Abstract

New optical arrangements with two single-mode input fibers and a fiber-optic coupler are devised to measure the instantaneous velocity difference and the local velocity. The fibers and the coupler are polarization-preserving to guarantee a high signal-to-noise ratio. When the two input fibers are used to collect the scattered light with the same momentum-transfer vector but from two spatially separated regions in a flow, the obtained signals interfere when combined via the fiber-optic coupler. The resultant light received by a photomultiplier tube contains a cross-beat frequency proportional to the velocity difference between the two measuring points. If the two input fibers are used to collect the scattered light from a common scattering region but with two different momentum-transfer vectors, then the resultant light contains a self-beat frequency proportional to the local velocity at the measuring point. The experiment shows that both the cross-beat and the self-beat signals are large and that the standard laser Doppler signal processor can be used to measure the velocity difference and the local velocity in real time. The new technique will have various applications in the general area of fluid dynamics.

© 2001 Optical Society of America

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References

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  1. U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge U. Press, Cambridge, UK, 1995).
  2. K. R. Sreenivasan, “Fluid turbulence,” Rev. Mod. Phys. 71, S383–395 (1999).
    [CrossRef]
  3. G. I. Taylor, “The spectrum of turbulence,” Proc. R. Soc. London Ser. A 164, 476–490 (1938).
    [CrossRef]
  4. 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]
  5. Y. Du, B. J. Ackerson, P. Tong, “Velocity difference measurement with a fiber-optic coupler,” J. Opt. Soc. Am. A 15, 2433–2439 (1998).
    [CrossRef]
  6. B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).
  7. 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]
  8. 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).
    [CrossRef] [PubMed]
  9. P. Tong, Y. Shen, “Relative velocity fluctuations in turbulent Rayleigh–Bénard convection,” Phys. Rev. Lett. 69, 2066–2069 (1992).
    [CrossRef] [PubMed]
  10. H. Kellay, X.-L. Wu, W. I. Goldburg, “Experiments with turbulent soap films,” Phys. Rev. Lett. 74, 3975–3978 (1995).
    [CrossRef] [PubMed]
  11. Oz Optics Ltd, 219 Westbrook Road, Carp, Ontario, Canada, K0A 1LO ( http://ozoptics.com ).
  12. P. Tong, K.-Q. Xia, B. J. Ackerson, “Incoherent cross-correlation spectroscopy,” J. Chem. Phys. 98, 9256–9264 (1993).
    [CrossRef]
  13. J. W. Daily, D. R. F. Harleman, Fluid Dynamics (Addison–Wesley, Reading, Mass., 1966), p. 421.
  14. F. M. White, Viscous Fluid Flow, (McGraw-Hill, New York, 1991), p. 470.
  15. F. Durst, J. H. Whitelaw, “Optimization of optical anemometers,” Proc. R. Soc. London Ser. A 324, 157–181 (1971).
  16. L. E. Drain, The Laser Doppler Technique (Wiley, New York, 1980).
  17. M. A. Rutgers, X.-L. Wu, R. Bagavatula, A. A. Peterson, W. I. Goldburg, “Two-dimensional velocity profiles and laminar boundary layers in flowing soap films,” Phys. Fluids 8, 2847 (1997).
    [CrossRef]
  18. W. I. Goldburg, A. Belmonte, X.-L. Wu, I. Zusman, “Flowing soap films: a laboratory for studying two-dimensional hydrodynamics,” Physica A 254, 231–247 (1998).
    [CrossRef]
  19. V. K. Horváth, R. Crassman, W. I. Goldburg, X.-L. Wu, “Hysteresis at low Reynolds number: onset of two-dimensional vortex shedding,” Phys. Rev. E 61, R4702–4705 (2000).
    [CrossRef]
  20. For a full description of the correlator board, see http://karman.phyast.pitt.edu/horvath/corr/ . The total cost for the correlator board is less than $100. The device can be duplicated for nonprofit applications without permission.
  21. See, e.g., W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, UK, 1992).
  22. J. D. Scargle, “Studies in astronomical time series analysis III,” Astrophys. J. 343, 874–887 (1989).
    [CrossRef]
  23. S. H. Yao, P. Tong, B. J. Ackerson, “Instantaneous vorticity measurements using fiber-optic couplers,” manuscript available from the authors, c/o P. Tong at Oklahoma State University, Physics Department, Stillwater, Okla. 74078-3072.

2000 (1)

V. K. Horváth, R. Crassman, W. I. Goldburg, X.-L. Wu, “Hysteresis at low Reynolds number: onset of two-dimensional vortex shedding,” Phys. Rev. E 61, R4702–4705 (2000).
[CrossRef]

1999 (1)

K. R. Sreenivasan, “Fluid turbulence,” Rev. Mod. Phys. 71, S383–395 (1999).
[CrossRef]

1998 (2)

Y. Du, B. J. Ackerson, P. Tong, “Velocity difference measurement with a fiber-optic coupler,” J. Opt. Soc. Am. A 15, 2433–2439 (1998).
[CrossRef]

W. I. Goldburg, A. Belmonte, X.-L. Wu, I. Zusman, “Flowing soap films: a laboratory for studying two-dimensional hydrodynamics,” Physica A 254, 231–247 (1998).
[CrossRef]

1997 (2)

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]

M. A. Rutgers, X.-L. Wu, R. Bagavatula, A. A. Peterson, W. I. Goldburg, “Two-dimensional velocity profiles and laminar boundary layers in flowing soap films,” Phys. Fluids 8, 2847 (1997).
[CrossRef]

1995 (1)

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

1993 (1)

P. Tong, K.-Q. Xia, B. J. Ackerson, “Incoherent cross-correlation spectroscopy,” J. Chem. Phys. 98, 9256–9264 (1993).
[CrossRef]

1992 (2)

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).
[CrossRef] [PubMed]

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

1989 (1)

J. D. Scargle, “Studies in astronomical time series analysis III,” Astrophys. J. 343, 874–887 (1989).
[CrossRef]

1988 (1)

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]

1971 (1)

F. Durst, J. H. Whitelaw, “Optimization of optical anemometers,” Proc. R. Soc. London Ser. A 324, 157–181 (1971).

1938 (1)

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

Ackerson, B. J.

Y. Du, B. J. Ackerson, P. Tong, “Velocity difference measurement with a fiber-optic coupler,” J. Opt. Soc. Am. A 15, 2433–2439 (1998).
[CrossRef]

P. Tong, K.-Q. Xia, B. J. Ackerson, “Incoherent cross-correlation spectroscopy,” J. Chem. Phys. 98, 9256–9264 (1993).
[CrossRef]

S. H. Yao, P. Tong, B. J. Ackerson, “Instantaneous vorticity measurements using fiber-optic couplers,” manuscript available from the authors, c/o P. Tong at Oklahoma State University, Physics Department, Stillwater, Okla. 74078-3072.

Bagavatula, R.

M. A. Rutgers, X.-L. Wu, R. Bagavatula, A. A. Peterson, W. I. Goldburg, “Two-dimensional velocity profiles and laminar boundary layers in flowing soap films,” Phys. Fluids 8, 2847 (1997).
[CrossRef]

Belmonte, A.

W. I. Goldburg, A. Belmonte, X.-L. Wu, I. Zusman, “Flowing soap films: a laboratory for studying two-dimensional hydrodynamics,” Physica A 254, 231–247 (1998).
[CrossRef]

Berne, B. J.

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

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.

Crassman, R.

V. K. Horváth, R. Crassman, W. I. Goldburg, X.-L. Wu, “Hysteresis at low Reynolds number: onset of two-dimensional vortex shedding,” Phys. Rev. E 61, R4702–4705 (2000).
[CrossRef]

Daily, J. W.

J. W. Daily, D. R. F. Harleman, Fluid Dynamics (Addison–Wesley, Reading, Mass., 1966), p. 421.

Drain, L. E.

L. E. Drain, The Laser Doppler Technique (Wiley, New York, 1980).

Du, Y.

Durst, F.

F. Durst, J. H. Whitelaw, “Optimization of optical anemometers,” Proc. R. Soc. London Ser. A 324, 157–181 (1971).

Flannery, B. P.

See, e.g., W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, UK, 1992).

Frisch, U.

U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge U. Press, Cambridge, UK, 1995).

Goldburg, W. I.

V. K. Horváth, R. Crassman, W. I. Goldburg, X.-L. Wu, “Hysteresis at low Reynolds number: onset of two-dimensional vortex shedding,” Phys. Rev. E 61, R4702–4705 (2000).
[CrossRef]

W. I. Goldburg, A. Belmonte, X.-L. Wu, I. Zusman, “Flowing soap films: a laboratory for studying two-dimensional hydrodynamics,” Physica A 254, 231–247 (1998).
[CrossRef]

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]

M. A. Rutgers, X.-L. Wu, R. Bagavatula, A. A. Peterson, W. I. Goldburg, “Two-dimensional velocity profiles and laminar boundary layers in flowing soap films,” Phys. Fluids 8, 2847 (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).
[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]

Harleman, D. R. F.

J. W. Daily, D. R. F. Harleman, Fluid Dynamics (Addison–Wesley, Reading, Mass., 1966), p. 421.

Horváth, V. K.

V. K. Horváth, R. Crassman, W. I. Goldburg, X.-L. Wu, “Hysteresis at low Reynolds number: onset of two-dimensional vortex shedding,” Phys. Rev. E 61, R4702–4705 (2000).
[CrossRef]

Kellay, H.

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

Narayanan, T.

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).
[CrossRef] [PubMed]

Pecora, R.

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

Peterson, A. A.

M. A. Rutgers, X.-L. Wu, R. Bagavatula, A. A. Peterson, W. I. Goldburg, “Two-dimensional velocity profiles and laminar boundary layers in flowing soap films,” Phys. Fluids 8, 2847 (1997).
[CrossRef]

Press, W. H.

See, e.g., W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, UK, 1992).

Rutgers, M. A.

M. A. Rutgers, X.-L. Wu, R. Bagavatula, A. A. Peterson, W. I. Goldburg, “Two-dimensional velocity profiles and laminar boundary layers in flowing soap films,” Phys. Fluids 8, 2847 (1997).
[CrossRef]

Scargle, J. D.

J. D. Scargle, “Studies in astronomical time series analysis III,” Astrophys. J. 343, 874–887 (1989).
[CrossRef]

Shen, Y.

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

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).
[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]

Sreenivasan, K. R.

K. R. Sreenivasan, “Fluid turbulence,” Rev. Mod. Phys. 71, S383–395 (1999).
[CrossRef]

Taylor, G. I.

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

Teukolsky, S. A.

See, e.g., W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, UK, 1992).

Tong, P.

Y. Du, B. J. Ackerson, P. Tong, “Velocity difference measurement with a fiber-optic coupler,” J. Opt. Soc. Am. A 15, 2433–2439 (1998).
[CrossRef]

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, K.-Q. Xia, B. J. Ackerson, “Incoherent cross-correlation spectroscopy,” J. Chem. Phys. 98, 9256–9264 (1993).
[CrossRef]

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

S. H. Yao, P. Tong, B. J. Ackerson, “Instantaneous vorticity measurements using fiber-optic couplers,” manuscript available from the authors, c/o P. Tong at Oklahoma State University, Physics Department, Stillwater, Okla. 74078-3072.

Vetterling, W. T.

See, e.g., W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, UK, 1992).

White, F. M.

F. M. White, Viscous Fluid Flow, (McGraw-Hill, New York, 1991), p. 470.

Whitelaw, J. H.

F. Durst, J. H. Whitelaw, “Optimization of optical anemometers,” Proc. R. Soc. London Ser. A 324, 157–181 (1971).

Wu, X.-L.

V. K. Horváth, R. Crassman, W. I. Goldburg, X.-L. Wu, “Hysteresis at low Reynolds number: onset of two-dimensional vortex shedding,” Phys. Rev. E 61, R4702–4705 (2000).
[CrossRef]

W. I. Goldburg, A. Belmonte, X.-L. Wu, I. Zusman, “Flowing soap films: a laboratory for studying two-dimensional hydrodynamics,” Physica A 254, 231–247 (1998).
[CrossRef]

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]

M. A. Rutgers, X.-L. Wu, R. Bagavatula, A. A. Peterson, W. I. Goldburg, “Two-dimensional velocity profiles and laminar boundary layers in flowing soap films,” Phys. Fluids 8, 2847 (1997).
[CrossRef]

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

Xia, K.-Q.

P. Tong, K.-Q. Xia, B. J. Ackerson, “Incoherent cross-correlation spectroscopy,” J. Chem. Phys. 98, 9256–9264 (1993).
[CrossRef]

Yao, S. H.

S. H. Yao, P. Tong, B. J. Ackerson, “Instantaneous vorticity measurements using fiber-optic couplers,” manuscript available from the authors, c/o P. Tong at Oklahoma State University, Physics Department, Stillwater, Okla. 74078-3072.

Zusman, I.

W. I. Goldburg, A. Belmonte, X.-L. Wu, I. Zusman, “Flowing soap films: a laboratory for studying two-dimensional hydrodynamics,” Physica A 254, 231–247 (1998).
[CrossRef]

Appl. Opt. (1)

Astrophys. J. (1)

J. D. Scargle, “Studies in astronomical time series analysis III,” Astrophys. J. 343, 874–887 (1989).
[CrossRef]

J. Chem. Phys. (1)

P. Tong, K.-Q. Xia, B. J. Ackerson, “Incoherent cross-correlation spectroscopy,” J. Chem. Phys. 98, 9256–9264 (1993).
[CrossRef]

J. Opt. Soc. Am. A (1)

Phys. Fluids (1)

M. A. Rutgers, X.-L. Wu, R. Bagavatula, A. A. Peterson, W. I. Goldburg, “Two-dimensional velocity profiles and laminar boundary layers in flowing soap films,” Phys. Fluids 8, 2847 (1997).
[CrossRef]

Phys. Rev. A (1)

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. E (1)

V. K. Horváth, R. Crassman, W. I. Goldburg, X.-L. Wu, “Hysteresis at low Reynolds number: onset of two-dimensional vortex shedding,” Phys. Rev. E 61, R4702–4705 (2000).
[CrossRef]

Phys. Rev. Lett. (3)

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).
[CrossRef] [PubMed]

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

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

Physica A (1)

W. I. Goldburg, A. Belmonte, X.-L. Wu, I. Zusman, “Flowing soap films: a laboratory for studying two-dimensional hydrodynamics,” Physica A 254, 231–247 (1998).
[CrossRef]

Proc. R. Soc. London Ser. A (2)

F. Durst, J. H. Whitelaw, “Optimization of optical anemometers,” Proc. R. Soc. London Ser. A 324, 157–181 (1971).

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

Rev. Mod. Phys. (1)

K. R. Sreenivasan, “Fluid turbulence,” Rev. Mod. Phys. 71, S383–395 (1999).
[CrossRef]

Other (9)

U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge U. Press, Cambridge, UK, 1995).

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

Oz Optics Ltd, 219 Westbrook Road, Carp, Ontario, Canada, K0A 1LO ( http://ozoptics.com ).

L. E. Drain, The Laser Doppler Technique (Wiley, New York, 1980).

J. W. Daily, D. R. F. Harleman, Fluid Dynamics (Addison–Wesley, Reading, Mass., 1966), p. 421.

F. M. White, Viscous Fluid Flow, (McGraw-Hill, New York, 1991), p. 470.

For a full description of the correlator board, see http://karman.phyast.pitt.edu/horvath/corr/ . The total cost for the correlator board is less than $100. The device can be duplicated for nonprofit applications without permission.

See, e.g., W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, UK, 1992).

S. H. Yao, P. Tong, B. J. Ackerson, “Instantaneous vorticity measurements using fiber-optic couplers,” manuscript available from the authors, c/o P. Tong at Oklahoma State University, Physics Department, Stillwater, Okla. 74078-3072.

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Figures (7)

Fig. 1
Fig. 1

(a) Scattering geometry for the velocity difference measurement. ki, incident wave vector; ks, scattered wave vector; q=ks-ki. (b) Experimental setup for the velocity difference measurement in rigid body rotation. (c) Flow cell and optical arrangement for a jet flow.

Fig. 2
Fig. 2

(a) One-beam scattering geometry for the local-velocity measurement. ki, incident wave vector; (ks)1 and (ks)2, two scattered wave vectors; q1=(ks)1-ki; q2=(ks)2-ki; Δq=q2-q1. (b) Two-beam scattering geometry for the local-velocity measurement. (ki)1 and (ki)2, two incident wave vectors; ks, scattered wave vector; q1=ks-(ki)1; q2=ks-(ki)2; Δq=q2-q1. (c) Schematic diagram of a one-beam probe for the local velocity measurement. S, measuring point; P, lens; M, frequency modulator; C, fiber-optic coupler.

Fig. 3
Fig. 3

(a) Oscilloscope trace of a typical beat burst between two moving particles separated by a distance L = 1.0 mm. The signal is obtained in rigid-body rotation. (b) Measured beat frequency Δω2 as a function of separation L. Circles, the oscilloscope trace; triangles, intensity correlation function g(τ). Solid line, a linear fit to the data points.

Fig. 4
Fig. 4

Measured histogram P(δv) of the velocity difference δv(L, t) in the jet flow. The values of L are 0.5 mm (circles) and 0.8 mm (squares). Solid curve, Gaussian fit to the circles.

Fig. 5
Fig. 5

Measured intensity correlation function g(τ)-1 as a function of delay time τ at L = 0.5 mm (circles). The measurements indicated by squares are obtained when one of the input fibers is blocked. The solid curve is a plot of Eq. (5).

Fig. 6
Fig. 6

Measured intensity autocorrelation function A(τ) as a function of delay time τ with the measuring time T = 30 ms. Inset (a) shows an enlarged portion of A(τ) for small values of τ up to τ=20 μs. Inset (b) shows the frequency power spectrum P(f) of the measured A(τ).

Fig. 7
Fig. 7

Measured intensity autocorrelation function A(τ) as a function of delay time τ with the measuring time T=50 μs. Inset (a) shows the frequency power spectrum P(f) obtained by FFT. Inset (b) shows the frequency spectrum Q(f) obtained by the Scargle–Lomb method.

Equations (5)

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)=1+I12+I22(I1+I2)2Gs(τ)+2I1I2(I1+I2)2Gc(τ)=1+bsGs(τ)+bcGc(τ),
Gc(τ)Gs(τ)-+dδvP(δv)cos(qδv(L)τ).
g(τ)=1+Gs(τ){bs+bccos(qδvτ)exp[-(qστ)2/2]}.

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