Abstract

Dynamic statistical properties of laser speckle produced in the diffraction field from an out-of-plane vibrating object are studied theoretically and experimentally. The equations expressing the averaged power spectrum and the autocorrelation function of vibrating laser speckle intensity fluctuations are derived for the first time and evaluated numerically. An analysis of those equations shows that the statistical properties of the averaged power spectrum and the autocorrelation function of vibrating speckle are determined by a product of both the derivative value (vibration slope) of vibrating amplitudes of the object and the laser beam width used for illumination of the object. The correlation length and the power spectrum of vibrating speckle are investigated in detail as a function of the vibration slope of the object and the beam width of illumination. It is shown from numerical evaluation of the equations that the power spectral width of vibrating speckles is linearly dependent on the product of the vibration slope and the beam width except for extremely small values. Experiments confirming the theory were performed and show excellent agreement with the theoretical results.

© 1978 Optical Society of America

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References

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  1. N. Takai, Jpn. J. Appl. Phys. 13, 2025 (1974).
    [CrossRef]
  2. E. Jakeman, J. G. McWhirter, J. Phys. A 9, 785 (1976).
    [CrossRef]
  3. P. N. Pusey, J. Phys. D 9, 1399 (1976).
    [CrossRef]
  4. G. Stavis, Instrum. Control Syst. 39, 99 (1966).
  5. H. Ogiwara, H. Ukita, Jpn. J. Appl. Phys. 14, Suppl. 1, 307 (1975).
    [CrossRef]
  6. S. Komatsu, I. Yamaguchi, H. Saito, Opt. Commun. 18, 314 (1976).
    [CrossRef]
  7. J. Ohtsubo, T. Asakura, Opt. Quantum. Electron. 8, 523 (1976).
    [CrossRef]
  8. A. W. Lohmann, G. P. Weigelt, Opt. Commun. 14, 252 (1975).
    [CrossRef]
  9. A. W. Lohmann, G. P. Weigelt, Opt. Commun. 17, 47 (1976).
    [CrossRef]
  10. A. W. Lohmann, G. P. Weigelt, J. Opt. Soc. Am. 66, 1271 (1976).
    [CrossRef]
  11. E. Archbold, A. E. Ennos, P. A. Taylor, Optical Instruments and Techniques (Oriel Press, Newcastle-upon-Tyne, 1969), p. 265.
  12. E. Archbold, A. E. Ennos, Opt. Laser Technol. 7, 17 (1975).
    [CrossRef]
  13. E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
    [CrossRef]
  14. A. E. Ennos, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), p. 203.
    [CrossRef]
  15. F. P. Chiang, R. M. Juang, Appl. Opt. 15, 2199 (1976).
    [CrossRef] [PubMed]
  16. F. P. Chiang, R. M. Juang, Opt. Acta 23, 997 (1976).
    [CrossRef]
  17. H. J. Tiziani, Opt. Acta 18, 891 (1971).
    [CrossRef]
  18. H. J. Tiziani, Opt. Commun. 5, 271 (1972).
    [CrossRef]
  19. L. Ek, N. E. Molin, Opt. Commun. 2, 419 (1971).
    [CrossRef]
  20. N. Fernelius, C. Tome, J. Opt. Soc. Am. 61, 566 (1971).
    [CrossRef]
  21. J. J. Hopkins, G. H. Tidbery, Opt. Acta 24, 773 (1977).
    [CrossRef]
  22. D. Joyeux, S. Lowenthal, Opt. Commun. 4, 108 (1971).
    [CrossRef]
  23. K. J. Ebeling, Opt. Commun. 24, 125 (1978).
    [CrossRef]
  24. B. Eliasson, F. M. Mottier, J. Opt. Soc. Am. 61, 559 (1971).
    [CrossRef]
  25. N. Takai, Opt. Commun. 25, 31 (1978).
    [CrossRef]

1978 (2)

K. J. Ebeling, Opt. Commun. 24, 125 (1978).
[CrossRef]

N. Takai, Opt. Commun. 25, 31 (1978).
[CrossRef]

1977 (1)

J. J. Hopkins, G. H. Tidbery, Opt. Acta 24, 773 (1977).
[CrossRef]

1976 (8)

S. Komatsu, I. Yamaguchi, H. Saito, Opt. Commun. 18, 314 (1976).
[CrossRef]

J. Ohtsubo, T. Asakura, Opt. Quantum. Electron. 8, 523 (1976).
[CrossRef]

F. P. Chiang, R. M. Juang, Appl. Opt. 15, 2199 (1976).
[CrossRef] [PubMed]

F. P. Chiang, R. M. Juang, Opt. Acta 23, 997 (1976).
[CrossRef]

E. Jakeman, J. G. McWhirter, J. Phys. A 9, 785 (1976).
[CrossRef]

P. N. Pusey, J. Phys. D 9, 1399 (1976).
[CrossRef]

A. W. Lohmann, G. P. Weigelt, Opt. Commun. 17, 47 (1976).
[CrossRef]

A. W. Lohmann, G. P. Weigelt, J. Opt. Soc. Am. 66, 1271 (1976).
[CrossRef]

1975 (3)

E. Archbold, A. E. Ennos, Opt. Laser Technol. 7, 17 (1975).
[CrossRef]

H. Ogiwara, H. Ukita, Jpn. J. Appl. Phys. 14, Suppl. 1, 307 (1975).
[CrossRef]

A. W. Lohmann, G. P. Weigelt, Opt. Commun. 14, 252 (1975).
[CrossRef]

1974 (1)

N. Takai, Jpn. J. Appl. Phys. 13, 2025 (1974).
[CrossRef]

1972 (2)

E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

H. J. Tiziani, Opt. Commun. 5, 271 (1972).
[CrossRef]

1971 (5)

L. Ek, N. E. Molin, Opt. Commun. 2, 419 (1971).
[CrossRef]

N. Fernelius, C. Tome, J. Opt. Soc. Am. 61, 566 (1971).
[CrossRef]

D. Joyeux, S. Lowenthal, Opt. Commun. 4, 108 (1971).
[CrossRef]

H. J. Tiziani, Opt. Acta 18, 891 (1971).
[CrossRef]

B. Eliasson, F. M. Mottier, J. Opt. Soc. Am. 61, 559 (1971).
[CrossRef]

1966 (1)

G. Stavis, Instrum. Control Syst. 39, 99 (1966).

Archbold, E.

E. Archbold, A. E. Ennos, Opt. Laser Technol. 7, 17 (1975).
[CrossRef]

E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

E. Archbold, A. E. Ennos, P. A. Taylor, Optical Instruments and Techniques (Oriel Press, Newcastle-upon-Tyne, 1969), p. 265.

Asakura, T.

J. Ohtsubo, T. Asakura, Opt. Quantum. Electron. 8, 523 (1976).
[CrossRef]

Chiang, F. P.

Ebeling, K. J.

K. J. Ebeling, Opt. Commun. 24, 125 (1978).
[CrossRef]

Ek, L.

L. Ek, N. E. Molin, Opt. Commun. 2, 419 (1971).
[CrossRef]

Eliasson, B.

Ennos, A. E.

E. Archbold, A. E. Ennos, Opt. Laser Technol. 7, 17 (1975).
[CrossRef]

E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

A. E. Ennos, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), p. 203.
[CrossRef]

E. Archbold, A. E. Ennos, P. A. Taylor, Optical Instruments and Techniques (Oriel Press, Newcastle-upon-Tyne, 1969), p. 265.

Fernelius, N.

Hopkins, J. J.

J. J. Hopkins, G. H. Tidbery, Opt. Acta 24, 773 (1977).
[CrossRef]

Jakeman, E.

E. Jakeman, J. G. McWhirter, J. Phys. A 9, 785 (1976).
[CrossRef]

Joyeux, D.

D. Joyeux, S. Lowenthal, Opt. Commun. 4, 108 (1971).
[CrossRef]

Juang, R. M.

Komatsu, S.

S. Komatsu, I. Yamaguchi, H. Saito, Opt. Commun. 18, 314 (1976).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, G. P. Weigelt, Opt. Commun. 17, 47 (1976).
[CrossRef]

A. W. Lohmann, G. P. Weigelt, J. Opt. Soc. Am. 66, 1271 (1976).
[CrossRef]

A. W. Lohmann, G. P. Weigelt, Opt. Commun. 14, 252 (1975).
[CrossRef]

Lowenthal, S.

D. Joyeux, S. Lowenthal, Opt. Commun. 4, 108 (1971).
[CrossRef]

McWhirter, J. G.

E. Jakeman, J. G. McWhirter, J. Phys. A 9, 785 (1976).
[CrossRef]

Molin, N. E.

L. Ek, N. E. Molin, Opt. Commun. 2, 419 (1971).
[CrossRef]

Mottier, F. M.

Ogiwara, H.

H. Ogiwara, H. Ukita, Jpn. J. Appl. Phys. 14, Suppl. 1, 307 (1975).
[CrossRef]

Ohtsubo, J.

J. Ohtsubo, T. Asakura, Opt. Quantum. Electron. 8, 523 (1976).
[CrossRef]

Pusey, P. N.

P. N. Pusey, J. Phys. D 9, 1399 (1976).
[CrossRef]

Saito, H.

S. Komatsu, I. Yamaguchi, H. Saito, Opt. Commun. 18, 314 (1976).
[CrossRef]

Stavis, G.

G. Stavis, Instrum. Control Syst. 39, 99 (1966).

Takai, N.

N. Takai, Opt. Commun. 25, 31 (1978).
[CrossRef]

N. Takai, Jpn. J. Appl. Phys. 13, 2025 (1974).
[CrossRef]

Taylor, P. A.

E. Archbold, A. E. Ennos, P. A. Taylor, Optical Instruments and Techniques (Oriel Press, Newcastle-upon-Tyne, 1969), p. 265.

Tidbery, G. H.

J. J. Hopkins, G. H. Tidbery, Opt. Acta 24, 773 (1977).
[CrossRef]

Tiziani, H. J.

H. J. Tiziani, Opt. Commun. 5, 271 (1972).
[CrossRef]

H. J. Tiziani, Opt. Acta 18, 891 (1971).
[CrossRef]

Tome, C.

Ukita, H.

H. Ogiwara, H. Ukita, Jpn. J. Appl. Phys. 14, Suppl. 1, 307 (1975).
[CrossRef]

Weigelt, G. P.

A. W. Lohmann, G. P. Weigelt, J. Opt. Soc. Am. 66, 1271 (1976).
[CrossRef]

A. W. Lohmann, G. P. Weigelt, Opt. Commun. 17, 47 (1976).
[CrossRef]

A. W. Lohmann, G. P. Weigelt, Opt. Commun. 14, 252 (1975).
[CrossRef]

Yamaguchi, I.

S. Komatsu, I. Yamaguchi, H. Saito, Opt. Commun. 18, 314 (1976).
[CrossRef]

Appl. Opt. (1)

Instrum. Control Syst. (1)

G. Stavis, Instrum. Control Syst. 39, 99 (1966).

J. Opt. Soc. Am. (3)

J. Phys. A (1)

E. Jakeman, J. G. McWhirter, J. Phys. A 9, 785 (1976).
[CrossRef]

J. Phys. D (1)

P. N. Pusey, J. Phys. D 9, 1399 (1976).
[CrossRef]

Jpn. J. Appl. Phys. (2)

H. Ogiwara, H. Ukita, Jpn. J. Appl. Phys. 14, Suppl. 1, 307 (1975).
[CrossRef]

N. Takai, Jpn. J. Appl. Phys. 13, 2025 (1974).
[CrossRef]

Opt. Acta (4)

F. P. Chiang, R. M. Juang, Opt. Acta 23, 997 (1976).
[CrossRef]

H. J. Tiziani, Opt. Acta 18, 891 (1971).
[CrossRef]

E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

J. J. Hopkins, G. H. Tidbery, Opt. Acta 24, 773 (1977).
[CrossRef]

Opt. Commun. (8)

D. Joyeux, S. Lowenthal, Opt. Commun. 4, 108 (1971).
[CrossRef]

K. J. Ebeling, Opt. Commun. 24, 125 (1978).
[CrossRef]

N. Takai, Opt. Commun. 25, 31 (1978).
[CrossRef]

H. J. Tiziani, Opt. Commun. 5, 271 (1972).
[CrossRef]

L. Ek, N. E. Molin, Opt. Commun. 2, 419 (1971).
[CrossRef]

S. Komatsu, I. Yamaguchi, H. Saito, Opt. Commun. 18, 314 (1976).
[CrossRef]

A. W. Lohmann, G. P. Weigelt, Opt. Commun. 14, 252 (1975).
[CrossRef]

A. W. Lohmann, G. P. Weigelt, Opt. Commun. 17, 47 (1976).
[CrossRef]

Opt. Laser Technol. (1)

E. Archbold, A. E. Ennos, Opt. Laser Technol. 7, 17 (1975).
[CrossRef]

Opt. Quantum. Electron. (1)

J. Ohtsubo, T. Asakura, Opt. Quantum. Electron. 8, 523 (1976).
[CrossRef]

Other (2)

E. Archbold, A. E. Ennos, P. A. Taylor, Optical Instruments and Techniques (Oriel Press, Newcastle-upon-Tyne, 1969), p. 265.

A. E. Ennos, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), p. 203.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram for the formation of vibrating speckle produced by an out-of-plane vibrating object.

Fig. 2
Fig. 2

Averaged power spectra obtained from numerical calculations for various values of σλ: (a) 0.01 μm, (b) 0.1 μm, (c) 0.2 μm, (d) 0.5 μm, and (e) 1.0 μm.

Fig. 3
Fig. 3

Autocorrelation functions obtained from numerical calculations for various values of σλ: (a) 0.01 μm, (b) 0.1 μm, (c) 0.2 μm, (d) 0.5 μm, and 1.0 μm.

Fig. 4
Fig. 4

Dependence of the correlation length (a) and the spectral width (b) of vibrating speckle intensity fluctuations on the value of σ.

Fig. 5
Fig. 5

Experimental arrangement for investigating time-varying vibrating speckle intensity fluctuations produced in the diffraction field by an out-of-plane vibrating cantilever.

Fig. 6
Fig. 6

Photocurrent signals of time-varying speckle intensity fluctuations in the diffraction field obtained for various values of σ (the scale in figures is 10 msec/division): (a) 0.02, (b) 0.17, (c) 0.35, and (d) 0.70.

Fig. 7
Fig. 7

Power spectra of time-varying speckle intensity fluctuations obtained for various values of σ: (a) 0.042, (b) 0.084, (c) 0.17, (d) 0.25, (e) 0.34, and (f) 0.42.

Fig. 8
Fig. 8

Autocorrelation functions of time-varying speckle intensity fluctuations obtained for various values of σ: (a) 0.064, (b) 0.085, (c) 0.11, (d) 0.14, (e) 0.17, and (f) 0.19.

Fig. 9
Fig. 9

Autocorrelation functions of time-varying speckle intensity fluctuations obtained under illumination of various Gaussian beams with different widths w: (a) 10 μm, (b) 100 μm, (c) 200 μm, (d) 400 μm, and (e) 600 μm.

Fig. 10
Fig. 10

Dependence of the correlation length (a) and the spectral width (b) on the width of Gaussian beams used for illumination. Open and closed circles indicate the experimental values for two different vibration slopes γ0 of 0.0016 rad and 0.0028 rad.

Equations (29)

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e ( t , x ) = E 0 ( ξ ) exp [ i ϕ ( ξ ) ] exp [ i ψ ( t , ξ ) ] K ( ξ , x ) d ξ ,
ψ ( t , ξ ) = ( 4 π / λ ) α ( ξ ) sin ( ω 0 t + θ 0 ) ,
i ( t , x ) = e ( t , x ) e * ( t , x ) = E 0 ( ξ 1 ) E 0 * ( ξ 2 ) exp { i [ ϕ ( ξ 1 ) ϕ ( ξ 2 ) ] } × exp { i [ ψ ( t , ξ 1 ) ψ ( t , ξ 2 ) ] } K ( ξ 1 , x ) K * ( ξ 2 , x ) d ξ 1 d ξ 2 .
i ( t , x ) = n = exp [ i n ( ω 0 t + θ 0 ) ] E 0 ( ξ 1 ) E 0 * ( ξ 2 ) × exp { i [ ϕ ( ξ 1 ) ϕ ( ξ 2 ) ] × J n { ( 4 π / λ ) [ α ( ξ 1 ) α ( ξ 2 ) ] } K ( ξ 1 , x ) K 0 * ( ξ 2 , x ) d ξ 1 d ξ 2 ,
α ( ξ 1 ) α ( ξ 2 ) = α ( ξ 0 ) ( ξ 1 ξ 2 ) + 1 2 α ( ξ 0 ) [ ( ξ 1 ξ 0 ) 2 ( ξ 2 ξ 0 ) 2 ] + 1 3 ! α ( ξ 0 ) [ ( ξ 1 ξ 0 ) 3 ( ξ 2 ξ 0 3 ] + ,
i ( t , x ) = n = exp [ i n ( ω 0 t + θ 0 ) ] E 0 ( ξ 1 ) E * ( ξ 2 ) × exp { i [ ϕ ( ξ 1 ) ϕ ( ξ 2 ) ] } × J n [ ( 4 π / λ ) γ 0 ( ξ 1 ξ 2 ) ] K ( ξ 1 , x ) K * ( ξ 2 , x ) d ξ 1 d ξ 2 ,
I ( ω , x ) = i ( t , x ) exp ( i ω t ) d t = m = δ ( ω n ω 0 ) exp ( i n θ 0 ) E 0 ( ξ 1 ) E 0 * ( ξ 2 ) × exp { i [ ϕ ( ξ 1 ) ϕ ( ξ 2 ) ] } J n [ ( 4 π / λ ) γ 0 ( ξ 1 ξ 2 ) ] × K ( ξ 1 , x ) K * ( ξ 2 , x ) d ξ 1 d ξ 2 ,
| I ( ω , x ) | 2 = n = n = δ ( ω n ω 0 ) δ ( ω n ω 0 ) exp [ i ( n n ) θ 0 ] × E 0 ( ξ 1 ) E 0 * ( ξ 2 ) E 0 * ( ξ 3 ) E 0 ( ξ 4 ) × exp { i [ ϕ ( ξ 1 ) ϕ ( ξ 2 ) ϕ ( ξ 3 ) + ϕ ( ξ 4 ) ] } × J n [ ( 4 π / λ ) γ 0 ( ξ 1 ξ 2 ) ] J n [ ( 4 π / λ ) γ 0 ( ξ 3 ξ 4 ) ] × K ( ξ 1 , x ) K * ( ξ 2 , x ) K * ( ξ 3 , x ) K ( ξ 4 , x ) d ξ 1 d ξ 2 d ξ 3 d ξ 4 ,
| I ( ω , x ) | 2 = n = δ ( ω n ω 0 ) × E 0 ( ξ 1 ) E 0 * ( ξ 2 ) E 0 * ( ξ 3 ) E 0 ( ξ 4 ) × exp { i [ ϕ ( ξ 1 ) ϕ ( ξ 2 ) ϕ ( ξ 3 ) + ϕ ( ξ 4 ) ] } × J n [ ( 4 π / λ ) γ 0 ( ξ 1 ξ 2 ) ] × J n [ ( 4 π / λ ) γ 0 ( ξ 3 ξ 4 ) ] K ( ξ 1 , x ) K * ( ξ 2 , x ) K * ( ξ 3 , x ) × K ( ξ 4 , x ) d ξ 1 d ξ 2 d ξ 3 d ξ 4 .
J n ( 0 ) = 0 ,
G exp { [ i [ ϕ ( ξ 1 ) ϕ ( ξ 2 ) ϕ ( ξ 3 ) + ϕ ( ξ 4 ) ] } .
G = exp { 1 2 [ ϕ ( ξ 1 ) ϕ ( ξ 2 ) ϕ ( ξ 3 ) + ϕ ( ξ 4 ) ] 2 } = exp ( 2 ϕ 2 ) exp { ϕ 2 [ ρ ( ξ 1 ξ 2 ) ρ ( ξ 2 ξ 3 ) + ρ ( ξ 3 ξ 4 ) ρ ( ξ 4 ξ 1 ) + ρ ( ξ 1 ξ 3 ) + ρ ( ξ 2 ξ 4 ) ] } ,
ϕ 2 = ϕ 2 ( ξ ) ,
ρ ( ξ l ξ m ) = ϕ ( ξ l ) ϕ ( ξ m ) / ϕ 2 .
ρ ( ξ l ξ m ) = { 1 , if ξ l = ξ m 0 , otherwise .
G exp ( 2 ϕ 2 ) exp { ϕ 2 [ ρ ( ξ 1 ξ 2 ) + ρ ( ξ 3 ξ 4 ) + ρ ( ξ 1 ξ 3 ) + ρ ( ξ 2 ξ 4 ) ] } .
G = exp ( 2 ϕ 2 ) exp { ϕ 2 [ ρ ( ξ 1 ξ 3 ) + ρ ( ξ 2 ξ 4 ) ] } ρ ( ξ 1 ξ 3 ) ρ ( ξ 2 ξ 4 ) ,
| I ( ω , x ) | 2 = n = , n 0 δ ( ω n ω 0 ) I 0 ( ξ 1 ) I 0 ( ξ 2 ) × J n 2 [ ( 4 π / λ ) γ 0 ( ξ 1 ξ 2 ) ] | K ( ξ 1 , x ) | 2 | K ( ξ 2 , x ) | 2 d ξ 1 d ξ 2 ,
K ( ξ , x ) = exp [ ( i π / λ R ) ( ξ x ) 2 ] ,
| I ( ω , x ) | 2 = n = , n 0 δ ( ω n ω 0 ) I 0 ( ξ 1 ) I 0 ( ξ 2 ) × J n 2 [ ( 4 π / λ ) γ 0 ( ξ 1 ξ 2 ) ] d ξ 1 d ξ 2 .
η 1 = ( ξ 1 + ξ 2 ) / 2 , η 2 = ( ξ 1 ξ 2 ) / 2 ,
| I ( ω , x ) | 2 = n = , n 0 δ ( ω n ω 0 ) C ( 2 η 2 ) J n 2 [ ( 8 π / λ ) γ 0 η 2 ] d η 2 ,
C ( τ ) = I 0 ( η 1 ) I 0 ( η 1 + τ ) d η 1 ,
I 0 ( ξ ) = A 0 exp [ ( ξ ξ 0 ) 2 / w 2 ] ,
C ( 2 η ) = A 0 2 exp ( [ ( 2 η 2 ) / w 2 ] ) .
| I ( ω , x ) | 2 = A 0 2 [ λ / ( 8 π γ 0 ) ] n = , n 0 δ ( ω n ω 0 ) × exp [ ( ζ 2 ) / ( 32 π 2 σ 2 ) ] J n 2 ( ζ ) d ζ ,
σ = ( γ 0 w ) / λ .
R ( τ ) = | I ( ω , x ) | 2 exp ( i ω τ ) d ω .
Δ f = ( 32.7 σ + 2.5 ) f 0 = [ 32.7 ( w γ 0 / λ ) + 2.5 ] f 0 .

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