Abstract

We study spatially coherent forward-scattered light propagating in a turbid medium of moderate optical depth (0–9 mean free paths). Coherent detection was achieved by using a tilted heterodyne geometry, which desensitizes coherent detection of the attenuated incident light. We show that the degree of spatial coherence is significantly higher for light scattered only once in comparison with that for multiply scattered light and that it approaches a small constant value for large numbers of scattering events.

© 1999 Optical Society of America

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References

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  1. P. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
    [CrossRef] [PubMed]
  2. W. L. Sha, C. Liu, R. R. Alfano, “Spectral and temporal measurements of laser action of Rhodamine 640 dye in strongly scattering media,” Opt. Lett. 19, 1922–1924 (1994).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
    [CrossRef] [PubMed]
  5. K. M. Yoo, R. R. Alfano, “Time-resolved coherent and incoherent components of forward light scattering in random media,” Opt. Lett. 15, 320–322 (1990).
    [CrossRef] [PubMed]
  6. L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
    [CrossRef] [PubMed]
  7. J. M. Schmitt, A. H. Gandjbakhche, R. F. Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. 31, 6535–6546 (1992).
    [CrossRef] [PubMed]
  8. O. Emile, F. Bretenaker, A. L. Floch, “Rotating polarization imaging in turbid media,” Opt. Lett. 21, 1706–1708 (1996).
    [CrossRef] [PubMed]
  9. M. R. Hee, J. A. Izatt, J. M. Jacobson, J. G. Fujimoto, E. Swanson, “Femtosecond transillumination optical coherence tomography,” Opt. Lett. 18, 950–952 (1993).
    [CrossRef] [PubMed]
  10. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.
  11. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.
  12. M. Toida, M. Kondo, T. Ichimura, H. Inaba, “Two-dimensional coherent detection imaging in multiple scattering media based on the directional resolution capability of the optical heterodyne method,” Appl. Phys. B 52, 391–394 (1991).
    [CrossRef]
  13. W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).
  14. The cross section derived from the experimental curve from the zero angular tilt is 28% smaller than that calculated from Mie theory. This is likely due to miscalibration of the turbid media concentrations during preparation.
  15. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965).

1996 (2)

1994 (3)

1993 (1)

1992 (1)

1991 (1)

M. Toida, M. Kondo, T. Ichimura, H. Inaba, “Two-dimensional coherent detection imaging in multiple scattering media based on the directional resolution capability of the optical heterodyne method,” Appl. Phys. B 52, 391–394 (1991).
[CrossRef]

1990 (1)

1985 (1)

P. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

1984 (1)

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Alfano, R. R.

Bonner, R. F.

Bretenaker, F.

de Rooij, W. A.

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Emile, O.

Feld, M. S.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Floch, A. L.

Fujimoto, J. G.

Gandjbakhche, A. H.

Hee, M. R.

Ichimura, T.

M. Toida, M. Kondo, T. Ichimura, H. Inaba, “Two-dimensional coherent detection imaging in multiple scattering media based on the directional resolution capability of the optical heterodyne method,” Appl. Phys. B 52, 391–394 (1991).
[CrossRef]

Inaba, H.

M. Toida, M. Kondo, T. Ichimura, H. Inaba, “Two-dimensional coherent detection imaging in multiple scattering media based on the directional resolution capability of the optical heterodyne method,” Appl. Phys. B 52, 391–394 (1991).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

Itzkan, I.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Izatt, J. A.

Jacobson, J. M.

Kondo, M.

M. Toida, M. Kondo, T. Ichimura, H. Inaba, “Two-dimensional coherent detection imaging in multiple scattering media based on the directional resolution capability of the optical heterodyne method,” Appl. Phys. B 52, 391–394 (1991).
[CrossRef]

Liu, C.

Maret, G.

P. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Owen, G. M.

Perelman, L. T.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Reif, F.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965).

Schmitt, J. M.

Sha, W. L.

Swanson, E.

Swanson, E. A.

Thomas, J. E.

Toida, M.

M. Toida, M. Kondo, T. Ichimura, H. Inaba, “Two-dimensional coherent detection imaging in multiple scattering media based on the directional resolution capability of the optical heterodyne method,” Appl. Phys. B 52, 391–394 (1991).
[CrossRef]

van der Stap, C. C. A. H.

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Wax, A.

Wolf, P.

P. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Wu, J.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Yoo, K. M.

Appl. Opt. (1)

Appl. Phys. B (1)

M. Toida, M. Kondo, T. Ichimura, H. Inaba, “Two-dimensional coherent detection imaging in multiple scattering media based on the directional resolution capability of the optical heterodyne method,” Appl. Phys. B 52, 391–394 (1991).
[CrossRef]

Astron. Astrophys. (1)

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Opt. Lett. (6)

Phys. Rev. Lett. (2)

P. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Other (4)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

The cross section derived from the experimental curve from the zero angular tilt is 28% smaller than that calculated from Mie theory. This is likely due to miscalibration of the turbid media concentrations during preparation.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965).

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Figures (5)

Fig. 1
Fig. 1

Tilted heterodyne geometry, schematic diagram.

Fig. 2
Fig. 2

(a) Total transmitted intensity versus number of MFPs, D, for various tilt angles. (b) S, computed from Eq. (4), versus number of MFPs, D, for various tilt angles; error bars denote the extent of fluctuations in the heterodyne signals.

Fig. 3
Fig. 3

Geometry used in the theoretical model. The distance along the z axis is measured in units of the optical depth D.

Fig. 4
Fig. 4

Plot of transmitted intensity versus number of MFP’s with theoretical fits.

Fig. 5
Fig. 5

Plot of S versus number of MFP’s for a tilt angle of 10 mrad with theoretical fit. The error bars denote the extent of fluctuations in the heterodyne signals. See text for details.

Equations (14)

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ET(t)exp[iϕT(ρ)]=j=0Ej exp[iϕj(ρ)]×exp[i(ω+Δω)t],
Ej exp[iϕj(ρ)]=trajEj,traj exp[iϕj,traj(ρ)],
Re{2Er*(t)ET(t)exp[iϕT(ρ)]}
=j=0ErEj{2 cos[Δωt+ϕj(ρ)]},
{2 cos[Δωt+ϕj(ρ)]}=aj cos(Δωt+ϕ¯j),
|Re{2Er*(t)ET(t)exp[iϕT(ρ)]}|amp=Erj=0(ajEj)2+i=0j=0jiaiajEiEj cos(ϕ¯i-ϕ¯j)1/2Erj=0(ajEj)21/2.
T={ET*(t)exp[-iϕT(ρ)]ET(t)exp[iϕT(ρ)]}=j=0Ej2+i=0j=0jiEiEj{2 Re(cos[ϕi(ρ)-ϕj(ρ)])}j=0Ej2.
S=|Re{2Er*(t)ET(t)exp[iϕT(ρ)]}|amp2a02Er2=E02+j=1aja02Ej2.
δ(Ej2(D, ϑ))=Ej-12(D, ϑ)×[PS(f(ϑ, ϑ))δD]exp-D-Dcos[f(ϑ, z)].
Ej2(D, θx, θy)
0D-π/2π/2-π/2π/2Ej-12(D, θx, θy)×kπexp{-k[(θx-θx)2+(θy-θy)2]}×exp[-(D-D)]dθxdθydD0D--Ej-12(D, θx, θy)×kπexp(-k[(θx-θx)2]+(θy-θy)2])×exp[-(D-D)]dθxdθydD.
E02(D=0, θx, θy)=Einput2 exp[-α(θx2+θy2)],
Ej2(D, θx, θy)Einput2kjα+k×exp-kαjα+k(θx2+θy2)×Djj!exp(-D).
Saa02j=0Ej2.

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