, 2066–2069 (1992).
[CrossRef]

1988

See, e.g., P. Tong, W. I. Goldburg, “Relative velocity fluctuations in turbulent flows at moderate Reynolds number II. Model calculation,” Phys. Fluids 31, 3253–3259 (1988).
[CrossRef]

P. Tong, W. I. Goldburg, C. K. Chan, A. Sirivat, “Turbulent transition by photon correlation spectroscopy,” Phys. Rev. A 37, 2125–2133 (1988).
[CrossRef] [PubMed]

1981

B. J. Ackerson, N. A. Clark, “Dynamic light scattering at low rates of shear,” J. Phys. (Paris) 42, 929–936 (1981).
[CrossRef]

1980

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in a laminar flow by homodyne light-scattering spectroscopy,” J. Fluid Mech. 100, 555–575 (1980).
[CrossRef]

1970

P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
[CrossRef]

1938

G. I. Taylor, “The spectrum of turbulence,” Proc. R. Soc. London, Ser. A 164, 476–490 (1938).
[CrossRef]

Ackerson, B. J.

Berne, B. J.

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

Bourke, P. J.

P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
[CrossRef]

Butterworth, J.

P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
[CrossRef]

Chan, C. K.

P. Tong, W. I. Goldburg, C. K. Chan, A. Sirivat, “Turbulent transition by photon correlation spectroscopy,” Phys. Rev. A 37, 2125–2133 (1988).
[CrossRef] [PubMed]

Cheung, C.

Clark, N. A.

B. J. Ackerson, N. A. Clark, “Dynamic light scattering at low rates of shear,” J. Phys. (Paris) 42, 929–936 (1981).
[CrossRef]

Cummins, H. Z.

H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 8, pp. 133–200.

Drain, L. E.

P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
[CrossRef]

Egelstaff, P. A.

P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
[CrossRef]

Fuller, G. G.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in a laminar flow by homodyne light-scattering spectroscopy,” J. Fluid Mech. 100, 555–575 (1980).
[CrossRef]

Goldburg, W. I.

T. Narayanan, C. Cheung, P. Tong, W. I. Goldburg, X.-L. Wu, “Measurement of the velocity difference by photon correlation spectroscopy: an improved scheme,” Appl. Opt. 36, 7639–7644 (1997).
[CrossRef]

H. Kellay, X.-L. Wu, W. I. Goldburg, “Experiments with turbulent soap films,” Phys. Rev. Lett. 74, 3975–3978 (1995).
[CrossRef] [PubMed]

H. K. Pak, W. I. Goldburg, A. Sirivat, “Measuring the probability distribution of the relative velocities in grid-generated turbulence,” Phys. Rev. Lett. 68, 938–941 (1992); “An experimental study of weak turbulence,” Fluid Dyn. Res. 8, 19–31 (1991).
[CrossRef] [PubMed]

K. J. Måløy, W. I. Goldburg, H. K. Pak, “Spatial coherence of homodyne light scattering from particles in a convective field,” Phys. Rev. A 46, 3288–3291 (1992).
[CrossRef]

See, e.g., P. Tong, W. I. Goldburg, “Relative velocity fluctuations in turbulent flows at moderate Reynolds number II. Model calculation,” Phys. Fluids 31, 3253–3259 (1988).
[CrossRef]

P. Tong, W. I. Goldburg, C. K. Chan, A. Sirivat, “Turbulent transition by photon correlation spectroscopy,” Phys. Rev. A 37, 2125–2133 (1988).
[CrossRef] [PubMed]

Hanson, S.

S. Hanson, “The laser gradient anemometer,” in Photon Correlation Techniques in Fluid Mechanics, E. O. Schulz-DuBois, ed. (Springer-Verlag, Berlin, 1983), pp. 212–220.

Hughes, A. J.

P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
[CrossRef]

Hutchinson, P.

P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
[CrossRef]

Jackson, D. A.

P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
[CrossRef]

Jakeman, E.

P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
[CrossRef]

Jones, S. W.

Kellay, H.

H. Kellay, X.-L. Wu, W. I. Goldburg, “Experiments with turbulent soap films,” Phys. Rev. Lett. 74, 3975–3978 (1995).
[CrossRef] [PubMed]

Keveloh, C.

C. Keveloh, W. Staude, “The application of cross-correlation measurements to the determination of velocity gradients in fluid flow,” in Photon Correlation Techniques in Fluid Mechanics, E. O. Schulz-DuBois, ed. (Springer-Verlag, Berlin, 1983), pp. 221–224.

Leal, L. G.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in a laminar flow by homodyne light-scattering spectroscopy,” J. Fluid Mech. 100, 555–575 (1980).
[CrossRef]

Måløy, K. J.

K. J. Måløy, W. I. Goldburg, H. K. Pak, “Spatial coherence of homodyne light scattering from particles in a convective field,” Phys. Rev. A 46, 3288–3291 (1992).
[CrossRef]

Moss, B.

P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
[CrossRef]

Narayanan, T.

Nobbmann, U.

O’Shaughnessy, J.

P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
[CrossRef]

Pak, H. K.

H. K. Pak, W. I. Goldburg, A. Sirivat, “Measuring the probability distribution of the relative velocities in grid-generated turbulence,” Phys. Rev. Lett. 68, 938–941 (1992); “An experimental study of weak turbulence,” Fluid Dyn. Res. 8, 19–31 (1991).
[CrossRef] [PubMed]

K. J. Måløy, W. I. Goldburg, H. K. Pak, “Spatial coherence of homodyne light scattering from particles in a convective field,” Phys. Rev. A 46, 3288–3291 (1992).
[CrossRef]

Pecora, R.

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

Pike, E. R.

P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
[CrossRef]

Rallison, J. M.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in a laminar flow by homodyne light-scattering spectroscopy,” J. Fluid Mech. 100, 555–575 (1980).
[CrossRef]

Schmidt, R. L.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in a laminar flow by homodyne light-scattering spectroscopy,” J. Fluid Mech. 100, 555–575 (1980).
[CrossRef]

Schofield, P.

P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
[CrossRef]

Shen, Y.

P. Tong, Y. Shen, “Relative velocity fluctuations in turbulent Rayleigh–Bénard convection,” Phys. Rev. Lett. 69, 2066–2069 (1992).
[CrossRef]

Sirivat, A.

H. K. Pak, W. I. Goldburg, A. Sirivat, “Measuring the probability distribution of the relative velocities in grid-generated turbulence,” Phys. Rev. Lett. 68, 938–941 (1992); “An experimental study of weak turbulence,” Fluid Dyn. Res. 8, 19–31 (1991).
[CrossRef] [PubMed]

P. Tong, W. I. Goldburg, C. K. Chan, A. Sirivat, “Turbulent transition by photon correlation spectroscopy,” Phys. Rev. A 37, 2125–2133 (1988).
[CrossRef] [PubMed]

Staude, W.

C. Keveloh, W. Staude, “The application of cross-correlation measurements to the determination of velocity gradients in fluid flow,” in Photon Correlation Techniques in Fluid Mechanics, E. O. Schulz-DuBois, ed. (Springer-Verlag, Berlin, 1983), pp. 221–224.

Swinney, H. L.

H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 8, pp. 133–200.

Taylor, G. I.

G. I. Taylor, “The spectrum of turbulence,” Proc. R. Soc. London, Ser. A 164, 476–490 (1938).
[CrossRef]

Tong, P.

T. Narayanan, C. Cheung, P. Tong, W. I. Goldburg, X.-L. Wu, “Measurement of the velocity difference by photon correlation spectroscopy: an improved scheme,” Appl. Opt. 36, 7639–7644 (1997).
[CrossRef]

P. Tong, Y. Shen, “Relative velocity fluctuations in turbulent Rayleigh–Bénard convection,” Phys. Rev. Lett. 69, 2066–2069 (1992).
[CrossRef]

P. Tong, W. I. Goldburg, C. K. Chan, A. Sirivat, “Turbulent transition by photon correlation spectroscopy,” Phys. Rev. A 37, 2125–2133 (1988).
[CrossRef] [PubMed]

See, e.g., P. Tong, W. I. Goldburg, “Relative velocity fluctuations in turbulent flows at moderate Reynolds number II. Model calculation,” Phys. Fluids 31, 3253–3259 (1988).
[CrossRef]

Wu, X.-L.

Appl. Opt.

J. Fluid Mech.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in a laminar flow by homodyne light-scattering spectroscopy,” J. Fluid Mech. 100, 555–575 (1980).
[CrossRef]

J. Phys. (Paris)

B. J. Ackerson, N. A. Clark, “Dynamic light scattering at low rates of shear,” J. Phys. (Paris) 42, 929–936 (1981).
[CrossRef]

J. Phys. A

P. J. Bourke, J. Butterworth, L. E. Drain, P. A. Egelstaff, A. J. Hughes, P. Hutchinson, D. A. Jackson, E. Jakeman, B. Moss, J. O’Shaughnessy, E. R. Pike, P. Schofield, “A study of the spatial structure of turbulent flow by intensity-fluctuation spectroscopy,” J. Phys. A 3, 216–228 (1970).
[CrossRef]

Phys. Fluids

See, e.g., P. Tong, W. I. Goldburg, “Relative velocity fluctuations in turbulent flows at moderate Reynolds number II. Model calculation,” Phys. Fluids 31, 3253–3259 (1988).
[CrossRef]

Phys. Rev. A

K. J. Måløy, W. I. Goldburg, H. K. Pak, “Spatial coherence of homodyne light scattering from particles in a convective field,” Phys. Rev. A 46, 3288–3291 (1992).
[CrossRef]

P. Tong, W. I. Goldburg, C. K. Chan, A. Sirivat, “Turbulent transition by photon correlation spectroscopy,” Phys. Rev. A 37, 2125–2133 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett.

H. K. Pak, W. I. Goldburg, A. Sirivat, “Measuring the probability distribution of the relative velocities in grid-generated turbulence,” Phys. Rev. Lett. 68, 938–941 (1992); “An experimental study of weak turbulence,” Fluid Dyn. Res. 8, 19–31 (1991).
[CrossRef] [PubMed]

P. Tong, Y. Shen, “Relative velocity fluctuations in turbulent Rayleigh–Bénard convection,” Phys. Rev. Lett. 69, 2066–2069 (1992).
[CrossRef]

H. Kellay, X.-L. Wu, W. I. Goldburg, “Experiments with turbulent soap films,” Phys. Rev. Lett. 74, 3975–3978 (1995).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A

G. I. Taylor, “The spectrum of turbulence,” Proc. R. Soc. London, Ser. A 164, 476–490 (1938).
[CrossRef]

Other

H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 8, pp. 133–200.

S. Hanson, “The laser gradient anemometer,” in Photon Correlation Techniques in Fluid Mechanics, E. O. Schulz-DuBois, ed. (Springer-Verlag, Berlin, 1983), pp. 212–220.

C. Keveloh, W. Staude, “The application of cross-correlation measurements to the determination of velocity gradients in fluid flow,” in Photon Correlation Techniques in Fluid Mechanics, E. O. Schulz-DuBois, ed. (Springer-Verlag, Berlin, 1983), pp. 221–224.

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

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of (a) the scattering geometry and (b) the experimental setup.

Fig. 2
Fig. 2

Measured intensity profile f(x, x1) as a function of x-x1. The solid curve is a fit by Eq. (10) with l0=0.83 mm.

Fig. 3
Fig. 3

(a) Measured g(τ) as a function of the delay time τ when ω=2.5 rad s-1 and L=4.33 mm (circles). The squares were obtained when one of the input optical fibers was blocked. (b) Measured g(τ) obtained with an additional lens when L=3.68 mm (circles). The squares were obtained when one of the input optical fibers was blocked. The lower solid curves in (a) and (b) show the fits of Eq. (13) to the circles. The upper solid curves are the fits by Eq. (13) with Gc(τ)=0.

Fig. 4
Fig. 4

Log–log plot of the fitted a2 as a function of L. The solid line shows the fitted linear function a2=41.74 (rad/ms/mm)L.

Fig. 5
Fig. 5

Measured g(τ) as a function of τ when ω=2.5 rad s-1 and L=9.68 mm. The solid curve is a fit by Eq. (13).

Equations (13)

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

g(τ)=I(t+τ)I(t)I(t)2=1+bG(τ),
G(τ)=0Ldrh(r)-+dδvP(δv, r)cos(qδvτ),
g(τ)=[E1(t+τ)+E2(t+τ)][E1(t+τ)+E2(t+τ)]*[E1(t)+E2(t)][E1(t)+E2(t)]*[E1(t)+E2(t)][E1(t)+E2(t)]*2,
E1*(t+τ)E2(t+τ)E2*(t)E1(t)+c.c.=2I1I2N2i,jNcos[q·δv(rij)τ],
g(τ)=1+I12+I22(I1+I2)2Gs(τ)+2I1I2(I1+I2)2Gc(τ)=1+bsGs(τ)+bcGc(τ).
Gc(τ)=1N2i,jNcos[q·δv(rij)τ]
Gs(τ)=1N2i,jNcos[q·δv(ξij)τ],
Gc(τ)=-+dxdxf(x, x1)f(x, x2)×-+dδvP(δv)×cos{q·[v(x)-v(x)]τ}=-+drh(r, x1, x2)-+dδvP(δv)×cos[q·δv(r)τ],
h(r, x1, x2)=-+dxf(r+x, x1)f(x, x2).
f(x, x1)=1(2πl02)1/2exp-(x-x1)2l02,
h(r, x1, x2)=0.2l0exp-(r+x2-x1)22l02.
Gc(τ)=12πl02-dxdx×exp-(x-x1)2+(x-x2)2l02×cos[ksω(x-x)τ]=cos(a2τ)exp[-(a1τ)2],
g(τ)=1+bs exp[-(a1τ)2]+bc cos(a2τ)exp[-(a1τ)2].

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