Abstract

We examine the time-domain statistics of randomly varying electric fields generated by multiple scattering of single-cycle electromagnetic pulses in a random medium. This analysis emphasizes the fact that these measured random fields are not stationary, as is commonly assumed for diffusing photons generated with a narrowband light source. We demonstrate that the nonstationarity is a consequence of the time dependence of the configurationally averaged intensity, and that the statistical properties of the random field can be predicted if this quantity is known. We also discuss an approach for describing the transition from nonstationary to stationary behavior by investigating the degree of stationarity during a short time window. A parameterization of the statistics using a gamma distribution provides a quantifiable measure of the approach to stationarity. Our predictions are in good agreement with experimental observations.

© 2006 Optical Society of America

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References

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  1. M. A. O'Leary, D. A. Boas, B. Chance, and A. G. Yodh, "Experimental images of heterogeneous turbid media by frequency-domain diffusing tomography," Opt. Lett. 20, 426-428 (1995).
    [CrossRef] [PubMed]
  2. D. A. Boas, L. E. Campbell, and A. G. Yodh, "Scattering and imaging with diffusing temporal field correlations," Phys. Rev. Lett. 75, 1855-1858 (1995).
    [CrossRef] [PubMed]
  3. D. A. Boas and A. G. Yodh, "Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation," J. Opt. Soc. Am. A 14, 192-215 (1997).
    [CrossRef]
  4. D. L. Lasocki, C. L. Matson, and P. J. Collins, "Analysis of forward scattering of diffuse photon-density waves in turbid media: a diffraction tomography approach to an analytic solution," Opt. Lett. 23, 558-560 (1998).
    [CrossRef]
  5. J. D. McKinney, M. A. Webster, K. J. Webb, and 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]
  6. J. C. Schotland and V. A. Markel, "Inverse scattering with diffusing waves," J. Opt. Soc. Am. A 18, 2767-2777 (2001).
    [CrossRef]
  7. R. Berkovits and S. Feng, "Correlations in coherent multiple scattering," Phys. Rep. 238, 135-172 (1994).
    [CrossRef]
  8. C. A. Thompson, K. J. Webb, and A. M. Weiner, "Diffusive media characterization with laser speckle," Appl. Opt. 36, 3726-3734 (1997).
    [CrossRef] [PubMed]
  9. M. A. Webster, K. J. Webb, and A. M. Weiner, "Temporal response of a random medium from third-order laser speckle frequency correlations," Phys. Rev. Lett. 88, 033901 (2002).
    [CrossRef] [PubMed]
  10. B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, "Delay-time statistics for diffuse waves," Phys. Rev. E 59, 7166-7172 (1999).
    [CrossRef]
  11. J. Rivas, R. Sprik, and A. Lagendijk, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
    [CrossRef]
  12. J. Pearce, Z. Jian, and D. Mittleman, "Statistics of multiply scattered broadband terahertz pulses," Phys. Rev. Lett. 91, 043903 (2003).
    [CrossRef] [PubMed]
  13. S. Mujumdar, K. J. Chau, and A. Y. Elezzabi, "Experimental and numerical investigation of terahertz transmission through strongly scattering sub-wavelength size spheres," Appl. Phys. Lett. 85, 6284-6286 (2004).
    [CrossRef]
  14. J. Pearce, Z. Jian, and D. Mittleman, "Spectral shifts as a signature of the onset of diffusion of broadband terahertz pulses," Opt. Lett. 29, 2926-2928 (2004).
    [CrossRef]
  15. A. A. Chabanov, B. Hu, and A. Z. Genack, "Dynamic correlation in wave propagation in random media," Phys. Rev. Lett. 93, 123901 (2004).
    [CrossRef] [PubMed]
  16. K. J. Chau, G. D. Dice, and A. Y. Elezzabi, "Coherent plasmonic enhanced terahertz transmission through random metallic media," Phys. Rev. Lett. 94, 173904 (2005).
    [CrossRef] [PubMed]
  17. K. J. Chau and A. Y. Elezzabi, "Terahertz transmission through ensembles of subwavelength-size metallic particles," Phys. Rev. B 72, 075110 (2005).
    [CrossRef]
  18. J. Pearce and D. Mittleman, "Bayesian approach to non-Gaussian field statistics for diffuse broadband terahertz pulses," Opt. Lett. 30, 2843-2845 (2005).
    [CrossRef] [PubMed]
  19. J. W. Goodman, Statistical Optics (Wiley, 2000).
  20. A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, "Statistics of wave dynamics in random media," Phys. Rev. Lett. 82, 715-718 (1999).
    [CrossRef]
  21. A. A. Chabanov and A. Z. Genack, "Field distributions in the crossover from ballistic to diffusive wave propagation," Phys. Rev. E 56, R1338-R1341 (1997).
    [CrossRef]
  22. K. M. Yoo and R. R. Alfano, "Time resolved depolarization of multiple back scattered light from random media," Phys. Lett. A 142, 531-536 (1989).
    [CrossRef]
  23. M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," Appl. Opt. 28, 2331-2336 (1989).
    [CrossRef] [PubMed]
  24. D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, "Time-resolved experiments on light diffusion in anisotropic random media," Phys. Rev. E 62, 6681-6687 (2000).
    [CrossRef]
  25. J. Pearce and D. Mittleman, "The propagation of single-cycle THz pulses in random media," Opt. Lett. 26, 2002-2004 (2001).
    [CrossRef]
  26. M. A. Webster, T. D. Gerke, A. M. Weiner, and K. J. Webb, "Spectral and temporal speckle field measurements of a random medium," Opt. Lett. 29, 1491-1493 (2004).
    [CrossRef] [PubMed]
  27. M. Bondani, D. Redaelli, A. Spinelli, A. Andreoni, G. Roberti, P. Riccio, and R. Liuzzi, "Photon time-of-flight distributions through turbid media directly measured with single-photon avalanche diodes," J. Opt. Soc. Am. B 20, 2383-2388 (2003).
    [CrossRef]
  28. Z. Jian, J. Pearce, and D. Mittleman, "Characterizing individual scattering events by measuring the amplitude and phase of the electric field diffusing through a random medium," Phys. Rev. Lett. 91, 033903 (2003).
    [CrossRef] [PubMed]

2005 (3)

K. J. Chau, G. D. Dice, and A. Y. Elezzabi, "Coherent plasmonic enhanced terahertz transmission through random metallic media," Phys. Rev. Lett. 94, 173904 (2005).
[CrossRef] [PubMed]

K. J. Chau and A. Y. Elezzabi, "Terahertz transmission through ensembles of subwavelength-size metallic particles," Phys. Rev. B 72, 075110 (2005).
[CrossRef]

J. Pearce and D. Mittleman, "Bayesian approach to non-Gaussian field statistics for diffuse broadband terahertz pulses," Opt. Lett. 30, 2843-2845 (2005).
[CrossRef] [PubMed]

2004 (4)

M. A. Webster, T. D. Gerke, A. M. Weiner, and K. J. Webb, "Spectral and temporal speckle field measurements of a random medium," Opt. Lett. 29, 1491-1493 (2004).
[CrossRef] [PubMed]

J. Pearce, Z. Jian, and D. Mittleman, "Spectral shifts as a signature of the onset of diffusion of broadband terahertz pulses," Opt. Lett. 29, 2926-2928 (2004).
[CrossRef]

S. Mujumdar, K. J. Chau, and A. Y. Elezzabi, "Experimental and numerical investigation of terahertz transmission through strongly scattering sub-wavelength size spheres," Appl. Phys. Lett. 85, 6284-6286 (2004).
[CrossRef]

A. A. Chabanov, B. Hu, and A. Z. Genack, "Dynamic correlation in wave propagation in random media," Phys. Rev. Lett. 93, 123901 (2004).
[CrossRef] [PubMed]

2003 (3)

J. Pearce, Z. Jian, and D. Mittleman, "Statistics of multiply scattered broadband terahertz pulses," Phys. Rev. Lett. 91, 043903 (2003).
[CrossRef] [PubMed]

Z. Jian, J. Pearce, and D. Mittleman, "Characterizing individual scattering events by measuring the amplitude and phase of the electric field diffusing through a random medium," Phys. Rev. Lett. 91, 033903 (2003).
[CrossRef] [PubMed]

M. Bondani, D. Redaelli, A. Spinelli, A. Andreoni, G. Roberti, P. Riccio, and R. Liuzzi, "Photon time-of-flight distributions through turbid media directly measured with single-photon avalanche diodes," J. Opt. Soc. Am. B 20, 2383-2388 (2003).
[CrossRef]

2002 (1)

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

2001 (3)

2000 (2)

D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, "Time-resolved experiments on light diffusion in anisotropic random media," Phys. Rev. E 62, 6681-6687 (2000).
[CrossRef]

J. D. McKinney, M. A. Webster, K. J. Webb, and 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]

1999 (2)

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, "Delay-time statistics for diffuse waves," Phys. Rev. E 59, 7166-7172 (1999).
[CrossRef]

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, "Statistics of wave dynamics in random media," Phys. Rev. Lett. 82, 715-718 (1999).
[CrossRef]

1998 (1)

1997 (3)

1995 (2)

D. A. Boas, L. E. Campbell, and A. G. Yodh, "Scattering and imaging with diffusing temporal field correlations," Phys. Rev. Lett. 75, 1855-1858 (1995).
[CrossRef] [PubMed]

M. A. O'Leary, D. A. Boas, B. Chance, and A. G. Yodh, "Experimental images of heterogeneous turbid media by frequency-domain diffusing tomography," Opt. Lett. 20, 426-428 (1995).
[CrossRef] [PubMed]

1994 (1)

R. Berkovits and S. Feng, "Correlations in coherent multiple scattering," Phys. Rep. 238, 135-172 (1994).
[CrossRef]

1989 (2)

M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," Appl. Opt. 28, 2331-2336 (1989).
[CrossRef] [PubMed]

K. M. Yoo and R. R. Alfano, "Time resolved depolarization of multiple back scattered light from random media," Phys. Lett. A 142, 531-536 (1989).
[CrossRef]

Alfano, R. R.

K. M. Yoo and R. R. Alfano, "Time resolved depolarization of multiple back scattered light from random media," Phys. Lett. A 142, 531-536 (1989).
[CrossRef]

Andreoni, A.

Berkovits, R.

R. Berkovits and S. Feng, "Correlations in coherent multiple scattering," Phys. Rep. 238, 135-172 (1994).
[CrossRef]

Boas, D. A.

Bondani, M.

Campbell, L. E.

D. A. Boas, L. E. Campbell, and A. G. Yodh, "Scattering and imaging with diffusing temporal field correlations," Phys. Rev. Lett. 75, 1855-1858 (1995).
[CrossRef] [PubMed]

Chabanov, A. A.

A. A. Chabanov, B. Hu, and A. Z. Genack, "Dynamic correlation in wave propagation in random media," Phys. Rev. Lett. 93, 123901 (2004).
[CrossRef] [PubMed]

A. A. Chabanov and A. Z. Genack, "Field distributions in the crossover from ballistic to diffusive wave propagation," Phys. Rev. E 56, R1338-R1341 (1997).
[CrossRef]

Chance, B.

Chau, K. J.

K. J. Chau, G. D. Dice, and A. Y. Elezzabi, "Coherent plasmonic enhanced terahertz transmission through random metallic media," Phys. Rev. Lett. 94, 173904 (2005).
[CrossRef] [PubMed]

K. J. Chau and A. Y. Elezzabi, "Terahertz transmission through ensembles of subwavelength-size metallic particles," Phys. Rev. B 72, 075110 (2005).
[CrossRef]

S. Mujumdar, K. J. Chau, and A. Y. Elezzabi, "Experimental and numerical investigation of terahertz transmission through strongly scattering sub-wavelength size spheres," Appl. Phys. Lett. 85, 6284-6286 (2004).
[CrossRef]

Collins, P. J.

Colocci, M.

D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, "Time-resolved experiments on light diffusion in anisotropic random media," Phys. Rev. E 62, 6681-6687 (2000).
[CrossRef]

Dice, G. D.

K. J. Chau, G. D. Dice, and A. Y. Elezzabi, "Coherent plasmonic enhanced terahertz transmission through random metallic media," Phys. Rev. Lett. 94, 173904 (2005).
[CrossRef] [PubMed]

Elezzabi, A. Y.

K. J. Chau, G. D. Dice, and A. Y. Elezzabi, "Coherent plasmonic enhanced terahertz transmission through random metallic media," Phys. Rev. Lett. 94, 173904 (2005).
[CrossRef] [PubMed]

K. J. Chau and A. Y. Elezzabi, "Terahertz transmission through ensembles of subwavelength-size metallic particles," Phys. Rev. B 72, 075110 (2005).
[CrossRef]

S. Mujumdar, K. J. Chau, and A. Y. Elezzabi, "Experimental and numerical investigation of terahertz transmission through strongly scattering sub-wavelength size spheres," Appl. Phys. Lett. 85, 6284-6286 (2004).
[CrossRef]

Feng, S.

R. Berkovits and S. Feng, "Correlations in coherent multiple scattering," Phys. Rep. 238, 135-172 (1994).
[CrossRef]

Genack, A. Z.

A. A. Chabanov, B. Hu, and A. Z. Genack, "Dynamic correlation in wave propagation in random media," Phys. Rev. Lett. 93, 123901 (2004).
[CrossRef] [PubMed]

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, "Delay-time statistics for diffuse waves," Phys. Rev. E 59, 7166-7172 (1999).
[CrossRef]

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, "Statistics of wave dynamics in random media," Phys. Rev. Lett. 82, 715-718 (1999).
[CrossRef]

A. A. Chabanov and A. Z. Genack, "Field distributions in the crossover from ballistic to diffusive wave propagation," Phys. Rev. E 56, R1338-R1341 (1997).
[CrossRef]

Gerke, T. D.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Hu, B.

A. A. Chabanov, B. Hu, and A. Z. Genack, "Dynamic correlation in wave propagation in random media," Phys. Rev. Lett. 93, 123901 (2004).
[CrossRef] [PubMed]

Jian, Z.

J. Pearce, Z. Jian, and D. Mittleman, "Spectral shifts as a signature of the onset of diffusion of broadband terahertz pulses," Opt. Lett. 29, 2926-2928 (2004).
[CrossRef]

J. Pearce, Z. Jian, and D. Mittleman, "Statistics of multiply scattered broadband terahertz pulses," Phys. Rev. Lett. 91, 043903 (2003).
[CrossRef] [PubMed]

Z. Jian, J. Pearce, and D. Mittleman, "Characterizing individual scattering events by measuring the amplitude and phase of the electric field diffusing through a random medium," Phys. Rev. Lett. 91, 033903 (2003).
[CrossRef] [PubMed]

Lagendijk, A.

J. Rivas, R. Sprik, and A. Lagendijk, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
[CrossRef]

Lasocki, D. L.

Liuzzi, R.

Markel, V. A.

Matson, C. L.

McKinney, J. D.

Mittleman, D.

J. Pearce and D. Mittleman, "Bayesian approach to non-Gaussian field statistics for diffuse broadband terahertz pulses," Opt. Lett. 30, 2843-2845 (2005).
[CrossRef] [PubMed]

J. Pearce, Z. Jian, and D. Mittleman, "Spectral shifts as a signature of the onset of diffusion of broadband terahertz pulses," Opt. Lett. 29, 2926-2928 (2004).
[CrossRef]

J. Pearce, Z. Jian, and D. Mittleman, "Statistics of multiply scattered broadband terahertz pulses," Phys. Rev. Lett. 91, 043903 (2003).
[CrossRef] [PubMed]

Z. Jian, J. Pearce, and D. Mittleman, "Characterizing individual scattering events by measuring the amplitude and phase of the electric field diffusing through a random medium," Phys. Rev. Lett. 91, 033903 (2003).
[CrossRef] [PubMed]

J. Pearce and D. Mittleman, "The propagation of single-cycle THz pulses in random media," Opt. Lett. 26, 2002-2004 (2001).
[CrossRef]

Mujumdar, S.

S. Mujumdar, K. J. Chau, and A. Y. Elezzabi, "Experimental and numerical investigation of terahertz transmission through strongly scattering sub-wavelength size spheres," Appl. Phys. Lett. 85, 6284-6286 (2004).
[CrossRef]

Muzzi, A.

D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, "Time-resolved experiments on light diffusion in anisotropic random media," Phys. Rev. E 62, 6681-6687 (2000).
[CrossRef]

O'Leary, M. A.

Patterson, M. S.

Pearce, J.

J. Pearce and D. Mittleman, "Bayesian approach to non-Gaussian field statistics for diffuse broadband terahertz pulses," Opt. Lett. 30, 2843-2845 (2005).
[CrossRef] [PubMed]

J. Pearce, Z. Jian, and D. Mittleman, "Spectral shifts as a signature of the onset of diffusion of broadband terahertz pulses," Opt. Lett. 29, 2926-2928 (2004).
[CrossRef]

J. Pearce, Z. Jian, and D. Mittleman, "Statistics of multiply scattered broadband terahertz pulses," Phys. Rev. Lett. 91, 043903 (2003).
[CrossRef] [PubMed]

Z. Jian, J. Pearce, and D. Mittleman, "Characterizing individual scattering events by measuring the amplitude and phase of the electric field diffusing through a random medium," Phys. Rev. Lett. 91, 033903 (2003).
[CrossRef] [PubMed]

J. Pearce and D. Mittleman, "The propagation of single-cycle THz pulses in random media," Opt. Lett. 26, 2002-2004 (2001).
[CrossRef]

Redaelli, D.

Riccio, P.

Righini, R.

D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, "Time-resolved experiments on light diffusion in anisotropic random media," Phys. Rev. E 62, 6681-6687 (2000).
[CrossRef]

Rivas, J.

J. Rivas, R. Sprik, and A. Lagendijk, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
[CrossRef]

Roberti, G.

Schotland, J. C.

Sebbah, P.

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, "Statistics of wave dynamics in random media," Phys. Rev. Lett. 82, 715-718 (1999).
[CrossRef]

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, "Delay-time statistics for diffuse waves," Phys. Rev. E 59, 7166-7172 (1999).
[CrossRef]

Spinelli, A.

Sprik, R.

J. Rivas, R. Sprik, and A. Lagendijk, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
[CrossRef]

Stoytchev, M.

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, "Statistics of wave dynamics in random media," Phys. Rev. Lett. 82, 715-718 (1999).
[CrossRef]

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, "Delay-time statistics for diffuse waves," Phys. Rev. E 59, 7166-7172 (1999).
[CrossRef]

Thompson, C. A.

van Tiggelen, B. A.

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, "Delay-time statistics for diffuse waves," Phys. Rev. E 59, 7166-7172 (1999).
[CrossRef]

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, "Statistics of wave dynamics in random media," Phys. Rev. Lett. 82, 715-718 (1999).
[CrossRef]

Webb, K. J.

Webster, M. A.

Weiner, A. M.

Wiersma, D. S.

D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, "Time-resolved experiments on light diffusion in anisotropic random media," Phys. Rev. E 62, 6681-6687 (2000).
[CrossRef]

Wilson, B. C.

Yodh, A. G.

Yoo, K. M.

K. M. Yoo and R. R. Alfano, "Time resolved depolarization of multiple back scattered light from random media," Phys. Lett. A 142, 531-536 (1989).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

S. Mujumdar, K. J. Chau, and A. Y. Elezzabi, "Experimental and numerical investigation of terahertz transmission through strongly scattering sub-wavelength size spheres," Appl. Phys. Lett. 85, 6284-6286 (2004).
[CrossRef]

J. Opt. Soc. Am. A (2)

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

Opt. Lett. (7)

Phys. Lett. A (1)

K. M. Yoo and R. R. Alfano, "Time resolved depolarization of multiple back scattered light from random media," Phys. Lett. A 142, 531-536 (1989).
[CrossRef]

Phys. Rep. (1)

R. Berkovits and S. Feng, "Correlations in coherent multiple scattering," Phys. Rep. 238, 135-172 (1994).
[CrossRef]

Phys. Rev. B (1)

K. J. Chau and A. Y. Elezzabi, "Terahertz transmission through ensembles of subwavelength-size metallic particles," Phys. Rev. B 72, 075110 (2005).
[CrossRef]

Phys. Rev. E (4)

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, "Delay-time statistics for diffuse waves," Phys. Rev. E 59, 7166-7172 (1999).
[CrossRef]

J. Rivas, R. Sprik, and A. Lagendijk, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
[CrossRef]

D. S. Wiersma, A. Muzzi, M. Colocci, and R. Righini, "Time-resolved experiments on light diffusion in anisotropic random media," Phys. Rev. E 62, 6681-6687 (2000).
[CrossRef]

A. A. Chabanov and A. Z. Genack, "Field distributions in the crossover from ballistic to diffusive wave propagation," Phys. Rev. E 56, R1338-R1341 (1997).
[CrossRef]

Phys. Rev. Lett. (7)

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, "Statistics of wave dynamics in random media," Phys. Rev. Lett. 82, 715-718 (1999).
[CrossRef]

Z. Jian, J. Pearce, and D. Mittleman, "Characterizing individual scattering events by measuring the amplitude and phase of the electric field diffusing through a random medium," Phys. Rev. Lett. 91, 033903 (2003).
[CrossRef] [PubMed]

J. Pearce, Z. Jian, and D. Mittleman, "Statistics of multiply scattered broadband terahertz pulses," Phys. Rev. Lett. 91, 043903 (2003).
[CrossRef] [PubMed]

D. A. Boas, L. E. Campbell, and A. G. Yodh, "Scattering and imaging with diffusing temporal field correlations," Phys. Rev. Lett. 75, 1855-1858 (1995).
[CrossRef] [PubMed]

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

A. A. Chabanov, B. Hu, and A. Z. Genack, "Dynamic correlation in wave propagation in random media," Phys. Rev. Lett. 93, 123901 (2004).
[CrossRef] [PubMed]

K. J. Chau, G. D. Dice, and A. Y. Elezzabi, "Coherent plasmonic enhanced terahertz transmission through random metallic media," Phys. Rev. Lett. 94, 173904 (2005).
[CrossRef] [PubMed]

Other (1)

J. W. Goodman, Statistical Optics (Wiley, 2000).

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Figures (5)

Fig. 1
Fig. 1

Probability distribution of the electric field at several different time delays (as labeled), each normalized to unity at its peak. The solid curves are zero-mean Gaussians, computed with σ 2 ( τ ) = I sc ( τ ) , the average intensity. The variance σ 2 ( τ ) depends on the time delay τ, demonstrating that this statistical process is nonstationary. The τ = 30 ps result corresponds to a delay when no photons have yet reached the detector. This characterizes the measurement noise, which is also Gaussian distributed.

Fig. 2
Fig. 2

Configurationally averaged intensity I sc ( τ ) , proportional to the photon TOF distribution. This is computed by averaging the measured intensities over all configurations of the random medium. The solid curve shows the result of adjacent averaging of I sc ( τ ) using a sliding 50 ps time window.

Fig. 3
Fig. 3

Probability distribution of I E [ I ] , the intensity normalized by the mean intensity. The plot shows the data and the prediction computed using Eq. (3). The inset shows the same result on a log–log scale.

Fig. 4
Fig. 4

Probability distribution of the (real) electric field normalized by its standard deviation. The plot shows the data (circles) and the prediction computed from Eq. (4). The dashed curve shows a Gaussian, which does not fit the wings of the distribution.

Fig. 5
Fig. 5

α ( τ ) parameter of a gamma distribution [Eq. (2)] extracted using the fitting procedure described in the text, as a function of the position of the center of the 50 ps time window τ. In fitting to the gamma distribution, β ( τ ) is set equal to the smoothed average intensity I sc ( τ ) , the solid curve in Fig. 2. The error bars denote 95% confidence intervals.

Equations (4)

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

P ( E τ ) = 1 2 π I sc ( τ ) exp [ E 2 2 I sc ( τ ) ] ,
P ( I τ ) = I α ( τ ) 1 exp [ I β ( τ ) ] β ( τ ) α ( τ ) Γ [ α ( τ ) ] ,
p ( I ) = 1 Δ t t 1 t 2 exp [ I I sc ( τ ) ] I sc ( τ ) d τ .
P ( E ) = 1 Δ t t 1 t 2 1 2 π I sc ( τ ) exp [ E 2 2 I sc ( τ ) ] d τ .

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