Abstract

We reconstruct the temporal response of a random medium by using speckle intensity frequency correlations. When the scattered field from a random medium is described by circular complex Gaussian statistics, we show that third-order correlations permit retrieval of the Fourier phase of the temporal response with bispectral techniques. Our experimental results for random media samples in the diffusion regime are in excellent agreement with the intensity temporal response measured directly with an ultrafast pulse laser and a streak camera. Our speckle correlation measurements also demonstrate sensitivity to inhomogeneous samples, highlighting the potential application for imaging within a scattering medium.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  2. V. V. Tuchin, in Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, Vol. TT38 of Tutorial Texts in Optical Engineering, D. C. O’Shea, ed. (SPIE Press, Bellingham, Wash., 2000).
  3. M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
    [CrossRef]
  4. J. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984).
  5. G. Parry, “Some effects of temporal coherence on the first order statistics of speckle,” Opt. Acta 21, 763–772 (1974).
    [CrossRef]
  6. T. Bellini, M. A. Glaser, N. A. Clark, “Effects of finite laser coherence in quasielastic multiple scattering,” Phys. Rev. A 44, 5215–5223 (1991).
    [CrossRef] [PubMed]
  7. A. Z. Genack, “Optical transmission in disordered media,” Phys. Rev. Lett. 58, 2043–2046 (1987).
    [CrossRef] [PubMed]
  8. C. A. Thompson, K. J. Webb, A. M. Weiner, “Diffusive media characterization using laser speckle,” Appl. Opt. 36, 3726–3734 (1997).
    [CrossRef] [PubMed]
  9. C. A. Thompson, K. J. Webb, A. M. Weiner, “Imaging in scattering media by use of laser speckle,” J. Opt. Soc. Am. A 14, 2269–2277 (1997).
    [CrossRef]
  10. J. D. McKinney, M. A. Webster, K. J. Webb, A. M. Weiner, “Characterization and imaging in optically scattering media by use of laser speckle and a variable-coherence source,” Opt. Lett. 25, 4–6 (2000).
    [CrossRef]
  11. J. C. Hebden, D. T. Delpy, “Enhanced time-resolved imaging with a diffusion model of photon transport,” Opt. Lett. 19, 311–313 (1994).
    [CrossRef] [PubMed]
  12. J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
    [CrossRef] [PubMed]
  13. B. B. Das, F. Liu, R. R. Alfano, “Time-resolved fluorescence and photon migration studies in biomedical and model random media,” Rep. Prog. Phys. 60, 227–292 (1997).
    [CrossRef]
  14. J. C. Ye, K. J. Webb, C. A. Bouman, R. P. Millane, “Optical diffusion tomography using iterative coordinate descent optimization in a Bayesian framework,” J. Opt. Soc. Am. A 16, 2400–2412 (1999).
    [CrossRef]
  15. A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, R. P. Millane, “Three-dimensional Bayesian optical diffusion tomography with experimental data,” Opt. Lett. 27, 95–97 (2002).
    [CrossRef]
  16. M. A. Webster, K. J. Webb, A. M. Weiner, “Temporal response of a random medium from third-order laser speckle frequency correlations,” Phys. Rev. Lett. 88, 033901 (2002).
    [CrossRef] [PubMed]
  17. A. Z. Genack, J. M. Drake, “Relationship between optical intensity, fluctuations and pulse propagation in random media,” Europhys. Lett. 11, 331–336 (1990).
    [CrossRef]
  18. R. Hanbury Brown, R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954).
  19. H. Gamo, “Triple correlator of photoelectric fluctuations as a spectroscopic tool,” J. Appl. Phys. 34, 875–876 (1963).
    [CrossRef]
  20. H. Gamo, “Phase determination of coherence functions by the intensity interferometer,” in Electromagnetic Theory and Antennas, Vol. 6 of International Series of Monographs on Electromagnetic Waves, E. C. Jordan, ed. (Pergamon, New York, 1963), pp. 801–810.
  21. A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [CrossRef] [PubMed]
  22. E. I. Blount, J. R. Klauder, “Recovery of laser intensity from correlation data,” J. Appl. Phys. 40, 2874–2875 (1969).
    [CrossRef]
  23. S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmissions through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
    [CrossRef] [PubMed]
  24. S. Feng, P. A. Lee, “Mesoscopic conductors and correlations in laser speckle patterns,” Science 251, 633–639 (1991).
    [CrossRef] [PubMed]
  25. J. H. Li, A. Z. Genack, “Correlation in laser speckle,” Phys. Rev. E 49, 4530–4533 (1994).
    [CrossRef]
  26. F. Scheffold, W. Hartl, G. Maret, E. Matijevic, “Observation of long-range correlations in temporal intensity fluctuations in light,” Phys. Rev. B 56, 10942–10952 (1997).
    [CrossRef]
  27. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  28. I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
    [CrossRef]
  29. A. Z. Genack, “Fluctuations, correlations and average transport of electromagnetic radiation in random media,” in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 207–311.
  30. A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
    [CrossRef]
  31. H. Bartelt, A. W. Lohmann, B. Wirnitzer, “Phase and amplitude recovery from bispectra,” Appl. Opt. 23, 3121–3129 (1984).
    [CrossRef] [PubMed]
  32. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  33. R. Pnini, B. Shapiro, “Fluctuations in transmission of waves through disordered slabs,” Phys. Rev. B 39, 6986–6994 (1989).
    [CrossRef]
  34. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).
  35. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  36. M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  37. C. A. Haniff, “Least-squares Fourier phase estimation from the modulo 2π bispectrum phase,” J. Opt. Soc. Am. A 8, 134–140 (1991).
    [CrossRef]
  38. C. L. Matson, “Weighted-least-squares phase reconstruction from the bispectrum,” J. Opt. Soc. Am. A 8, 1905–1913 (1991).
    [CrossRef]
  39. J. C. Marron, P. P. Sanchez, R. C. Sullivan, “Unwrapping algorithm for least-squares phase recovery from the modulo 2π bispectrum phase,” J. Opt. Soc. Am. A 7, 14–20 (1990).
    [CrossRef]
  40. A. A. Scribot, “First-order probability density functions of speckle measured with a finite aperture,” Opt. Commun. 11, 238–241 (1974).
    [CrossRef]
  41. B. C. Park, M. S. Chung, “First-order probability density function of the integrated speckle,” Opt. Commun. 83, 5–9 (1991).
    [CrossRef]
  42. N. George, “Speckle at various planes in an optical system,” Opt. Eng. 25, 754–764 (1986).
    [CrossRef]
  43. H. Fujii, T. Asakura, “Effect of the point spread function on the average contrast of image speckle patterns,” Opt. Commun. 21, 80–84 (1977).
    [CrossRef]
  44. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, Boston, Mass., 1991).
  45. M. A. Berger, An Introduction to Probability and Stochastic Processes (Springer-Verlag, New York, 1993).
  46. P. Billingsley, Probability and Measure, 3rd ed. (Wiley, New York, 1995).
  47. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  48. R. A. Wooding, “The multivariate distribution of complex normal variables,” Biometrika 43, 212–215 (1956).
    [CrossRef]
  49. T. L. Grettenberg, “A representation theorem for complex normal processes,” IEEE Trans. Inf. Theory IT-11, 305–306 (1965).
    [CrossRef]

2002 (2)

A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, R. P. Millane, “Three-dimensional Bayesian optical diffusion tomography with experimental data,” Opt. Lett. 27, 95–97 (2002).
[CrossRef]

M. A. Webster, K. J. Webb, A. M. Weiner, “Temporal response of a random medium from third-order laser speckle frequency correlations,” Phys. Rev. Lett. 88, 033901 (2002).
[CrossRef] [PubMed]

2000 (1)

1999 (2)

M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

J. C. Ye, K. J. Webb, C. A. Bouman, R. P. Millane, “Optical diffusion tomography using iterative coordinate descent optimization in a Bayesian framework,” J. Opt. Soc. Am. A 16, 2400–2412 (1999).
[CrossRef]

1997 (5)

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

B. B. Das, F. Liu, R. R. Alfano, “Time-resolved fluorescence and photon migration studies in biomedical and model random media,” Rep. Prog. Phys. 60, 227–292 (1997).
[CrossRef]

C. A. Thompson, K. J. Webb, A. M. Weiner, “Diffusive media characterization using laser speckle,” Appl. Opt. 36, 3726–3734 (1997).
[CrossRef] [PubMed]

C. A. Thompson, K. J. Webb, A. M. Weiner, “Imaging in scattering media by use of laser speckle,” J. Opt. Soc. Am. A 14, 2269–2277 (1997).
[CrossRef]

F. Scheffold, W. Hartl, G. Maret, E. Matijevic, “Observation of long-range correlations in temporal intensity fluctuations in light,” Phys. Rev. B 56, 10942–10952 (1997).
[CrossRef]

1994 (3)

1991 (5)

T. Bellini, M. A. Glaser, N. A. Clark, “Effects of finite laser coherence in quasielastic multiple scattering,” Phys. Rev. A 44, 5215–5223 (1991).
[CrossRef] [PubMed]

C. A. Haniff, “Least-squares Fourier phase estimation from the modulo 2π bispectrum phase,” J. Opt. Soc. Am. A 8, 134–140 (1991).
[CrossRef]

C. L. Matson, “Weighted-least-squares phase reconstruction from the bispectrum,” J. Opt. Soc. Am. A 8, 1905–1913 (1991).
[CrossRef]

S. Feng, P. A. Lee, “Mesoscopic conductors and correlations in laser speckle patterns,” Science 251, 633–639 (1991).
[CrossRef] [PubMed]

B. C. Park, M. S. Chung, “First-order probability density function of the integrated speckle,” Opt. Commun. 83, 5–9 (1991).
[CrossRef]

1990 (2)

J. C. Marron, P. P. Sanchez, R. C. Sullivan, “Unwrapping algorithm for least-squares phase recovery from the modulo 2π bispectrum phase,” J. Opt. Soc. Am. A 7, 14–20 (1990).
[CrossRef]

A. Z. Genack, J. M. Drake, “Relationship between optical intensity, fluctuations and pulse propagation in random media,” Europhys. Lett. 11, 331–336 (1990).
[CrossRef]

1989 (2)

1988 (1)

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmissions through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

1987 (1)

A. Z. Genack, “Optical transmission in disordered media,” Phys. Rev. Lett. 58, 2043–2046 (1987).
[CrossRef] [PubMed]

1986 (1)

N. George, “Speckle at various planes in an optical system,” Opt. Eng. 25, 754–764 (1986).
[CrossRef]

1984 (2)

1983 (1)

1977 (1)

H. Fujii, T. Asakura, “Effect of the point spread function on the average contrast of image speckle patterns,” Opt. Commun. 21, 80–84 (1977).
[CrossRef]

1974 (2)

A. A. Scribot, “First-order probability density functions of speckle measured with a finite aperture,” Opt. Commun. 11, 238–241 (1974).
[CrossRef]

G. Parry, “Some effects of temporal coherence on the first order statistics of speckle,” Opt. Acta 21, 763–772 (1974).
[CrossRef]

1969 (1)

E. I. Blount, J. R. Klauder, “Recovery of laser intensity from correlation data,” J. Appl. Phys. 40, 2874–2875 (1969).
[CrossRef]

1965 (1)

T. L. Grettenberg, “A representation theorem for complex normal processes,” IEEE Trans. Inf. Theory IT-11, 305–306 (1965).
[CrossRef]

1963 (1)

H. Gamo, “Triple correlator of photoelectric fluctuations as a spectroscopic tool,” J. Appl. Phys. 34, 875–876 (1963).
[CrossRef]

1962 (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

1956 (1)

R. A. Wooding, “The multivariate distribution of complex normal variables,” Biometrika 43, 212–215 (1956).
[CrossRef]

1954 (1)

R. Hanbury Brown, R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954).

Alfano, R. R.

B. B. Das, F. Liu, R. R. Alfano, “Time-resolved fluorescence and photon migration studies in biomedical and model random media,” Rep. Prog. Phys. 60, 227–292 (1997).
[CrossRef]

Arridge, S. R.

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

Asakura, T.

H. Fujii, T. Asakura, “Effect of the point spread function on the average contrast of image speckle patterns,” Opt. Commun. 21, 80–84 (1977).
[CrossRef]

Bartelt, H.

Bellini, T.

T. Bellini, M. A. Glaser, N. A. Clark, “Effects of finite laser coherence in quasielastic multiple scattering,” Phys. Rev. A 44, 5215–5223 (1991).
[CrossRef] [PubMed]

Berger, M. A.

M. A. Berger, An Introduction to Probability and Stochastic Processes (Springer-Verlag, New York, 1993).

Billingsley, P.

P. Billingsley, Probability and Measure, 3rd ed. (Wiley, New York, 1995).

Blount, E. I.

E. I. Blount, J. R. Klauder, “Recovery of laser intensity from correlation data,” J. Appl. Phys. 40, 2874–2875 (1969).
[CrossRef]

Bouman, C. A.

Chance, B.

Chung, M. S.

B. C. Park, M. S. Chung, “First-order probability density function of the integrated speckle,” Opt. Commun. 83, 5–9 (1991).
[CrossRef]

Clark, N. A.

T. Bellini, M. A. Glaser, N. A. Clark, “Effects of finite laser coherence in quasielastic multiple scattering,” Phys. Rev. A 44, 5215–5223 (1991).
[CrossRef] [PubMed]

Das, B. B.

B. B. Das, F. Liu, R. R. Alfano, “Time-resolved fluorescence and photon migration studies in biomedical and model random media,” Rep. Prog. Phys. 60, 227–292 (1997).
[CrossRef]

Delpy, D. T.

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

J. C. Hebden, D. T. Delpy, “Enhanced time-resolved imaging with a diffusion model of photon transport,” Opt. Lett. 19, 311–313 (1994).
[CrossRef] [PubMed]

Drake, J. M.

A. Z. Genack, J. M. Drake, “Relationship between optical intensity, fluctuations and pulse propagation in random media,” Europhys. Lett. 11, 331–336 (1990).
[CrossRef]

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Feng, S.

S. Feng, P. A. Lee, “Mesoscopic conductors and correlations in laser speckle patterns,” Science 251, 633–639 (1991).
[CrossRef] [PubMed]

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmissions through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Feng, T. C.

Fujii, H.

H. Fujii, T. Asakura, “Effect of the point spread function on the average contrast of image speckle patterns,” Opt. Commun. 21, 80–84 (1977).
[CrossRef]

Gamo, H.

H. Gamo, “Triple correlator of photoelectric fluctuations as a spectroscopic tool,” J. Appl. Phys. 34, 875–876 (1963).
[CrossRef]

H. Gamo, “Phase determination of coherence functions by the intensity interferometer,” in Electromagnetic Theory and Antennas, Vol. 6 of International Series of Monographs on Electromagnetic Waves, E. C. Jordan, ed. (Pergamon, New York, 1963), pp. 801–810.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Genack, A. Z.

J. H. Li, A. Z. Genack, “Correlation in laser speckle,” Phys. Rev. E 49, 4530–4533 (1994).
[CrossRef]

A. Z. Genack, J. M. Drake, “Relationship between optical intensity, fluctuations and pulse propagation in random media,” Europhys. Lett. 11, 331–336 (1990).
[CrossRef]

A. Z. Genack, “Optical transmission in disordered media,” Phys. Rev. Lett. 58, 2043–2046 (1987).
[CrossRef] [PubMed]

A. Z. Genack, “Fluctuations, correlations and average transport of electromagnetic radiation in random media,” in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 207–311.

George, N.

N. George, “Speckle at various planes in an optical system,” Opt. Eng. 25, 754–764 (1986).
[CrossRef]

Glaser, M. A.

T. Bellini, M. A. Glaser, N. A. Clark, “Effects of finite laser coherence in quasielastic multiple scattering,” Phys. Rev. A 44, 5215–5223 (1991).
[CrossRef] [PubMed]

Goodman, J.

J. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984).

Goodman, J. W.

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

Grettenberg, T. L.

T. L. Grettenberg, “A representation theorem for complex normal processes,” IEEE Trans. Inf. Theory IT-11, 305–306 (1965).
[CrossRef]

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Hanbury Brown, R.

R. Hanbury Brown, R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954).

Haniff, C. A.

Hartl, W.

F. Scheffold, W. Hartl, G. Maret, E. Matijevic, “Observation of long-range correlations in temporal intensity fluctuations in light,” Phys. Rev. B 56, 10942–10952 (1997).
[CrossRef]

Haskell, R. C.

Hebden, J. C.

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

J. C. Hebden, D. T. Delpy, “Enhanced time-resolved imaging with a diffusion model of photon transport,” Opt. Lett. 19, 311–313 (1994).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Kane, C.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmissions through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Klauder, J. R.

E. I. Blount, J. R. Klauder, “Recovery of laser intensity from correlation data,” J. Appl. Phys. 40, 2874–2875 (1969).
[CrossRef]

Lee, P. A.

S. Feng, P. A. Lee, “Mesoscopic conductors and correlations in laser speckle patterns,” Science 251, 633–639 (1991).
[CrossRef] [PubMed]

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmissions through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Li, J. H.

J. H. Li, A. Z. Genack, “Correlation in laser speckle,” Phys. Rev. E 49, 4530–4533 (1994).
[CrossRef]

Liu, F.

B. B. Das, F. Liu, R. R. Alfano, “Time-resolved fluorescence and photon migration studies in biomedical and model random media,” Rep. Prog. Phys. 60, 227–292 (1997).
[CrossRef]

Lohmann, A. W.

Maret, G.

F. Scheffold, W. Hartl, G. Maret, E. Matijevic, “Observation of long-range correlations in temporal intensity fluctuations in light,” Phys. Rev. B 56, 10942–10952 (1997).
[CrossRef]

Marron, J. C.

Matijevic, E.

F. Scheffold, W. Hartl, G. Maret, E. Matijevic, “Observation of long-range correlations in temporal intensity fluctuations in light,” Phys. Rev. B 56, 10942–10952 (1997).
[CrossRef]

Matson, C. L.

McAdams, M. S.

McKinney, J. D.

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Millane, R. P.

Milstein, A. B.

Nieuwenhuizen, T. M.

M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

Oh, S.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, Boston, Mass., 1991).

Park, B. C.

B. C. Park, M. S. Chung, “First-order probability density function of the integrated speckle,” Opt. Commun. 83, 5–9 (1991).
[CrossRef]

Parry, G.

G. Parry, “Some effects of temporal coherence on the first order statistics of speckle,” Opt. Acta 21, 763–772 (1974).
[CrossRef]

Patterson, M. S.

Pnini, R.

R. Pnini, B. Shapiro, “Fluctuations in transmission of waves through disordered slabs,” Phys. Rev. B 39, 6986–6994 (1989).
[CrossRef]

Reed, I. S.

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

Reynolds, J. S.

Sanchez, P. P.

Scheffold, F.

F. Scheffold, W. Hartl, G. Maret, E. Matijevic, “Observation of long-range correlations in temporal intensity fluctuations in light,” Phys. Rev. B 56, 10942–10952 (1997).
[CrossRef]

Scribot, A. A.

A. A. Scribot, “First-order probability density functions of speckle measured with a finite aperture,” Opt. Commun. 11, 238–241 (1974).
[CrossRef]

Shapiro, B.

R. Pnini, B. Shapiro, “Fluctuations in transmission of waves through disordered slabs,” Phys. Rev. B 39, 6986–6994 (1989).
[CrossRef]

Stone, A. D.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmissions through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Sullivan, R. C.

Svaasand, L. O.

Thompson, C. A.

Tromberg, B. J.

Tsay, T. T.

Tuchin, V. V.

V. V. Tuchin, in Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, Vol. TT38 of Tutorial Texts in Optical Engineering, D. C. O’Shea, ed. (SPIE Press, Bellingham, Wash., 2000).

Twiss, R. Q.

R. Hanbury Brown, R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954).

van Rossum, M. C. W.

M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

Webb, K. J.

Webster, M. A.

M. A. Webster, K. J. Webb, A. M. Weiner, “Temporal response of a random medium from third-order laser speckle frequency correlations,” Phys. Rev. Lett. 88, 033901 (2002).
[CrossRef] [PubMed]

J. D. McKinney, M. A. Webster, K. J. Webb, A. M. Weiner, “Characterization and imaging in optically scattering media by use of laser speckle and a variable-coherence source,” Opt. Lett. 25, 4–6 (2000).
[CrossRef]

Weigelt, G.

Weiner, A. M.

Wilson, B. C.

Wirnitzer, B.

Wooding, R. A.

R. A. Wooding, “The multivariate distribution of complex normal variables,” Biometrika 43, 212–215 (1956).
[CrossRef]

Ye, J. C.

Appl. Opt. (4)

Biometrika (1)

R. A. Wooding, “The multivariate distribution of complex normal variables,” Biometrika 43, 212–215 (1956).
[CrossRef]

Europhys. Lett. (1)

A. Z. Genack, J. M. Drake, “Relationship between optical intensity, fluctuations and pulse propagation in random media,” Europhys. Lett. 11, 331–336 (1990).
[CrossRef]

IEEE Trans. Inf. Theory (1)

T. L. Grettenberg, “A representation theorem for complex normal processes,” IEEE Trans. Inf. Theory IT-11, 305–306 (1965).
[CrossRef]

IRE Trans. Inf. Theory (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

J. Appl. Phys. (2)

E. I. Blount, J. R. Klauder, “Recovery of laser intensity from correlation data,” J. Appl. Phys. 40, 2874–2875 (1969).
[CrossRef]

H. Gamo, “Triple correlator of photoelectric fluctuations as a spectroscopic tool,” J. Appl. Phys. 34, 875–876 (1963).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Acta (1)

G. Parry, “Some effects of temporal coherence on the first order statistics of speckle,” Opt. Acta 21, 763–772 (1974).
[CrossRef]

Opt. Commun. (3)

A. A. Scribot, “First-order probability density functions of speckle measured with a finite aperture,” Opt. Commun. 11, 238–241 (1974).
[CrossRef]

B. C. Park, M. S. Chung, “First-order probability density function of the integrated speckle,” Opt. Commun. 83, 5–9 (1991).
[CrossRef]

H. Fujii, T. Asakura, “Effect of the point spread function on the average contrast of image speckle patterns,” Opt. Commun. 21, 80–84 (1977).
[CrossRef]

Opt. Eng. (1)

N. George, “Speckle at various planes in an optical system,” Opt. Eng. 25, 754–764 (1986).
[CrossRef]

Opt. Lett. (3)

Philos. Mag. (1)

R. Hanbury Brown, R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954).

Phys. Med. Biol. (1)

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

Phys. Rev. A (1)

T. Bellini, M. A. Glaser, N. A. Clark, “Effects of finite laser coherence in quasielastic multiple scattering,” Phys. Rev. A 44, 5215–5223 (1991).
[CrossRef] [PubMed]

Phys. Rev. B (2)

F. Scheffold, W. Hartl, G. Maret, E. Matijevic, “Observation of long-range correlations in temporal intensity fluctuations in light,” Phys. Rev. B 56, 10942–10952 (1997).
[CrossRef]

R. Pnini, B. Shapiro, “Fluctuations in transmission of waves through disordered slabs,” Phys. Rev. B 39, 6986–6994 (1989).
[CrossRef]

Phys. Rev. E (1)

J. H. Li, A. Z. Genack, “Correlation in laser speckle,” Phys. Rev. E 49, 4530–4533 (1994).
[CrossRef]

Phys. Rev. Lett. (3)

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmissions through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

A. Z. Genack, “Optical transmission in disordered media,” Phys. Rev. Lett. 58, 2043–2046 (1987).
[CrossRef] [PubMed]

M. A. Webster, K. J. Webb, A. M. Weiner, “Temporal response of a random medium from third-order laser speckle frequency correlations,” Phys. Rev. Lett. 88, 033901 (2002).
[CrossRef] [PubMed]

Proc. IEEE (1)

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[CrossRef]

Rep. Prog. Phys. (1)

B. B. Das, F. Liu, R. R. Alfano, “Time-resolved fluorescence and photon migration studies in biomedical and model random media,” Rep. Prog. Phys. 60, 227–292 (1997).
[CrossRef]

Rev. Mod. Phys. (1)

M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

Science (1)

S. Feng, P. A. Lee, “Mesoscopic conductors and correlations in laser speckle patterns,” Science 251, 633–639 (1991).
[CrossRef] [PubMed]

Other (12)

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

A. Z. Genack, “Fluctuations, correlations and average transport of electromagnetic radiation in random media,” in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 207–311.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

J. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

V. V. Tuchin, in Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, Vol. TT38 of Tutorial Texts in Optical Engineering, D. C. O’Shea, ed. (SPIE Press, Bellingham, Wash., 2000).

H. Gamo, “Phase determination of coherence functions by the intensity interferometer,” in Electromagnetic Theory and Antennas, Vol. 6 of International Series of Monographs on Electromagnetic Waves, E. C. Jordan, ed. (Pergamon, New York, 1963), pp. 801–810.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, Boston, Mass., 1991).

M. A. Berger, An Introduction to Probability and Stochastic Processes (Springer-Verlag, New York, 1993).

P. Billingsley, Probability and Measure, 3rd ed. (Wiley, New York, 1995).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Experimental setup used to measure the speckle intensity patterns as a function of the laser diode center frequency. The Fabry–Perot interferometer is used to monitor the change in the laser diode center frequency as it is tuned. Lens L1 (fL1=50 mm) focuses the laser output onto the front face of the scattering random medium. The spatial structure of the speckle pattern at plane P1 is controlled by the unity magnification spatial filter. Lens L2 (fL2=75 mm) provides a magnification factor of M=10 from plane P1 to the CCD image plane, where the resultant frequency-dependent speckle pattern is obtained.

Fig. 2
Fig. 2

Typical speckle image of the output intensity from the random medium obtained by the CCD.

Fig. 3
Fig. 3

Intensity histogram of the speckle pattern given in Fig. 2 plotted on a semilogarithmic scale (solid curve). Also shown is the ideal negative exponential intensity probability density function in Eq. (37) expected for zero-mean circular complex Gaussian field statistics (dashed curve).

Fig. 4
Fig. 4

Measured intensity spatial autocorrelation function of the speckle image shown in Fig. 2 (circles). The theoretical result in Eq. (39) is also plotted, showing good agreement (dashed curve). The average speckle diameter, taken as the distance from Δr=0 to the first minimum, is estimated to be approximately 250 µm.

Fig. 5
Fig. 5

Plot of the measured second-order intensity frequency correlation defined in Eq. (40) for two slab thicknesses of a scattering random medium (symbols). Excellent agreement with the second-order intensity correlation in Eq. (14), calculated with an analytic diffusion model for the temporal response p(t), with values of μs=13 cm-1 and negligible absorption (μa=0 cm-1) was obtained (dashed curves).

Fig. 6
Fig. 6

Plot of the measured third-order intensity correlation defined by Eq. (41) for the sample of thickness d=6 mm. These data are equal to twice the real part of the bispectrum of p(t).

Fig. 7
Fig. 7

(a) Reconstructed Fourier magnitude of the temporal response for the two sample thicknesses with the use of measured data (symbols) and the Fourier magnitude calculated with an analytic diffusion model for the temporal response with μs=13 cm-1 and μa=0 cm-1 for each thickness (dashed curves). (b) Reconstructed Fourier phase of the temporal response with the use of measured data (symbols) and the Fourier phase calculated with the diffusion model for each sample thickness (dashed curves).

Fig. 8
Fig. 8

Reconstructed temporal response obtained by taking an inverse fast Fourier transform of the Fourier magnitude and phase data presented in Fig. 7 (solid curves). Each sample thickness gives excellent agreement with a diffusion approximation model for the temporal response (dashed curves).

Fig. 9
Fig. 9

Plots of the intensity temporal response [Eq. (25)] directly measured with a streak camera for each sample thickness (solid curves). Overlaid are the temporal responses reconstructed by using third-order speckle correlations, given in Fig. 8, showing excellent agreement (dotted curves).

Fig. 10
Fig. 10

Cross section of inhomogeneous random medium sample. The background has a scattering coefficient of μs=13 cm-1, and the lower-scattering inhomogeneity has an estimated scattering coefficient of μs=4 cm-1. The source–detector location combinations used are A–A, B–B, and C–C, each separated by 5 mm.

Fig. 11
Fig. 11

(a) Measured second-order intensity correlations for the inhomogeneous sample shown in Fig. 10 for the source–detector location combinations A–A, B–B, and C–C. (b) Reconstructed temporal responses for the inhomogeneous sample for the source–detector locations A–A and C–C.

Equations (69)

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

ein(t)=Ei exp(j2πνt)+c.c.,
eout(t)=Eo(ν)exp(j2πνt)+c.c.,
Eo(ν)=1Nk=1NAk exp[-jϕk(ν)],
ϕk(ν)=2πνtk.
pxy(x, y)=12πσ2exp-12x2+y2σ2,
σ2=1Nk=1NAk22.
I=A2.
z=Eo(ν1)Eo(ν2)Eo(νM).
pz(z)=1πM|Cz|exp(-zHCz-1z),
Eo(ν+Δν)Eo*(ν)=IP(Δν),
P(Δν)=-dt p(t)exp(-j2πΔνt).
I(ν+Δν)I(ν)=I2+I2|P(Δν)|2.
I(ν)I(ν+Δν1)I(ν+Δν2)
=I3+I3|P(Δν1)|2+I3|P(Δν2)|2
+I3|P(Δν1+Δν2)|2+2I3 Re{P(Δν1)
×P(Δν2)P*(Δν1+Δν2)}.
I˜(ν+Δν)I˜(ν)=|P(Δν)|2,
I˜(ν)I˜(ν+Δν1)I˜(ν+Δν1+Δν2)=2 Re{P(Δν1)P(Δν2)×P*(Δν1+Δν2)}.
ain(t)=u(t)exp(j2πν0t)+c.c.,
aout(t)=v(t)exp(j2πν0t)+c.c.,
I(t)=|v(t)|2.
H(ν+Δν)H*(ν)=|H(ν)|2P(Δν),
v(t)=-dν U(ν)H(ν+ν0)exp(j2πνt).
I(t)=-dν U(ν)H(ν+ν0)exp(j2πνt)×-dν U*(ν)H*(ν+ν0)exp(-j2πνt).
I(t)=-dν-dΔνH(ν+ν0+Δν)×H*(ν+ν0)U(ν+Δν)U*(ν)exp(j2πΔνt).
I(t)=-dν|V(ν)|2-dΔν P(Δν)exp(j2πΔνt).
-dν|V(ν)|2=-dt|v(t)|2.
p(t)=I(t)-dtI(t).
tΦ(r, t)+vμaΦ(r, t)-·DΦ(r, t)=vS0(r, t),
J(r, t)=-DvΦ(r, t).
pD(t)=ID(t)-dt ID(t),
g(3)(τ1, τ2)=-dt f(t)f(t+τ1)f(t+τ2).
G(3)(ν1, ν2)=-dτ1-dτ2 g(3)(τ1, τ2)×exp[-j2π(ν1τ1+ν2τ2)],
f(t)=-dt F(ν)exp(j2πνt).
G(3)(ν1, ν2)=F(ν1)F(ν2)F(-ν1-ν2).
G(3)(ν1, ν2)=F(ν1)F(ν2)F*(ν1+ν2).
ψ(ν1, ν2)=ϕ(ν1)+ϕ(ν2)-ϕ(ν1+ν2).
ϕk=1k-1i=1k-1ϕi+ϕk-i-ψi,k-i,k=2,, Ns,
ψ(Δν1, Δν2)=±cos-1I˜1I˜2I˜32(I˜1I˜2I˜2I˜3I˜3I˜1)1/2.
pI(I)=1Iexp(-I/I),
RI(Δr)=I(r0+Δr)I(r0)I2,
RI(Δr)=1+2J1πDΔrMλfπDΔrMλf2,
I˜(ν)I˜(ν+Δν)meas=1N-ij=0N-1-iI˜jI˜j+i,
I˜(ν)I˜(ν+Δν1)I˜(ν+Δν1+Δν2)meas=1N-i-jk=0N-1-i-jI˜kI˜k+iI˜k+i+j,
Eo(νm)=1Nk=1NAk exp[-jϕk(νm)].
u=limN1Nk=1Nuk
pu(u)=1(2π)M|Cu|1/2exp-12uTCu-1u.
u1=x1(ν1)x1(νM)y1(ν1)y1(νM),u2=x2(ν1)x2(νM)y2(ν1)y2(νM),,uN=xN(ν1)xN(νM)yN(ν1)yN(νM).
Cu=CxxCxyCyxCyy.
uk2=m=1MAk2 cos2 ϕk(νm)+m=1MAk2 sin2 ϕk(νm).
uk2=m=1MAk2cos2 ϕk(νm)+m=1MAk2×sin2 ϕk(νm).
uk2=MA2.
Cxx=Cyy,Cyx=-Cxy.
z=x(ν1)+jy(ν1)x(νM)+jy(νM),
pz(z)=1πM|Cz|exp(-zHCz-1z),
x(νi)x(νj)=1Nk=1Nl=1NAkAl cos ϕk(νi)cos ϕl(νj).
x(νi)x(νj)=1Nk=1NAk2cos ϕk(νi)cos ϕk(νj)+1Nk=1Nl=1lkNAkAlcos ϕk(νi)×cos ϕl(νj).
x(νi)x(νj)=12Nk=1NAk2[cos 2π(νi-νj)tk+cos 2π(νi+νj)tk].
x(νi)x(νj)=A2cos 2π(νi-νj)t/2.
y(νi)y(νj)=A2cos 2π(νi-νj)t/2,
x(νi)y(νj)=A2sin 2π(νi-νj)t/2,
y(νi)x(νj)=-A2sin 2π(νi-νj)t/2.
Eo(ν+Δν)=1Nk=1NAk exp[-jϕk(ν+Δν)],
Eo(ν)=1Nl=1NAl exp[-jϕl(ν)],
Eo(ν+Δν)Eo*(ν)=1Nk=1Nl=1NAkAl×exp[-jϕk(ν+Δν)+jϕl(ν)].
Eo(ν+Δν)Eo*(ν)=1Nk=1NAk2exp[-jϕk(ν+Δν)+jϕk(ν)]+1Nk=1Nl=1lkNAkAl×exp(-jϕk(ν+Δν)×exp(jϕl(ν).
Eo(ν+Δν)Eo*(ν)=A2exp(-j2πΔνt).
exp(-j2πΔνt)=-dt p(t)exp(-j2πΔνt)=P(Δν),
Eo(ν+Δν)Eo*(ν)=IP(Δν).

Metrics