Abstract

We develop a closed-form expression for the probability distribution function for the field components of a diffusive broadband wave propagating through a random medium. We consider each spectral component to provide an individual observation of a random variable, the configurationally averaged spectral intensity. Since the intensity determines the variance of the field distribution at each frequency, this random variable serves as the Bayesian prior that determines the form of the non-Gaussian field statistics. This model agrees well with experimental results.

© 2005 Optical Society of America

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  1. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, Opt. Lett. 20, 426 (1995).
    [CrossRef]
  2. D. A. Boas, L. E. Campbell, and A. G. Yodh, Phys. Rev. Lett. 75, 1855 (1995).
    [CrossRef] [PubMed]
  3. D. A. Boas and A. G. Yodh, J. Opt. Soc. Am. A 14, 192 (1997).
    [CrossRef]
  4. D. L. Lasocki, C. L. Matson, and P. J. Collins, Opt. Lett. 23, 558 (1998).
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  5. J. D. McKinney, M. A. Webster, K. J. Webb, and A. M. Weiner, Opt. Lett. 25, 4 (2000).
    [CrossRef]
  6. J. C. Schotland and V. A. Markel, J. Opt. Soc. Am. A 18, 2767 (2001).
    [CrossRef]
  7. R. Berkovits and S. Feng, Phys. Rep. 238, 135 (1994).
    [CrossRef]
  8. C. A. Thompson, K. J. Webb, and A. M. Weiner, Appl. Opt. 36, 3726 (1997).
    [CrossRef] [PubMed]
  9. M. A. Webster, K. J. Webb, and A. M. Weiner, Phys. Rev. Lett. 88, 033901 (2002).
    [CrossRef]
  10. B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, Phys. Rev. E 59, 7166 (1999).
    [CrossRef]
  11. J. Rivas, R. Sprik, and A. Lagendijk, Phys. Rev. E 63, 046613 (2001).
    [CrossRef]
  12. J. Pearce, Z. Jian, and D. Mittleman, Phys. Rev. Lett. 91, 043903 (2003).
    [CrossRef]
  13. S. Mujumdar, K. J. Chau, and A. Y. Elezzabi, Appl. Phys. Lett. 85, 6284 (2004).
    [CrossRef]
  14. J. Pearce, Z. Jian, and D. Mittleman, Opt. Lett. 29, 2926 (2004).
    [CrossRef]
  15. K. J. Chau, G. D. Dice, and A. Y. Elezzabi, Phys. Rev. Lett. 94, 173904 (2005).
    [CrossRef]
  16. J. Pearce and D. Mittleman, Opt. Lett. 26, 2002 (2001).
    [CrossRef]
  17. A. A. Chabanov and A. Z. Genack, Phys. Rev. E 56, R1338 (1997).
    [CrossRef]
  18. J. W. Goodman, Statistical Optics (Wiley, 2000).
  19. A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian Data Analysis (Chapman & Hall, 2003).
  20. N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions (Wiley, New York, 1994).
  21. J. Pearce, K. Doyle, Z. Jian, J. Deibel, and D. M. Mittleman, “Non-stationary time-domain statistics of multiply scattered broadband terahertz pulses,” submitted to Phys. Rev. E.

2005 (1)

K. J. Chau, G. D. Dice, and A. Y. Elezzabi, Phys. Rev. Lett. 94, 173904 (2005).
[CrossRef]

2004 (2)

S. Mujumdar, K. J. Chau, and A. Y. Elezzabi, Appl. Phys. Lett. 85, 6284 (2004).
[CrossRef]

J. Pearce, Z. Jian, and D. Mittleman, Opt. Lett. 29, 2926 (2004).
[CrossRef]

2003 (1)

J. Pearce, Z. Jian, and D. Mittleman, Phys. Rev. Lett. 91, 043903 (2003).
[CrossRef]

2002 (1)

M. A. Webster, K. J. Webb, and A. M. Weiner, Phys. Rev. Lett. 88, 033901 (2002).
[CrossRef]

2001 (3)

2000 (1)

1999 (1)

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, Phys. Rev. E 59, 7166 (1999).
[CrossRef]

1998 (1)

1997 (3)

1995 (2)

D. A. Boas, L. E. Campbell, and A. G. Yodh, Phys. Rev. Lett. 75, 1855 (1995).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, Opt. Lett. 20, 426 (1995).
[CrossRef]

1994 (1)

R. Berkovits and S. Feng, Phys. Rep. 238, 135 (1994).
[CrossRef]

Balakrishnan, N.

N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions (Wiley, New York, 1994).

Berkovits, R.

R. Berkovits and S. Feng, Phys. Rep. 238, 135 (1994).
[CrossRef]

Boas, D. A.

Campbell, L. E.

D. A. Boas, L. E. Campbell, and A. G. Yodh, Phys. Rev. Lett. 75, 1855 (1995).
[CrossRef] [PubMed]

Carlin, J.

A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian Data Analysis (Chapman & Hall, 2003).

Chabanov, A. A.

A. A. Chabanov and A. Z. Genack, Phys. Rev. E 56, R1338 (1997).
[CrossRef]

Chance, B.

Chau, K. J.

K. J. Chau, G. D. Dice, and A. Y. Elezzabi, Phys. Rev. Lett. 94, 173904 (2005).
[CrossRef]

S. Mujumdar, K. J. Chau, and A. Y. Elezzabi, Appl. Phys. Lett. 85, 6284 (2004).
[CrossRef]

Collins, P. J.

Deibel, J.

J. Pearce, K. Doyle, Z. Jian, J. Deibel, and D. M. Mittleman, “Non-stationary time-domain statistics of multiply scattered broadband terahertz pulses,” submitted to Phys. Rev. E.

Dice, G. D.

K. J. Chau, G. D. Dice, and A. Y. Elezzabi, Phys. Rev. Lett. 94, 173904 (2005).
[CrossRef]

Doyle, K.

J. Pearce, K. Doyle, Z. Jian, J. Deibel, and D. M. Mittleman, “Non-stationary time-domain statistics of multiply scattered broadband terahertz pulses,” submitted to Phys. Rev. E.

Elezzabi, A. Y.

K. J. Chau, G. D. Dice, and A. Y. Elezzabi, Phys. Rev. Lett. 94, 173904 (2005).
[CrossRef]

S. Mujumdar, K. J. Chau, and A. Y. Elezzabi, Appl. Phys. Lett. 85, 6284 (2004).
[CrossRef]

Feng, S.

R. Berkovits and S. Feng, Phys. Rep. 238, 135 (1994).
[CrossRef]

Gelman, A.

A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian Data Analysis (Chapman & Hall, 2003).

Genack, A. Z.

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, Phys. Rev. E 59, 7166 (1999).
[CrossRef]

A. A. Chabanov and A. Z. Genack, Phys. Rev. E 56, R1338 (1997).
[CrossRef]

Goodman, J. W.

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

Jian, Z.

J. Pearce, Z. Jian, and D. Mittleman, Opt. Lett. 29, 2926 (2004).
[CrossRef]

J. Pearce, Z. Jian, and D. Mittleman, Phys. Rev. Lett. 91, 043903 (2003).
[CrossRef]

J. Pearce, K. Doyle, Z. Jian, J. Deibel, and D. M. Mittleman, “Non-stationary time-domain statistics of multiply scattered broadband terahertz pulses,” submitted to Phys. Rev. E.

Johnson, N. L.

N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions (Wiley, New York, 1994).

Kotz, S.

N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions (Wiley, New York, 1994).

Lagendijk, A.

J. Rivas, R. Sprik, and A. Lagendijk, Phys. Rev. E 63, 046613 (2001).
[CrossRef]

Lasocki, D. L.

Markel, V. A.

Matson, C. L.

McKinney, J. D.

Mittleman, D.

Mittleman, D. M.

J. Pearce, K. Doyle, Z. Jian, J. Deibel, and D. M. Mittleman, “Non-stationary time-domain statistics of multiply scattered broadband terahertz pulses,” submitted to Phys. Rev. E.

Mujumdar, S.

S. Mujumdar, K. J. Chau, and A. Y. Elezzabi, Appl. Phys. Lett. 85, 6284 (2004).
[CrossRef]

O’Leary, M. A.

Pearce, J.

J. Pearce, Z. Jian, and D. Mittleman, Opt. Lett. 29, 2926 (2004).
[CrossRef]

J. Pearce, Z. Jian, and D. Mittleman, Phys. Rev. Lett. 91, 043903 (2003).
[CrossRef]

J. Pearce and D. Mittleman, Opt. Lett. 26, 2002 (2001).
[CrossRef]

J. Pearce, K. Doyle, Z. Jian, J. Deibel, and D. M. Mittleman, “Non-stationary time-domain statistics of multiply scattered broadband terahertz pulses,” submitted to Phys. Rev. E.

Rivas, J.

J. Rivas, R. Sprik, and A. Lagendijk, Phys. Rev. E 63, 046613 (2001).
[CrossRef]

Rubin, D.

A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian Data Analysis (Chapman & Hall, 2003).

Schotland, J. C.

Sebbah, P.

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, Phys. Rev. E 59, 7166 (1999).
[CrossRef]

Sprik, R.

J. Rivas, R. Sprik, and A. Lagendijk, Phys. Rev. E 63, 046613 (2001).
[CrossRef]

Stern, H.

A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian Data Analysis (Chapman & Hall, 2003).

Stoytchev, M.

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, Phys. Rev. E 59, 7166 (1999).
[CrossRef]

Thompson, C. A.

van Tiggelen, B. A.

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, Phys. Rev. E 59, 7166 (1999).
[CrossRef]

Webb, K. J.

Webster, M. A.

M. A. Webster, K. J. Webb, and A. M. Weiner, Phys. Rev. Lett. 88, 033901 (2002).
[CrossRef]

J. D. McKinney, M. A. Webster, K. J. Webb, and A. M. Weiner, Opt. Lett. 25, 4 (2000).
[CrossRef]

Weiner, A. M.

Yodh, A. G.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

S. Mujumdar, K. J. Chau, and A. Y. Elezzabi, Appl. Phys. Lett. 85, 6284 (2004).
[CrossRef]

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

Opt. Lett. (5)

Phys. Rep. (1)

R. Berkovits and S. Feng, Phys. Rep. 238, 135 (1994).
[CrossRef]

Phys. Rev. E (3)

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, Phys. Rev. E 59, 7166 (1999).
[CrossRef]

J. Rivas, R. Sprik, and A. Lagendijk, Phys. Rev. E 63, 046613 (2001).
[CrossRef]

A. A. Chabanov and A. Z. Genack, Phys. Rev. E 56, R1338 (1997).
[CrossRef]

Phys. Rev. Lett. (4)

J. Pearce, Z. Jian, and D. Mittleman, Phys. Rev. Lett. 91, 043903 (2003).
[CrossRef]

D. A. Boas, L. E. Campbell, and A. G. Yodh, Phys. Rev. Lett. 75, 1855 (1995).
[CrossRef] [PubMed]

M. A. Webster, K. J. Webb, and A. M. Weiner, Phys. Rev. Lett. 88, 033901 (2002).
[CrossRef]

K. J. Chau, G. D. Dice, and A. Y. Elezzabi, Phys. Rev. Lett. 94, 173904 (2005).
[CrossRef]

Other (4)

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

A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian Data Analysis (Chapman & Hall, 2003).

N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions (Wiley, New York, 1994).

J. Pearce, K. Doyle, Z. Jian, J. Deibel, and D. M. Mittleman, “Non-stationary time-domain statistics of multiply scattered broadband terahertz pulses,” submitted to Phys. Rev. E.

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Figures (3)

Fig. 1
Fig. 1

Configurationally averaged intensity I ( ω ) . Inset, the same data on a log scale, normalized to unity at its peak. The horizontal line indicates the noise level, which we use to determine the value of the γ parameter in Eq. (3).

Fig. 2
Fig. 2

Probability distribution P ( S ) of the configurationally averaged intensity normalized to its expectation value E [ S ] , computed by taking a histogram of the estimate of I ( ω ) shown in Fig. 1. Solid curve, a gamma distribution [Eq. (3)], with parameters α = 0.689 , β = 1.38 , and γ = 0.05 , determined as described in the text.

Fig. 3
Fig. 3

Probability distribution of the normalized real (triangles) and imaginary (circles) parts of the complex scattered electric field, P ( r σ ) and P ( i σ ) , normalized to their standard deviation σ. Solid curve, prediction of Eq. (2) with the gamma probability distribution function in Eq. (3) shown as the prior distribution. Inset, the same result plotted on a log scale. The predicted distribution agrees with the measurements over nearly four decades and over ± 6 standard deviations in the field amplitude. The dashed curve is a best fit to a Gaussian distribution.

Equations (5)

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P ( a ) = 1 Δ ω ω 1 ω 2 1 [ π I ( ω ) ] 1 2 exp [ a 2 I ( ω ) ] d ω ,
P ( a ) = 1 π S exp ( a 2 S ) P ( S ) d S ,
P ( S ) = 1 β Γ ( α ) ( S γ β ) α 1 exp ( S γ β ) ,
E [ S ] = α β + γ ,
E [ S 2 ] = α 2 β 2 + γ 2 + α β ( β + 2 γ ) ,

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