Abstract

We show that a scalar Bethe–Salpeter equation model captures the measured copolarized electric field frequency correlation magnitude for randomly scattering slabs in the weakly scattering, nondiffusive regime. Consequently, the model could be used to form images of tissue on the millimeter and submillimeter length scale, and for environmental sensing with comparable scatter, as dictated by the optical scattering length in relation to the scattering domain size.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, Phys. Rev. E 49, 1767 (1994).
    [CrossRef]
  2. T. D. Gerke, M. A. Webster, A. M. Weiner, and K. J. Webb, J. Opt. Soc. Am. A 22, 2691 (2005).
    [CrossRef]
  3. Z. Wang, M. A. Webster, A. M. Weiner, and K. J. Webb, J. Opt. Soc. Am. A 23, 3045 (2006).
    [CrossRef]
  4. A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, Science 287, 287 (2000).
    [CrossRef]
  5. S. R. Arridge, Appl. Opt. 34, 7395 (1995).
    [CrossRef]
  6. M. A. Webster, K. J. Webb, and A. M. Weiner, Phys. Rev. Lett. 88, 033901 (2002).
    [CrossRef]
  7. M. J. Stephen and G. Cwilich, Phys. Rev. B 34, 7564 (1986).
    [CrossRef]
  8. K. J. Peters, Phys. Rev. B 46, 801 (1992).
    [CrossRef]
  9. U. Frisch, “Wave propagation in random media,” in Probability Methods in Applied Mathematics, A. Bharucha-Reid, ed. (Academic, 1968), Vol. 1, pp. 176–198.
  10. Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
    [CrossRef]
  11. A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, and R. P. Millane, Opt. Lett. 27, 95 (2002).
    [CrossRef]
  12. M. I. Mishchenko, Appl. Opt. 41, 7114 (2002).
    [CrossRef]
  13. L. A. Apresyan and Y. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects (Gordon and Breach, 1996).
  14. A. Z. Genack and J. M. Drake, Europhys. Lett. 11, 331 (1990).
    [CrossRef]
  15. M. B. van der Mark, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 37, 3575 (1988).
    [CrossRef]
  16. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1977), Vol. 2.
  17. J. Goodman, Statistical Optics (Wiley, 1985).
  18. M. S. Patterson, B. Chance, and B. C. Wilson, Appl. Opt. 28, 2331 (1989).
    [CrossRef]
  19. C. A. Thompson, K. J. Webb, and A. M. Weiner, Appl. Opt. 36, 3726 (1997).
    [CrossRef]
  20. V. L. Kuz’min and I. V. Meglinskiĭ, J. Exp. Theor. Phys. 110, 742 (2010).
    [CrossRef]
  21. I. S. Reed, IRE Tran. Inf. Theory 8, 194 (1962).
    [CrossRef]

2010 (1)

V. L. Kuz’min and I. V. Meglinskiĭ, J. Exp. Theor. Phys. 110, 742 (2010).
[CrossRef]

2006 (1)

2005 (1)

2002 (3)

2000 (1)

A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, Science 287, 287 (2000).
[CrossRef]

1999 (1)

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

1997 (1)

1995 (1)

1994 (1)

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, Phys. Rev. E 49, 1767 (1994).
[CrossRef]

1992 (1)

K. J. Peters, Phys. Rev. B 46, 801 (1992).
[CrossRef]

1990 (1)

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

1989 (1)

1988 (1)

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 37, 3575 (1988).
[CrossRef]

1986 (1)

M. J. Stephen and G. Cwilich, Phys. Rev. B 34, 7564 (1986).
[CrossRef]

1962 (1)

I. S. Reed, IRE Tran. Inf. Theory 8, 194 (1962).
[CrossRef]

Apresyan, L. A.

L. A. Apresyan and Y. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects (Gordon and Breach, 1996).

Arridge, S. R.

Balents, L.

A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, Science 287, 287 (2000).
[CrossRef]

Baranger, H. U.

A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, Science 287, 287 (2000).
[CrossRef]

Bicout, D.

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, Phys. Rev. E 49, 1767 (1994).
[CrossRef]

Bouman, C. A.

Brosseau, C.

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, Phys. Rev. E 49, 1767 (1994).
[CrossRef]

Chance, B.

Cwilich, G.

M. J. Stephen and G. Cwilich, Phys. Rev. B 34, 7564 (1986).
[CrossRef]

Drake, J. M.

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

Frisch, U.

U. Frisch, “Wave propagation in random media,” in Probability Methods in Applied Mathematics, A. Bharucha-Reid, ed. (Academic, 1968), Vol. 1, pp. 176–198.

Genack, A. Z.

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

Gerke, T. D.

Goodman, J.

J. Goodman, Statistical Optics (Wiley, 1985).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1977), Vol. 2.

Jones, I. P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

Kravtsov, Y. A.

L. A. Apresyan and Y. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects (Gordon and Breach, 1996).

Kuz’min, V. L.

V. L. Kuz’min and I. V. Meglinskiĭ, J. Exp. Theor. Phys. 110, 742 (2010).
[CrossRef]

Lagendijk, A.

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 37, 3575 (1988).
[CrossRef]

Martinez, A. S.

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, Phys. Rev. E 49, 1767 (1994).
[CrossRef]

Meglinskii, I. V.

V. L. Kuz’min and I. V. Meglinskiĭ, J. Exp. Theor. Phys. 110, 742 (2010).
[CrossRef]

Millane, R. P.

Milstein, A. B.

Mishchenko, M. I.

Moustakas, A. L.

A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, Science 287, 287 (2000).
[CrossRef]

Oh, S.

Page, J. H.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

Patterson, M. S.

Peters, K. J.

K. J. Peters, Phys. Rev. B 46, 801 (1992).
[CrossRef]

Reed, I. S.

I. S. Reed, IRE Tran. Inf. Theory 8, 194 (1962).
[CrossRef]

Reynolds, J. S.

Schmitt, J. M.

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, Phys. Rev. E 49, 1767 (1994).
[CrossRef]

Schriemer, H. P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

Sengupta, A. M.

A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, Science 287, 287 (2000).
[CrossRef]

Sheng, P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

Simon, S. H.

A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, Science 287, 287 (2000).
[CrossRef]

Stephen, M. J.

M. J. Stephen and G. Cwilich, Phys. Rev. B 34, 7564 (1986).
[CrossRef]

Thompson, C. A.

van Albada, M. P.

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 37, 3575 (1988).
[CrossRef]

van der Mark, M. B.

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 37, 3575 (1988).
[CrossRef]

Wang, Z.

Webb, K. J.

Webster, M. A.

Weiner, A. M.

Weitz, D. A.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

Wilson, B. C.

Zhang, Z. Q.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

Appl. Opt. (4)

Europhys. Lett. (1)

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

IRE Tran. Inf. Theory (1)

I. S. Reed, IRE Tran. Inf. Theory 8, 194 (1962).
[CrossRef]

J. Exp. Theor. Phys. (1)

V. L. Kuz’min and I. V. Meglinskiĭ, J. Exp. Theor. Phys. 110, 742 (2010).
[CrossRef]

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

Opt. Lett. (1)

Phys. Rev. B (3)

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 37, 3575 (1988).
[CrossRef]

M. J. Stephen and G. Cwilich, Phys. Rev. B 34, 7564 (1986).
[CrossRef]

K. J. Peters, Phys. Rev. B 46, 801 (1992).
[CrossRef]

Phys. Rev. E (2)

D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, Phys. Rev. E 49, 1767 (1994).
[CrossRef]

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

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

Science (1)

A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, Science 287, 287 (2000).
[CrossRef]

Other (4)

U. Frisch, “Wave propagation in random media,” in Probability Methods in Applied Mathematics, A. Bharucha-Reid, ed. (Academic, 1968), Vol. 1, pp. 176–198.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1977), Vol. 2.

J. Goodman, Statistical Optics (Wiley, 1985).

L. A. Apresyan and Y. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects (Gordon and Breach, 1996).

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 (3)

Fig. 1.
Fig. 1.

Spectral interferometer for measuring the randomly scattered electric field. HWP is a half-wave plate, LP is a linear polarizer, PBS is a polarizing beam splitter, and the attenuator is composed of a HWP–LP pair. The reference interferometer gives the scan frequency. The copolarized or cross-polarized light is selected by the LP in front of the photodetector.

Fig. 2.
Fig. 2.

Comparison of the measured electric field frequency correlation magnitude for copolarized and cross-polarized light with the BSE and diffusion models for a slab with μs=4cm1 and a thickness of (a) 6 mm, (b) 9 mm, (c) 12 mm, and (d) 18 mm. Standard error bars for estimation of the mean from the experimental data are given.

Fig. 3.
Fig. 3.

Comparison of the measured electric field correlation magnitude for copolarized and cross-polarized light with the BSE and diffusion models for a slab with μs=14cm1 and a thickness of (a) 3 mm and (b) 6 mm. Error bars are shown.

Equations (9)

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

F{p(t)}=E(ν)E*(ν+Δν)/I,
G(rd,rs;ν)=G0(rd,rs;ν)+G0(rd,r;ν)·M(r,r;ν)G(r,rs;ν)drdr,
G(rd,rs;ν)=G0(rd,rs;ν)+j=1Nf[G0(rd,r;ν)·δ(rRj)δ(rRj)G(r,rs;ν)·p(Rj)drdrdRj].
G(rd,rs;ν)=Ae(ik|rdrs|)4π|rdrs|,
E(rd1;ν)E*(rd2;ν+Δν)=E(rd1;ν)E*(rd2;ν+Δν)+G(rd1,r;ν)G*(rd2,r;ν+Δν)U(r,r¯;r,r¯;ν,ν+Δν)E(r¯;ν)E*(r¯;ν+Δν)drdr¯drdr¯,
U1(r,r¯;r,r¯;ν,ν+Δν)=j=1NS(rRj,r¯Rj;ν)·S*(rRj,r¯Rj;ν+Δν)p(Rj)dRj=μsδ(rR)δ(r¯R)δ(rR)·δ(r¯R)dR,
E(r;ν)E*(r;ν+Δν)=E(r;ν)E*(r;ν+Δν)+μsG(r,R;ν)G*(r,R;ν+Δν)E(R;ν)E*(R;ν+Δν)dR.
E(r,z,ν)=exp(|r|2/σ2)·exp(ikbz)·exp(z/2ls),
ϕ(q,z;ν,ν+Δν)=ψ(q,z;ν,ν+Δν)+μs0d[K(q,z,Z;ν,ν+Δν)·ϕ(q,Z;ν,ν+Δν)]dZ,

Metrics