Abstract

The intensity fluctuations of scattered laser light are reinvestigated. The objective is to show that useful spectral information can be obtained by correlation of the recurrence rate of a fixed, arbitrary intensity level. In developing the concept, we first analyze the arrival and departure statistics of the scatterers in certain light-scattering experiments. Kolmogorov’s forward equation is solved for the transition probability of the scatterer occupation number in the detection volume. It then becomes apparent that in the absence of configuration periodicities the scattered light intensity of a many-scatterer system is a process with statistically independent increments. It reduces to a sequence of independent pulses when the occupation number fluctuates between zero and one. All intensity levels recur at exponentially distributed time intervals, the mean recurrence rate depending on the level. Furthermore, the recurrence rate autocorrelation function becomes an important and easily accessible observable. In the absence of scatterer configuration periodicities it equals the scatterer velocity correlation function times a constant factor. Several experimental examples utilizing commerical digital correlators are presented which demonstrate the application of recurrence rate correlation. A case of continuing interest is the measurement of turbulence spectra in fluid flow.

© 1978 Optical Society of America

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References

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  1. Photon Correlation and Light Beating Spectroscopy, edited by H. Z. Cummins and E. R. Pike (Plenum, New York, 1974).
  2. B. Crosignani, P. DiPorto, and M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).
  3. E. Jakeman, “Photon Correlation,” in Ref. 1, p. 75–149.
  4. C. J. Oliver, “Correlation Techniques,” in Ref. 1, p. 151–223.
  5. E. R. Pike, “The Application of Photon-Correlation Spectroscopy to Laser Doppler Velocimetry,” in The Use of the Laser Doppler Velocimeter, edited by W. H. Stevenson and H. D. Thompson, Proceedings of a Workshop, Purdue University, March1972, p. 133–146.
  6. G. B. Benedek, J. B. Lastovka, K. Fritsch, and T. Greytak, “Brillouin scattering in liquids and solids using low-power lasers,” J. Opt. Soc. Am. 54, 1284–1285 (1964).
    [Crossref]
  7. N. George, “Speckle from rough, moving objects,” J. Opt. Soc. Am. 66, 1182–1194 (1976).
    [Crossref]
  8. J. C. Erdmann and R. I. Gellert, “Speckle field of curved, rotating surfaces of Gaussian roughness illuminated by a laser light spot,” J. Opt. Soc. Am. 66, 1194–1204 (1976).
    [Crossref]
  9. H. Z. Cummins, “Applications of Light Beating Spectroscopy to Biology,” in Ref. 1, p. 285–330.
  10. D. W. Schaefer and B. J. Berne, “Light scattering from non-Gaussian concentration fluctuations,” Phys. Rev. Lett. 28, 475–478 (1971).
    [Crossref]
  11. S. H. Chen, P. Tartaglia, and P. N. Pusey, “Light scattering from independent particles—non-Gaussian correction to the clipped intensity correlation function, J. Phys. A 6, 490–495 (1973).
    [Crossref]
  12. D. W. Schaefer and P. N. Pusey, “Statistics of light scattered by non-Gaussian fluctuations,” in Coherence and Quantum Optics edited by L. Mandel and E. Wolf (Plenum, New York, 1973), p. 839–850.
    [Crossref]
  13. E. Jakeman, J. G. McWhirter, and P. N. Pusey, “Enhanced fluctuations in radiation scattered by a moving random phase screen,” J. Opt. Soc. Am. 66, 1175–1182 (1976).
    [Crossref]
  14. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1 and 2 (Wiley, New York, 1968, 1971).
  15. S. Chandrasekhar, “Stochastic Problems in Physics and Astronomy,” Revs. Mod. Phys. 15, 1–89 (1943).
    [Crossref]
  16. A. Kolmogoroff, “Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung,” Math. Ann. 104, 415–458 (1931).
    [Crossref]
  17. S. Karlin and J. L. McGregor, “The differential equations of birth-and-death processes and the Stieltjes moment problem,” Trans. Am. Math. Soc. 85, 489–546 (1957).
    [Crossref]
  18. C. Palm, “Intensitätsschwankungen im Fernsprechverkehr,” Ericsson Technics (Stockholm) 44, 1–189 (1943).
  19. Reference 14, Vol. 2, p. 98.
  20. J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958), p. 214.
  21. J. C. Erdmann and R. I. Gellert, “Particle arrival statistics in laser anemometry of turbulent flow,” Appl. Phys. Lett. 29, 408–411 (1976).
    [Crossref]
  22. Reference 14, Vol. 1, p. 146.
  23. Reference 14, Vol. 1, p. 164.
  24. Reference 20, p. 383.
  25. Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer, Berlin, 1975).
  26. F. Durst, A. Melling, and J. H. Whitelaw, Principles and Practices of Laser-Doppler Anemometry (Academic, New York, 1976).
  27. D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Benjamin, Reading, Mass., 1975).
  28. We have not made any assumptions about shape, size, or size distributions of the scatterers. In special cases where the simultaneous arrival or departure of scatterers is a fact, the above analysis still holds for the particular scatterer aggregates.

1976 (4)

1973 (1)

S. H. Chen, P. Tartaglia, and P. N. Pusey, “Light scattering from independent particles—non-Gaussian correction to the clipped intensity correlation function, J. Phys. A 6, 490–495 (1973).
[Crossref]

1971 (1)

D. W. Schaefer and B. J. Berne, “Light scattering from non-Gaussian concentration fluctuations,” Phys. Rev. Lett. 28, 475–478 (1971).
[Crossref]

1964 (1)

1957 (1)

S. Karlin and J. L. McGregor, “The differential equations of birth-and-death processes and the Stieltjes moment problem,” Trans. Am. Math. Soc. 85, 489–546 (1957).
[Crossref]

1943 (2)

C. Palm, “Intensitätsschwankungen im Fernsprechverkehr,” Ericsson Technics (Stockholm) 44, 1–189 (1943).

S. Chandrasekhar, “Stochastic Problems in Physics and Astronomy,” Revs. Mod. Phys. 15, 1–89 (1943).
[Crossref]

1931 (1)

A. Kolmogoroff, “Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung,” Math. Ann. 104, 415–458 (1931).
[Crossref]

Bendat, J. S.

J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958), p. 214.

Benedek, G. B.

Berne, B. J.

D. W. Schaefer and B. J. Berne, “Light scattering from non-Gaussian concentration fluctuations,” Phys. Rev. Lett. 28, 475–478 (1971).
[Crossref]

Bertolotti, M.

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

Chandrasekhar, S.

S. Chandrasekhar, “Stochastic Problems in Physics and Astronomy,” Revs. Mod. Phys. 15, 1–89 (1943).
[Crossref]

Chen, S. H.

S. H. Chen, P. Tartaglia, and P. N. Pusey, “Light scattering from independent particles—non-Gaussian correction to the clipped intensity correlation function, J. Phys. A 6, 490–495 (1973).
[Crossref]

Crosignani, B.

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

Cummins, H. Z.

H. Z. Cummins, “Applications of Light Beating Spectroscopy to Biology,” in Ref. 1, p. 285–330.

DiPorto, P.

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

Durst, F.

F. Durst, A. Melling, and J. H. Whitelaw, Principles and Practices of Laser-Doppler Anemometry (Academic, New York, 1976).

Erdmann, J. C.

J. C. Erdmann and R. I. Gellert, “Speckle field of curved, rotating surfaces of Gaussian roughness illuminated by a laser light spot,” J. Opt. Soc. Am. 66, 1194–1204 (1976).
[Crossref]

J. C. Erdmann and R. I. Gellert, “Particle arrival statistics in laser anemometry of turbulent flow,” Appl. Phys. Lett. 29, 408–411 (1976).
[Crossref]

Feller, W.

W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1 and 2 (Wiley, New York, 1968, 1971).

Forster, D.

D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Benjamin, Reading, Mass., 1975).

Fritsch, K.

Gellert, R. I.

J. C. Erdmann and R. I. Gellert, “Speckle field of curved, rotating surfaces of Gaussian roughness illuminated by a laser light spot,” J. Opt. Soc. Am. 66, 1194–1204 (1976).
[Crossref]

J. C. Erdmann and R. I. Gellert, “Particle arrival statistics in laser anemometry of turbulent flow,” Appl. Phys. Lett. 29, 408–411 (1976).
[Crossref]

George, N.

Greytak, T.

Jakeman, E.

Karlin, S.

S. Karlin and J. L. McGregor, “The differential equations of birth-and-death processes and the Stieltjes moment problem,” Trans. Am. Math. Soc. 85, 489–546 (1957).
[Crossref]

Kolmogoroff, A.

A. Kolmogoroff, “Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung,” Math. Ann. 104, 415–458 (1931).
[Crossref]

Lastovka, J. B.

McGregor, J. L.

S. Karlin and J. L. McGregor, “The differential equations of birth-and-death processes and the Stieltjes moment problem,” Trans. Am. Math. Soc. 85, 489–546 (1957).
[Crossref]

McWhirter, J. G.

Melling, A.

F. Durst, A. Melling, and J. H. Whitelaw, Principles and Practices of Laser-Doppler Anemometry (Academic, New York, 1976).

Oliver, C. J.

C. J. Oliver, “Correlation Techniques,” in Ref. 1, p. 151–223.

Palm, C.

C. Palm, “Intensitätsschwankungen im Fernsprechverkehr,” Ericsson Technics (Stockholm) 44, 1–189 (1943).

Pike, E. R.

E. R. Pike, “The Application of Photon-Correlation Spectroscopy to Laser Doppler Velocimetry,” in The Use of the Laser Doppler Velocimeter, edited by W. H. Stevenson and H. D. Thompson, Proceedings of a Workshop, Purdue University, March1972, p. 133–146.

Pusey, P. N.

E. Jakeman, J. G. McWhirter, and P. N. Pusey, “Enhanced fluctuations in radiation scattered by a moving random phase screen,” J. Opt. Soc. Am. 66, 1175–1182 (1976).
[Crossref]

S. H. Chen, P. Tartaglia, and P. N. Pusey, “Light scattering from independent particles—non-Gaussian correction to the clipped intensity correlation function, J. Phys. A 6, 490–495 (1973).
[Crossref]

D. W. Schaefer and P. N. Pusey, “Statistics of light scattered by non-Gaussian fluctuations,” in Coherence and Quantum Optics edited by L. Mandel and E. Wolf (Plenum, New York, 1973), p. 839–850.
[Crossref]

Schaefer, D. W.

D. W. Schaefer and B. J. Berne, “Light scattering from non-Gaussian concentration fluctuations,” Phys. Rev. Lett. 28, 475–478 (1971).
[Crossref]

D. W. Schaefer and P. N. Pusey, “Statistics of light scattered by non-Gaussian fluctuations,” in Coherence and Quantum Optics edited by L. Mandel and E. Wolf (Plenum, New York, 1973), p. 839–850.
[Crossref]

Tartaglia, P.

S. H. Chen, P. Tartaglia, and P. N. Pusey, “Light scattering from independent particles—non-Gaussian correction to the clipped intensity correlation function, J. Phys. A 6, 490–495 (1973).
[Crossref]

Whitelaw, J. H.

F. Durst, A. Melling, and J. H. Whitelaw, Principles and Practices of Laser-Doppler Anemometry (Academic, New York, 1976).

Appl. Phys. Lett. (1)

J. C. Erdmann and R. I. Gellert, “Particle arrival statistics in laser anemometry of turbulent flow,” Appl. Phys. Lett. 29, 408–411 (1976).
[Crossref]

Ericsson Technics (Stockholm) (1)

C. Palm, “Intensitätsschwankungen im Fernsprechverkehr,” Ericsson Technics (Stockholm) 44, 1–189 (1943).

J. Opt. Soc. Am. (4)

J. Phys. A (1)

S. H. Chen, P. Tartaglia, and P. N. Pusey, “Light scattering from independent particles—non-Gaussian correction to the clipped intensity correlation function, J. Phys. A 6, 490–495 (1973).
[Crossref]

Math. Ann. (1)

A. Kolmogoroff, “Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung,” Math. Ann. 104, 415–458 (1931).
[Crossref]

Phys. Rev. Lett. (1)

D. W. Schaefer and B. J. Berne, “Light scattering from non-Gaussian concentration fluctuations,” Phys. Rev. Lett. 28, 475–478 (1971).
[Crossref]

Revs. Mod. Phys. (1)

S. Chandrasekhar, “Stochastic Problems in Physics and Astronomy,” Revs. Mod. Phys. 15, 1–89 (1943).
[Crossref]

Trans. Am. Math. Soc. (1)

S. Karlin and J. L. McGregor, “The differential equations of birth-and-death processes and the Stieltjes moment problem,” Trans. Am. Math. Soc. 85, 489–546 (1957).
[Crossref]

Other (17)

Reference 14, Vol. 2, p. 98.

J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958), p. 214.

D. W. Schaefer and P. N. Pusey, “Statistics of light scattered by non-Gaussian fluctuations,” in Coherence and Quantum Optics edited by L. Mandel and E. Wolf (Plenum, New York, 1973), p. 839–850.
[Crossref]

W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1 and 2 (Wiley, New York, 1968, 1971).

Photon Correlation and Light Beating Spectroscopy, edited by H. Z. Cummins and E. R. Pike (Plenum, New York, 1974).

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

E. Jakeman, “Photon Correlation,” in Ref. 1, p. 75–149.

C. J. Oliver, “Correlation Techniques,” in Ref. 1, p. 151–223.

E. R. Pike, “The Application of Photon-Correlation Spectroscopy to Laser Doppler Velocimetry,” in The Use of the Laser Doppler Velocimeter, edited by W. H. Stevenson and H. D. Thompson, Proceedings of a Workshop, Purdue University, March1972, p. 133–146.

H. Z. Cummins, “Applications of Light Beating Spectroscopy to Biology,” in Ref. 1, p. 285–330.

Reference 14, Vol. 1, p. 146.

Reference 14, Vol. 1, p. 164.

Reference 20, p. 383.

Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer, Berlin, 1975).

F. Durst, A. Melling, and J. H. Whitelaw, Principles and Practices of Laser-Doppler Anemometry (Academic, New York, 1976).

D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Benjamin, Reading, Mass., 1975).

We have not made any assumptions about shape, size, or size distributions of the scatterers. In special cases where the simultaneous arrival or departure of scatterers is a fact, the above analysis still holds for the particular scatterer aggregates.

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Figures (7)

FIG. 1
FIG. 1

Typical record of intensity fluctuations (upper trace). Two pulse sequences (solid and dashed lines, respectively, in the lower trace) are derived by crossing of a threshold.

FIG. 2
FIG. 2

Signal processor, used to generate displays of the recurrence rate correlation function and the probability density function of the recurrence interval.

FIG. 3
FIG. 3

(a) Schematic of vortex generator. (b) Intensity recurrence rate correlation function displaying vortex periodicity. (c) Intensity correlation function for comparison.

FIG. 4
FIG. 4

Laser light scattering from a rotating disk with rough surface. (a) Scattering configuration. (b) Probability density function of the recurrence interval. (c) Recurrence rate correlation function.

FIG. 5
FIG. 5

Laser light scattering from a cardboard model mounted on a vibrating speaker. (a) Scattering configuration. (b) Velocity correlation function, 10 Hz vibration frequency. (c) Same as (b), but for 20 Hz.

FIG. 6
FIG. 6

Laser light scattering from particles embedded in a periodic air flow. (a) Configuration. (b) Velocity correlation function displaying 20 kHz velocity modulation periodicity.

FIG. 7
FIG. 7

Laser light scattering from particles embedded in a turbulent air flow. (a) Configuration. (b) Velocity correlation functions. The water seed particles are larger than the oil particles (approximately 10–100 μm compared with approximately 0.5–5 μm diam.).

Equations (64)

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A ( r ) = i = 1 M a i ( r ) b i ( r ) exp [ i ϕ i ( r ) ] ,
P { N = N } = P N = e - N N N / N !
I ( r ) = A ( r ) A * ( r )
I = i = 1 M b i ( a i 2 + A i cos ψ i ) ,             i = 1 M b i = N
A i cos ψ i = a i j i a j b j cos ( ϕ i - ϕ j )
i = 1 M b i A i sin ψ i = 0 ,
S J , N = S 0 + S 1 + + S N ,
S 0 = X 01 + X 02 + + X 0 N ,
S m = X m 1 + X m 2 + + X m ( N - 1 ) ,             ( m = 1 , 2 , , N ) .
X 0 = a i 2 ,             0 < a i 2 < ,
X m = a i a j cos ( ϕ i - ϕ j ) = a i j 2 ,             i j .
P { J > I } = e - I / I
P N , N * = e - N p 0 k = 0 J ( N k ) q 0 k p 0 N - k N N - k ( N - k ) ! ;             ( N k ) = N ! ( N - k ) ! k ! ,
P N , N * = lim p 0 0 P N , N .
b ( k ; N ; p 0 ) = ( N k ) q 0 k p 0 N - k ,             q 0 = 1 - p 0
p ( N ; N ) = e - N N N / N ! .
P ( s ) = j P j s j ,
k = 0 N ( N k ) ( q 0 s ) k p 0 N - k = ( q 0 s + p 0 ) N
k = 0 e - N ( N s ) k k ! = e - N ( 1 - s ) .
C ( s ) = A ( s ) B ( s ) .
P N , N * ( s ) = e - N ( 1 - s ) p 0 [ 1 - ( 1 - s ) q 0 ] N .
p j k ( 2 ) = ν p j ν p ν k
p j k ( n + 1 ) = ν p j ν p ν k ( n ) ,
p j k ( m + n ) = ν p j ν ( m ) p ν k ( n ) .
lim h 0 [ 1 - P N , N ( t , t + h ) ] / h = c N ( t ) = λ + N p 0
c N ( t ) π N , N ( t ) = lim h o P N , N ( t , t + h ) / h .
P N , N + 1 * N p 0 ,
P N , N - 1 * N p 0 ,
P N , N * 1 - ( N + N ) p 0 ,
P N , N * 0.
λ h = N p 0 .
c N ( t ) = ( N + N ) p 0 ( t ) ,
π N , N + 1 = N / ( N + N ) ,
π N , N - 1 = N / ( N + N ) ,
π N , N = 0.
P N , N ( τ , t + h ) = N P N , N ( τ , t ) P N , N ( t , t + h ) .
[ P N , N ( τ , t + h ) - P N , N ( τ , t ) ] / h = - c N ( t ) P N , N ( τ , t ) + h - 1 N N P N , N ( τ , t ) P N , N ( t , t + h ) + o ( h ) .
P N , N ( τ , t ) t = - c N ( t ) P N , N ( τ , t ) + N N P N , N ( τ , t ) c N ( t ) π N , N ( t ) .
P N , N ( τ , t ) t = { - ( N + N ) P N , N ( τ , t ) + N P N , N - 1 ( τ , t ) + ( N + 1 ) P N , N + 1 ( τ , t ) } p 0 ( t ) .
P N , N ( τ , τ ) = { 1 , for N = N 0 , otherwise .
P N ( s , τ , t ) = N P N , N ( τ , t ) s N ,
P N t = { - N ( N + N ) P N , N s N + N N P N , N - 1 s N + N ( N + 1 ) P N , N + 1 s N } p 0
P N t = ( 1 - s ) ( - N P N + P N s ) p 0 ( t )
P N ( s , τ , t ) = e - N ( 1 - s ) p [ 1 - ( 1 - s ) q ] N ,
p ( t ) = 1 - exp ( - τ t p 0 ( t ) d t )
I ( t 1 ) I ( t 2 ) 0 ,             t 1 t 2 ,
f ( k ; r ; p e ) = ( r + k - 1 k ) p e r q e k .
f ( k ; 1 ; p e ) = p e q e k
μ e = m = 1 m h p e ( 1 - p e ) m - 1 = h p e .
P { T e > k h } = ( 1 - p e ) k
P { T e > t 0 } = ( 1 - h / μ e ) t 0 / h e - t 0 / μ e
p e / h = 1 / μ e = ν e / T .
d ( I ) = e - I / I / I
g ( İ ) = e - İ 2 / 2 D 2 / D 2 π ,
ν e = T - + İ d ( I t ) g ( İ ) d İ = ( 2 / π ) ω s T e - I t / I ,             N 1
p ( r 1 ; r 2 ; ν e ) = p ( r 1 ; ν e ) p ( r 2 ; ν e )
ν e = A v ( t ) ,
Q ( r 1 ; r 2 ; ν e ) = - + - + p ( r 1 ; ν e ( v 1 ) ) × p ( r 2 ; ν e ( v 2 ) ) f ( v 1 ; v 2 ) d v 1 d v 2 ,
T f - 1
r 1 r 2 = r ( t 1 ) r ( t 2 ) = r 1 = 0 r 2 = 0 r 1 r 2 Q ( r 1 ; r 2 ; ν e )
r 1 r 2 = - + - + [ r 1 = 0 r 1 p ( r 1 ; ν e ( v 1 ) ) ] × [ r 2 = 0 r 2 p ( r 2 ; ν e ( v 2 ) ) ] f ( v 1 ; v 2 ) d v 1 d v 2 .
r ( t 1 ) r ( t 2 ) = A 2 - + - + v 1 v 2 f ( v 1 ; v 2 ) d v 1 d v 2 = A 2 v ( t 1 ) v ( t 2 ) .
p 0 = { A / ( π r 0 2 ) , u τ 2 r 0 1 , u τ > 2 r 0
A = ( 1 / 2 ) [ 4 r 0 2 sin - 1 ( u τ / 2 r 0 ) + u τ ( 4 r 0 2 - u 2 τ 2 ) 1 / 2 ]