Abstract

We have observed two-dimensional speckle in the space and the time domains; the speckle appears when a coherent short optical pulse propagates through a disordered solid in the multiple-scattering regime. The conventional speckle theory is extended into both the space and the time domains. The possibility of application of this speckle to optical measurements is discussed.

© 1995 Optical Society of America

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References

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  1. J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, 1984).
  2. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  3. M. Tomita and M. Matsuoka, Phys. Rev. B 43, 13579 (1991).
    [Crossref]
  4. R. Pecora, ed., Dynamic Light Scattering (Plenum, New York, 1985).
    [Crossref]
  5. D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988).
    [Crossref] [PubMed]
  6. R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1973).
  7. G. H. Watson, P. A. Fleury, and S. L. McCall, Phys. Rev. Lett. 58, 945 (1987).
    [Crossref] [PubMed]
  8. R. Vreeker, M. P. van Albada, R. Sprik, and A. Lagendijk, Phys. Lett. 132, 51 (1988).
    [Crossref]
  9. J. M. Drake and A. Z. Genack, Phys. Rev. Lett. 63, 259 (1989).
    [Crossref] [PubMed]
  10. L. M. Davis, Phys. Rev. Lett. 60, 1258 (1988).
    [Crossref] [PubMed]
  11. M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, Opt. Commun. 65, 315 (1988).
    [Crossref]
  12. A. Z. Genack, N. Garica, and W. Polkosnik, Phys. Rev. Lett. 65, 2129 (1990).
    [Crossref] [PubMed]
  13. M. P. van Albada, J. F. de Boer, and A. Lagendijk, Phys. Rev. Lett. 64, 2787 (1990).
    [Crossref] [PubMed]
  14. R. Pnini and B. Shapiro, Phys. Rev. B 39, 6986 (1989).
    [Crossref]
  15. J. F. de Boer, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 45, 658 (1992).
    [Crossref]
  16. J. W. Goodman, Proc. IEEE 53, 1688 (1965).
    [Crossref]
  17. A. Z. Genack and J. M. Drake, Europhys. Lett. 11, 331 (1990).
    [Crossref]
  18. P. G. Simpkins and T. D. Dudderar, J. Fluid Mech. 89, 665 (1978).
    [Crossref]
  19. R. K. Erf, ed., Speckle Metrology (Academic, New York, 1978).
  20. E. Wolf, ed., Progress in Optics XVI (North-Holland, Amsterdam, 1978).
  21. T. Nakajima, Jpn. J. Appl. Phys. 10, 394 (1971).
    [Crossref]
  22. S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 61, 834 (1988).
    [Crossref] [PubMed]
  23. I. Freund, M. Rousenbluh, and S. Feng, Phys. Rev. Lett. 61, 2328 (1988).
    [Crossref] [PubMed]
  24. M. Tomita, Phys. Rev. B 45, 1045 (1992).
    [Crossref]

1992 (2)

J. F. de Boer, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 45, 658 (1992).
[Crossref]

M. Tomita, Phys. Rev. B 45, 1045 (1992).
[Crossref]

1991 (1)

M. Tomita and M. Matsuoka, Phys. Rev. B 43, 13579 (1991).
[Crossref]

1990 (3)

A. Z. Genack, N. Garica, and W. Polkosnik, Phys. Rev. Lett. 65, 2129 (1990).
[Crossref] [PubMed]

M. P. van Albada, J. F. de Boer, and A. Lagendijk, Phys. Rev. Lett. 64, 2787 (1990).
[Crossref] [PubMed]

A. Z. Genack and J. M. Drake, Europhys. Lett. 11, 331 (1990).
[Crossref]

1989 (2)

R. Pnini and B. Shapiro, Phys. Rev. B 39, 6986 (1989).
[Crossref]

J. M. Drake and A. Z. Genack, Phys. Rev. Lett. 63, 259 (1989).
[Crossref] [PubMed]

1988 (6)

L. M. Davis, Phys. Rev. Lett. 60, 1258 (1988).
[Crossref] [PubMed]

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, Opt. Commun. 65, 315 (1988).
[Crossref]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988).
[Crossref] [PubMed]

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 61, 834 (1988).
[Crossref] [PubMed]

I. Freund, M. Rousenbluh, and S. Feng, Phys. Rev. Lett. 61, 2328 (1988).
[Crossref] [PubMed]

R. Vreeker, M. P. van Albada, R. Sprik, and A. Lagendijk, Phys. Lett. 132, 51 (1988).
[Crossref]

1987 (1)

G. H. Watson, P. A. Fleury, and S. L. McCall, Phys. Rev. Lett. 58, 945 (1987).
[Crossref] [PubMed]

1978 (1)

P. G. Simpkins and T. D. Dudderar, J. Fluid Mech. 89, 665 (1978).
[Crossref]

1971 (1)

T. Nakajima, Jpn. J. Appl. Phys. 10, 394 (1971).
[Crossref]

1965 (1)

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[Crossref]

Chaikin, P. M.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988).
[Crossref] [PubMed]

Davis, L. M.

L. M. Davis, Phys. Rev. Lett. 60, 1258 (1988).
[Crossref] [PubMed]

de Boer, J. F.

J. F. de Boer, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 45, 658 (1992).
[Crossref]

M. P. van Albada, J. F. de Boer, and A. Lagendijk, Phys. Rev. Lett. 64, 2787 (1990).
[Crossref] [PubMed]

Drake, J. M.

A. Z. Genack and J. M. Drake, Europhys. Lett. 11, 331 (1990).
[Crossref]

J. M. Drake and A. Z. Genack, Phys. Rev. Lett. 63, 259 (1989).
[Crossref] [PubMed]

Dudderar, T. D.

P. G. Simpkins and T. D. Dudderar, J. Fluid Mech. 89, 665 (1978).
[Crossref]

Feng, S.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 61, 834 (1988).
[Crossref] [PubMed]

I. Freund, M. Rousenbluh, and S. Feng, Phys. Rev. Lett. 61, 2328 (1988).
[Crossref] [PubMed]

Fleury, P. A.

G. H. Watson, P. A. Fleury, and S. L. McCall, Phys. Rev. Lett. 58, 945 (1987).
[Crossref] [PubMed]

Freund, I.

I. Freund, M. Rousenbluh, and S. Feng, Phys. Rev. Lett. 61, 2328 (1988).
[Crossref] [PubMed]

Garica, N.

A. Z. Genack, N. Garica, and W. Polkosnik, Phys. Rev. Lett. 65, 2129 (1990).
[Crossref] [PubMed]

Genack, A. Z.

A. Z. Genack, N. Garica, and W. Polkosnik, Phys. Rev. Lett. 65, 2129 (1990).
[Crossref] [PubMed]

A. Z. Genack and J. M. Drake, Europhys. Lett. 11, 331 (1990).
[Crossref]

J. M. Drake and A. Z. Genack, Phys. Rev. Lett. 63, 259 (1989).
[Crossref] [PubMed]

Goodman, J. W.

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[Crossref]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988).
[Crossref] [PubMed]

Kane, C.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 61, 834 (1988).
[Crossref] [PubMed]

Kuwata, M.

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, Opt. Commun. 65, 315 (1988).
[Crossref]

Lagendijk, A.

J. F. de Boer, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 45, 658 (1992).
[Crossref]

M. P. van Albada, J. F. de Boer, and A. Lagendijk, Phys. Rev. Lett. 64, 2787 (1990).
[Crossref] [PubMed]

R. Vreeker, M. P. van Albada, R. Sprik, and A. Lagendijk, Phys. Lett. 132, 51 (1988).
[Crossref]

Lee, P. A.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 61, 834 (1988).
[Crossref] [PubMed]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1973).

Matsuoka, M.

M. Tomita and M. Matsuoka, Phys. Rev. B 43, 13579 (1991).
[Crossref]

McCall, S. L.

G. H. Watson, P. A. Fleury, and S. L. McCall, Phys. Rev. Lett. 58, 945 (1987).
[Crossref] [PubMed]

Nagasawa, N.

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, Opt. Commun. 65, 315 (1988).
[Crossref]

Nakajima, T.

T. Nakajima, Jpn. J. Appl. Phys. 10, 394 (1971).
[Crossref]

Pine, D. J.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988).
[Crossref] [PubMed]

Pnini, R.

R. Pnini and B. Shapiro, Phys. Rev. B 39, 6986 (1989).
[Crossref]

Polkosnik, W.

A. Z. Genack, N. Garica, and W. Polkosnik, Phys. Rev. Lett. 65, 2129 (1990).
[Crossref] [PubMed]

Rousenbluh, M.

I. Freund, M. Rousenbluh, and S. Feng, Phys. Rev. Lett. 61, 2328 (1988).
[Crossref] [PubMed]

Shapiro, B.

R. Pnini and B. Shapiro, Phys. Rev. B 39, 6986 (1989).
[Crossref]

Simpkins, P. G.

P. G. Simpkins and T. D. Dudderar, J. Fluid Mech. 89, 665 (1978).
[Crossref]

Sprik, R.

R. Vreeker, M. P. van Albada, R. Sprik, and A. Lagendijk, Phys. Lett. 132, 51 (1988).
[Crossref]

Stone, A. D.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 61, 834 (1988).
[Crossref] [PubMed]

Takiguchi, Y.

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, Opt. Commun. 65, 315 (1988).
[Crossref]

Tomita, M.

M. Tomita, Phys. Rev. B 45, 1045 (1992).
[Crossref]

M. Tomita and M. Matsuoka, Phys. Rev. B 43, 13579 (1991).
[Crossref]

Tsuchiya, Y.

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, Opt. Commun. 65, 315 (1988).
[Crossref]

Ueda, M.

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, Opt. Commun. 65, 315 (1988).
[Crossref]

Urakami, T.

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, Opt. Commun. 65, 315 (1988).
[Crossref]

van Albada, M. P.

J. F. de Boer, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 45, 658 (1992).
[Crossref]

M. P. van Albada, J. F. de Boer, and A. Lagendijk, Phys. Rev. Lett. 64, 2787 (1990).
[Crossref] [PubMed]

R. Vreeker, M. P. van Albada, R. Sprik, and A. Lagendijk, Phys. Lett. 132, 51 (1988).
[Crossref]

Vreeker, R.

R. Vreeker, M. P. van Albada, R. Sprik, and A. Lagendijk, Phys. Lett. 132, 51 (1988).
[Crossref]

Watson, G. H.

G. H. Watson, P. A. Fleury, and S. L. McCall, Phys. Rev. Lett. 58, 945 (1987).
[Crossref] [PubMed]

Weitz, D. A.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988).
[Crossref] [PubMed]

Europhys. Lett. (1)

A. Z. Genack and J. M. Drake, Europhys. Lett. 11, 331 (1990).
[Crossref]

J. Fluid Mech. (1)

P. G. Simpkins and T. D. Dudderar, J. Fluid Mech. 89, 665 (1978).
[Crossref]

Jpn. J. Appl. Phys. (1)

T. Nakajima, Jpn. J. Appl. Phys. 10, 394 (1971).
[Crossref]

Opt. Commun. (1)

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, Opt. Commun. 65, 315 (1988).
[Crossref]

Phys. Lett. (1)

R. Vreeker, M. P. van Albada, R. Sprik, and A. Lagendijk, Phys. Lett. 132, 51 (1988).
[Crossref]

Phys. Rev. B (4)

M. Tomita and M. Matsuoka, Phys. Rev. B 43, 13579 (1991).
[Crossref]

R. Pnini and B. Shapiro, Phys. Rev. B 39, 6986 (1989).
[Crossref]

J. F. de Boer, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 45, 658 (1992).
[Crossref]

M. Tomita, Phys. Rev. B 45, 1045 (1992).
[Crossref]

Phys. Rev. Lett. (8)

G. H. Watson, P. A. Fleury, and S. L. McCall, Phys. Rev. Lett. 58, 945 (1987).
[Crossref] [PubMed]

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 61, 834 (1988).
[Crossref] [PubMed]

I. Freund, M. Rousenbluh, and S. Feng, Phys. Rev. Lett. 61, 2328 (1988).
[Crossref] [PubMed]

A. Z. Genack, N. Garica, and W. Polkosnik, Phys. Rev. Lett. 65, 2129 (1990).
[Crossref] [PubMed]

M. P. van Albada, J. F. de Boer, and A. Lagendijk, Phys. Rev. Lett. 64, 2787 (1990).
[Crossref] [PubMed]

J. M. Drake and A. Z. Genack, Phys. Rev. Lett. 63, 259 (1989).
[Crossref] [PubMed]

L. M. Davis, Phys. Rev. Lett. 60, 1258 (1988).
[Crossref] [PubMed]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988).
[Crossref] [PubMed]

Proc. IEEE (1)

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[Crossref]

Other (6)

R. K. Erf, ed., Speckle Metrology (Academic, New York, 1978).

E. Wolf, ed., Progress in Optics XVI (North-Holland, Amsterdam, 1978).

R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1973).

R. Pecora, ed., Dynamic Light Scattering (Plenum, New York, 1985).
[Crossref]

J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

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Figures (2)

Fig. 1
Fig. 1

Schematic diagram of the experiment for the observation of space–time speckle. M’s, mirrors; L, lens; P, polarizer

Fig. 2
Fig. 2

Example of space–time speckle

Equations (21)

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E out ( t , ξ , η ) = t d t d ξ d η × L ( t t , ξ , ξ , η , η ) E in ( t , ξ , η ) ,
I out ( t , ξ , η ) = t d t d t d ξ d η d ξ d η × L ( t t , ξ , ξ , η , η ) E in ( t , ξ , η ) × L ( t t , ξ , ξ , η , η ) E in * ( t , ξ , η ) ,
L ( t , ξ , ξ , η , η ) = ζ | P ζ | δ ( t s ζ / c ) ,
L ( t , ξ , ξ , η , η ) L ( t , ξ , ξ , η , η ) = T ( t , ξ , ξ , η , η ) ,
E in ( t , ξ , η ) = 0 d ω E in ( ω , ξ , η ) exp [ i ( ω t ) ] .
E out ( t , ξ , η ) = 0 d ω d ξ d η L ( ω , ξ , ξ , η , η ) × E in ( ω , ξ , η ) exp [ i ( ω t ) ] ,
L ( ω , ξ , ξ , η , η ) = 0 d t L ( t , ξ , ξ , η , η ) exp [ + i ( ω t ) ] .
ν β ( Δ ω , ξ , η ) L ( ω , ξ , η ) L * ( ω + Δ ω , ξ , η ) L ( ω , ξ , η ) L * ( ω , ξ , η ) = 0 d t T ( t , ξ , η ) exp ( i Δ ω t ) 0 d t T ( t , ξ , η )
B ( t , x , y ) = 0 d ω d ξ d η β ( ω , ξ , η ) × 1 λ z exp [ i π λ z ( x 2 + y 2 ) ] × exp [ i π λ z ( ξ 2 + η 2 ) ] × exp [ i 2 π λ z ( x ξ + y η ) ] exp ( i ω t ) ,
β ( ω , ξ , η ) = d ξ d η L ( ω , ξ , ξ , η , η ) E in ( ω , ξ , η )
A ( x , y ) = d ξ d η α ( ξ , η ) × 1 λ z exp [ i π λ z ( x 2 + y 2 ) ] × exp [ i π λ z ( ξ 2 + η 2 ) ] exp [ i 2 π λ z ( x ξ + y η ) ] .
α ( ξ , η ) = K ( ξ , η ) E in ( ξ , η ) .
μ α ( Δ ξ , Δ η ) K ( ξ , η ) K * ( ξ + Δ ξ , η + Δ η ) .
ν β ( Δ ω ) = Re { ( i Δ ω / ω c ) 1 / 2 sinh [ ( i Δ ω / ω c ) 1 / 2 ] } ,
J B ( t 1 , x 1 , y 1 , t 2 , x 2 , y 2 ) = B ( t 1 , x 1 , y 1 ) B * ( t 2 , x 2 , y 2 ) = 0 d ω 1 d ω 2 d ξ 1 d η 1 d ξ 2 d η 2 × 1 λ ¯ 2 z 2 exp { i π z [ 1 λ 1 ( x 1 2 + y 1 2 ) 1 λ 2 ( x 2 2 + y 2 2 ) ] } × J ( ω 1 , ξ 1 , η 1 , ω 2 , ξ 2 , η 2 ) × exp { i π z [ 1 λ 1 ( ξ 1 2 + η 1 2 ) 1 λ 2 ( ξ 2 2 + η 2 2 ) ] × exp { i 2 π z [ 1 λ 1 ( x 1 ξ 1 + y 1 η 1 ) 1 λ 2 ( x 2 ξ 2 + y 2 η 2 ) ] } × exp [ i ( ω 1 t 1 ω 2 t 2 ) ] ,
J ( ω 1 , ξ 1 , η 1 , ω 2 , ξ 2 , η 2 ) = β ( ω 1 , ξ 1 , η 1 ) β * ( ω 2 , ξ 2 , η 2 ) ,
L ( ω 1 , ξ 1 , η 1 ) L * ( ω 2 , ξ 2 , η 2 ) = κ | L ( ω 1 , ξ 1 , η 1 ) | 2 δ ( Δ ω , Δ ξ , Δ η ) ,
J ( ω 1 , ξ 1 , η 1 , ω 2 , ξ 2 , η 2 ) = d ξ d η L ( ω 1 , ξ 1 , ξ , η 1 , η ) × E in ( ω 1 , ξ , η ) d ξ d η L ( ω 2 , ξ 2 , ξ , η 2 , η ) × E in * ( ω 2 , ξ , η ) = κ d ξ d η | L ( ω 1 , ξ 1 , ξ , η 1 , η ) | 2 | E in ( ω 1 , ξ , η ) | 2 | J 0 ( ω , ξ , η ) | 2 .
J B ( t 1 , x 1 , y 1 , t 2 , x 2 , y 2 ) = 1 λ ¯ 2 z 2 exp { i π 2 [ 1 λ 1 ( x 1 2 + y 1 2 ) 1 λ 2 ( x 2 2 + y 2 2 ) ] } × 0 d ω d ξ d η | J 0 ( ω , ξ , η ) | 2 × exp [ i 2 π λ ( ξ Δ x + η Δ y ) ] exp ( i ω Δ t ) ,
ν B ( Δ t , Δ x , Δ y , t , x , y ) 0 d ω d ξ d η | J 0 ( ω , ξ , η ) | 2 = exp [ i π λ ( ξ Δ x + η Δ y ) ] exp ( i ω Δ t ) 0 d ω d ξ d η | J 0 ( ω , ξ , η ) | 2 ,
R I ( Δ t , Δ x , Δ y , t , x , y ) = 1 + | ν B ( Δ t , Δ x , Δ y , t , x , y ) | 2 ,

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