Abstract

We follow the trajectories of phase singularities at nulls of intensity in the speckle pattern of waves transmitted through random media as the frequency of the incident radiation is scanned in microwave experiments and numerical simulations. Phase singularities are observed to diffuse with a linear increase of the square displacement R2 with frequency shift. The product of the diffusion coefficient of phase singularities in the transmitted speckle pattern and the photon diffusion coefficient through the random medium is proportional to the square of the effective sample length. This provides the photon diffusion coefficient and a method for characterizing the motion of dynamic material systems.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Pecora, Ann. Rev. Biophys. Bioeng. 1, 257 (1972).
    [CrossRef]
  2. M. D. Stern, Nature 254, 56 (1975).
    [CrossRef]
  3. A. F. Fercher and J. D. Briers, Opt. Commun. 37, 326 (1981).
    [CrossRef]
  4. A. K. Dunn and D. A. Boas, J. Biomed. Opt. 15, 011109 (2010).
    [CrossRef]
  5. A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, J. Cereb. Blood Flow Metab. 21, 195201 (2001).
  6. A. Z. Genack and J. M. Drake, Europhys. Lett. 11, 331 (1990).
    [CrossRef]
  7. M. V. Berry and M. R. Dennis, Proc. R. Soc. London, Ser. A 456, 2059 (2000).
    [CrossRef]
  8. M. S. Soskin and M. V. Vasnetsov, Progress in OpticsE. Wolf, ed. (Elsevier, 2001), Vol. 42, p. 219.
  9. W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, Phys. Rev. Lett. 94, 103902 (2005).
    [CrossRef]
  10. S. Zhang, B. Hu, P. Sebbah, and A. Z. Genack, Phys. Rev. Lett. 99, 063902 (2007).
    [CrossRef]
  11. S. Zhang, Y. Lockerman, and A. Z. Genack, Phys. Rev. E 82, 051114 (2010).
    [CrossRef]
  12. I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
    [CrossRef]
  13. S. J. Kirkpatrick, K. Khaksari, D. Thomas, and D. D. Duncan, J. Biomed. Opt. 17, 050504 (2012).
    [CrossRef]
  14. J. W. Goodman, ed., Introduction to Fourier Optics, (McGraw-Hill, 1968).
  15. R. Kubo, Rep. Prog. Phys. 29, 255 (1966).
    [CrossRef]
  16. N. Garcia, A. Z. Genack, and A. A. Lisyansky, Phys. Rev. B 46, 14475 (1992).
    [CrossRef]
  17. P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, 2006).
  18. R. Landauer and M. Büttiker, Phys. Rev. B 36, 6255 (1987).
    [CrossRef]

2012 (1)

S. J. Kirkpatrick, K. Khaksari, D. Thomas, and D. D. Duncan, J. Biomed. Opt. 17, 050504 (2012).
[CrossRef]

2010 (2)

S. Zhang, Y. Lockerman, and A. Z. Genack, Phys. Rev. E 82, 051114 (2010).
[CrossRef]

A. K. Dunn and D. A. Boas, J. Biomed. Opt. 15, 011109 (2010).
[CrossRef]

2007 (1)

S. Zhang, B. Hu, P. Sebbah, and A. Z. Genack, Phys. Rev. Lett. 99, 063902 (2007).
[CrossRef]

2005 (1)

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef]

2001 (1)

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, J. Cereb. Blood Flow Metab. 21, 195201 (2001).

2000 (1)

M. V. Berry and M. R. Dennis, Proc. R. Soc. London, Ser. A 456, 2059 (2000).
[CrossRef]

1994 (1)

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef]

1992 (1)

N. Garcia, A. Z. Genack, and A. A. Lisyansky, Phys. Rev. B 46, 14475 (1992).
[CrossRef]

1990 (1)

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

1987 (1)

R. Landauer and M. Büttiker, Phys. Rev. B 36, 6255 (1987).
[CrossRef]

1981 (1)

A. F. Fercher and J. D. Briers, Opt. Commun. 37, 326 (1981).
[CrossRef]

1975 (1)

M. D. Stern, Nature 254, 56 (1975).
[CrossRef]

1972 (1)

R. Pecora, Ann. Rev. Biophys. Bioeng. 1, 257 (1972).
[CrossRef]

1966 (1)

R. Kubo, Rep. Prog. Phys. 29, 255 (1966).
[CrossRef]

Berry, M. V.

M. V. Berry and M. R. Dennis, Proc. R. Soc. London, Ser. A 456, 2059 (2000).
[CrossRef]

Boas, D. A.

A. K. Dunn and D. A. Boas, J. Biomed. Opt. 15, 011109 (2010).
[CrossRef]

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, J. Cereb. Blood Flow Metab. 21, 195201 (2001).

Bolay, H.

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, J. Cereb. Blood Flow Metab. 21, 195201 (2001).

Briers, J. D.

A. F. Fercher and J. D. Briers, Opt. Commun. 37, 326 (1981).
[CrossRef]

Büttiker, M.

R. Landauer and M. Büttiker, Phys. Rev. B 36, 6255 (1987).
[CrossRef]

Dennis, M. R.

M. V. Berry and M. R. Dennis, Proc. R. Soc. London, Ser. A 456, 2059 (2000).
[CrossRef]

Drake, J. M.

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

Duncan, D. D.

S. J. Kirkpatrick, K. Khaksari, D. Thomas, and D. D. Duncan, J. Biomed. Opt. 17, 050504 (2012).
[CrossRef]

Dunn, A. K.

A. K. Dunn and D. A. Boas, J. Biomed. Opt. 15, 011109 (2010).
[CrossRef]

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, J. Cereb. Blood Flow Metab. 21, 195201 (2001).

Fercher, A. F.

A. F. Fercher and J. D. Briers, Opt. Commun. 37, 326 (1981).
[CrossRef]

Freund, I.

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef]

Garcia, N.

N. Garcia, A. Z. Genack, and A. A. Lisyansky, Phys. Rev. B 46, 14475 (1992).
[CrossRef]

Genack, A. Z.

S. Zhang, Y. Lockerman, and A. Z. Genack, Phys. Rev. E 82, 051114 (2010).
[CrossRef]

S. Zhang, B. Hu, P. Sebbah, and A. Z. Genack, Phys. Rev. Lett. 99, 063902 (2007).
[CrossRef]

N. Garcia, A. Z. Genack, and A. A. Lisyansky, Phys. Rev. B 46, 14475 (1992).
[CrossRef]

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

Hanson, S. G.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef]

Hu, B.

S. Zhang, B. Hu, P. Sebbah, and A. Z. Genack, Phys. Rev. Lett. 99, 063902 (2007).
[CrossRef]

Khaksari, K.

S. J. Kirkpatrick, K. Khaksari, D. Thomas, and D. D. Duncan, J. Biomed. Opt. 17, 050504 (2012).
[CrossRef]

Kirkpatrick, S. J.

S. J. Kirkpatrick, K. Khaksari, D. Thomas, and D. D. Duncan, J. Biomed. Opt. 17, 050504 (2012).
[CrossRef]

Kubo, R.

R. Kubo, Rep. Prog. Phys. 29, 255 (1966).
[CrossRef]

Landauer, R.

R. Landauer and M. Büttiker, Phys. Rev. B 36, 6255 (1987).
[CrossRef]

Lisyansky, A. A.

N. Garcia, A. Z. Genack, and A. A. Lisyansky, Phys. Rev. B 46, 14475 (1992).
[CrossRef]

Lockerman, Y.

S. Zhang, Y. Lockerman, and A. Z. Genack, Phys. Rev. E 82, 051114 (2010).
[CrossRef]

Miyamoto, Y.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef]

Moskowitz, M. A.

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, J. Cereb. Blood Flow Metab. 21, 195201 (2001).

Pecora, R.

R. Pecora, Ann. Rev. Biophys. Bioeng. 1, 257 (1972).
[CrossRef]

Sebbah, P.

S. Zhang, B. Hu, P. Sebbah, and A. Z. Genack, Phys. Rev. Lett. 99, 063902 (2007).
[CrossRef]

Sheng, P.

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, 2006).

Shvartsman, N.

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, Progress in OpticsE. Wolf, ed. (Elsevier, 2001), Vol. 42, p. 219.

Stern, M. D.

M. D. Stern, Nature 254, 56 (1975).
[CrossRef]

Takeda, M.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef]

Thomas, D.

S. J. Kirkpatrick, K. Khaksari, D. Thomas, and D. D. Duncan, J. Biomed. Opt. 17, 050504 (2012).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, Progress in OpticsE. Wolf, ed. (Elsevier, 2001), Vol. 42, p. 219.

Wang, W.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef]

Zhang, S.

S. Zhang, Y. Lockerman, and A. Z. Genack, Phys. Rev. E 82, 051114 (2010).
[CrossRef]

S. Zhang, B. Hu, P. Sebbah, and A. Z. Genack, Phys. Rev. Lett. 99, 063902 (2007).
[CrossRef]

Ann. Rev. Biophys. Bioeng. (1)

R. Pecora, Ann. Rev. Biophys. Bioeng. 1, 257 (1972).
[CrossRef]

Europhys. Lett. (1)

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

J. Biomed. Opt. (2)

A. K. Dunn and D. A. Boas, J. Biomed. Opt. 15, 011109 (2010).
[CrossRef]

S. J. Kirkpatrick, K. Khaksari, D. Thomas, and D. D. Duncan, J. Biomed. Opt. 17, 050504 (2012).
[CrossRef]

J. Cereb. Blood Flow Metab. (1)

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, J. Cereb. Blood Flow Metab. 21, 195201 (2001).

Nature (1)

M. D. Stern, Nature 254, 56 (1975).
[CrossRef]

Opt. Commun. (1)

A. F. Fercher and J. D. Briers, Opt. Commun. 37, 326 (1981).
[CrossRef]

Phys. Rev. A (1)

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef]

Phys. Rev. B (2)

N. Garcia, A. Z. Genack, and A. A. Lisyansky, Phys. Rev. B 46, 14475 (1992).
[CrossRef]

R. Landauer and M. Büttiker, Phys. Rev. B 36, 6255 (1987).
[CrossRef]

Phys. Rev. E (1)

S. Zhang, Y. Lockerman, and A. Z. Genack, Phys. Rev. E 82, 051114 (2010).
[CrossRef]

Phys. Rev. Lett. (2)

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef]

S. Zhang, B. Hu, P. Sebbah, and A. Z. Genack, Phys. Rev. Lett. 99, 063902 (2007).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

M. V. Berry and M. R. Dennis, Proc. R. Soc. London, Ser. A 456, 2059 (2000).
[CrossRef]

Rep. Prog. Phys. (1)

R. Kubo, Rep. Prog. Phys. 29, 255 (1966).
[CrossRef]

Other (3)

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, 2006).

J. W. Goodman, ed., Introduction to Fourier Optics, (McGraw-Hill, 1968).

M. S. Soskin and M. V. Vasnetsov, Progress in OpticsE. Wolf, ed. (Elsevier, 2001), Vol. 42, p. 219.

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 (7)

Fig. 1.
Fig. 1.

Singularities in the speckle pattern in microwave experiments. Singularities with positive and negative charges are represented by yellow dots and yellow squares, respectively. We show trajectories of two singularities as examples. Note that since singularities are annihilated after a typical decaying frequency, as will be shown in Fig. 4, the comparably longer trajectory as in this figure is a relatively rare event.

Fig. 2.
Fig. 2.

Average of the square of singularity displacement with frequency shift. Slope of R2 approaches a constant.

Fig. 3.
Fig. 3.

Measurement of the velocity auto-correlation function with frequency shift.

Fig. 4.
Fig. 4.

Decay of the number N of phase singularities with frequency shift in microwave experiments. N0 is the initial number of singularities. N/N0 falls to 1/e in a frequency shift ΔνN=20.4MHz.

Fig. 5.
Fig. 5.

Comparison of the square displacement of phase singularities in microwave experiments in the frequency domain and simulations in the time domain. The displacement, frequency shift, and time are normalized in terms of the singularity density and survival as explained in the text.

Fig. 6.
Fig. 6.

Linear decay of intensity averaged over the cross section with depth x into the sample I(x) for sample with L=40cm determined from FDTD simulations. I0 is the value to which the average intensity within the sample extrapolates at the input surface x=0. The intensity profile within the sample extrapolates to zero at a length zb=3.85cm beyond the output surface.

Fig. 7.
Fig. 7.

Relation between the diffusion coefficient of singularities in the output plane Ds, the diffusion coefficient of photons through the sample Dp, and the effective sample length Leff in FDTD simulations. (a) Results are shown for three values of L with the same structure, and thus the same value of Dp. Ds is proportional to Leff2. (b) The relationship between Ds and Dp at L=23cm for samples 1, 2, and 3 corresponds to filling fractions of alumina spheres of 0.068, 0.1, and 0.15. Here C=4Ds(Dp/Leff2).

Equations (3)

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

R2(t)=(r⃗(t)r⃗(0))2=0tdt0tdtu⃗(t)u⃗(t),
R2=4DsΔν.
Dp=CLeff24Ds.

Metrics