Abstract

We study Wigner phase-space distributions W(x, p) in position (x) and momentum (p) for light undergoing multiple small-angle scattering in a turbid medium. Smoothed Wigner phase-space distributions are measured by using a heterodyne technique that achieves position and momentum resolution determined by the width and the diffraction angle of the local oscillator beam. The sample consists of 5.7-µm-radius polystyrene spheres suspended in a water–glycerol mixture. The momentum distribution of the transmitted light is found to contain a ballistic peak, a narrow diffractive pedestal, and a broad background. The narrow diffractive pedestal is found to decay more slowly than the ballistic peak as the concentration of scatterers is increased. The data are in excellent agreement with a simple theoretical model that explains the behavior of the narrow pedestal by including multiple diffractive scattering and treating large-angle scattering as a loss.

© 1998 Optical Society of America

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    [CrossRef]
  2. M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, “Propagation of Wigner coherence functions in multiple scattering media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 236–238.
  3. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
    [CrossRef] [PubMed]
  4. S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
    [CrossRef] [PubMed]
  5. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  6. M. Hillery, R. F. O’Connel, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
    [CrossRef]
  7. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [CrossRef] [PubMed]
  8. J. A. Izatt, M. D. Kulkerni, K. Kobayashi, M. S. Sivak, J. K. Barton, A. J. Welch, “Optical coherence tomography for biodiagnostics,” Opt. Photon. News 8(5), 41–47, 65 (1997).
    [CrossRef]
  9. 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).
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  11. For a review see M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
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  12. C. Iaconis, I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996).
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    [CrossRef] [PubMed]
  14. A. Wax, J. E. Thomas, “Heterodyne measurement of Wigner phase space distributions in turbid media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 238–242.
  15. M. Beck, M. E. Anderson, M. Raymer, “Imaging through scattering media using pulsed homodyne detection,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 257–260.
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  18. 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);L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Photon paths in turbid media: theory and experimental observation,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 153–155.
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  20. The mean square beat is positive definite and takes the form of a smoothed Wigner distribution. See N. D. Cartwright, “A non-negative Wigner-type distribution,” Physica (Utrecht) 83A, 210–212 (1976).
  21. H. P. Yuen, V. W. S. Chan, “Noise in homodyne and heterodyne detection,” Opt. Lett. 8, 177–179 (1983).
    [CrossRef] [PubMed]
  22. This method has been used in light beating spectroscopy. See H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” in Progress in Optics VIII, E. Wolf, ed. (North-Holland, New York, 1970), Chap. 3, pp. 133–200.
  23. This method has been used by G. L. Abbas, V. W. S. Chan, T. K. Yee, “A dual detector optical heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. 3, 1110–1112 (1985).
    [CrossRef]
  24. See, for example, V. J. Corcoran, “Directional characteristics in optical heterodyne detection processes,” J. Appl. Phys. 36, 1819–1825 (1965); A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Appl. Opt. 5, 1588–1594 (1966); S. Cohen, “Heterodyne detection: phase front alignment, beam spot size, and detector uniformity,” Appl. Opt. 14, 1953–1959 (1975); A. L. Migdall, B. Roop, Y. C. Zheng, J. E. Hardis, Gu Jun Xia, “Use of heterodyne detection to measure optical transmittance over a wide range,” Appl. Opt. 29, 5136–5144 (1990).
    [CrossRef] [PubMed]
  25. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).
  26. The idea of making measurements of Wigner distributions in this regime was suggested to us by R. J. Glauber, Department of Physics, Harvard University, Cambridge, Massachusetts 02138 (personal communication, May1996).
  27. Yu. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969).
  28. A. Zardecki, S. A. W. Gerstl, “Multi-Gaussian phase function model for off-axis laser beam scattering,” Appl. Opt. 26, 3000–3004 (1987).
    [CrossRef] [PubMed]
  29. A. Ishimaru, Y. Kuga, R. Cheung, K. Shimizu, “Scattering and diffusion of a beam wave in randomly distributed scatterers,” J. Opt. Soc. Am. 73, 131–136 (1983).
    [CrossRef]
  30. J. Cooper, P. Zoller, “Radiative transfer equations in broad-band, time-varying fields,” Astrophys. J. 277, 813–819 (1984).
    [CrossRef]
  31. See, for example, A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. I and II.

1997 (1)

J. A. Izatt, M. D. Kulkerni, K. Kobayashi, M. S. Sivak, J. K. Barton, A. J. Welch, “Optical coherence tomography for biodiagnostics,” Opt. Photon. News 8(5), 41–47, 65 (1997).
[CrossRef]

1996 (3)

1995 (5)

1994 (2)

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]

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);L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Photon paths in turbid media: theory and experimental observation,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 153–155.
[CrossRef] [PubMed]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

1987 (1)

1986 (2)

1985 (1)

This method has been used by G. L. Abbas, V. W. S. Chan, T. K. Yee, “A dual detector optical heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. 3, 1110–1112 (1985).
[CrossRef]

1984 (2)

M. Hillery, R. F. O’Connel, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

J. Cooper, P. Zoller, “Radiative transfer equations in broad-band, time-varying fields,” Astrophys. J. 277, 813–819 (1984).
[CrossRef]

1983 (2)

1976 (1)

The mean square beat is positive definite and takes the form of a smoothed Wigner distribution. See N. D. Cartwright, “A non-negative Wigner-type distribution,” Physica (Utrecht) 83A, 210–212 (1976).

1969 (1)

Yu. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969).

1965 (1)

See, for example, V. J. Corcoran, “Directional characteristics in optical heterodyne detection processes,” J. Appl. Phys. 36, 1819–1825 (1965); A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Appl. Opt. 5, 1588–1594 (1966); S. Cohen, “Heterodyne detection: phase front alignment, beam spot size, and detector uniformity,” Appl. Opt. 14, 1953–1959 (1975); A. L. Migdall, B. Roop, Y. C. Zheng, J. E. Hardis, Gu Jun Xia, “Use of heterodyne detection to measure optical transmittance over a wide range,” Appl. Opt. 29, 5136–5144 (1990).
[CrossRef] [PubMed]

1932 (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Abbas, G. L.

This method has been used by G. L. Abbas, V. W. S. Chan, T. K. Yee, “A dual detector optical heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. 3, 1110–1112 (1985).
[CrossRef]

Alfano, R. R.

Anderson, M.

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, “Propagation of Wigner coherence functions in multiple scattering media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 236–238.

Anderson, M. E.

M. Beck, M. E. Anderson, M. Raymer, “Imaging through scattering media using pulsed homodyne detection,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 257–260.

Barabanenkov, Yu. N.

Yu. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969).

Barton, J. K.

J. A. Izatt, M. D. Kulkerni, K. Kobayashi, M. S. Sivak, J. K. Barton, A. J. Welch, “Optical coherence tomography for biodiagnostics,” Opt. Photon. News 8(5), 41–47, 65 (1997).
[CrossRef]

Bastiaans, M. J.

Beck, M.

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
[CrossRef] [PubMed]

M. Beck, M. E. Anderson, M. Raymer, “Imaging through scattering media using pulsed homodyne detection,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 257–260.

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, “Propagation of Wigner coherence functions in multiple scattering media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 236–238.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Bonner, R. F.

Cartwright, N. D.

The mean square beat is positive definite and takes the form of a smoothed Wigner distribution. See N. D. Cartwright, “A non-negative Wigner-type distribution,” Physica (Utrecht) 83A, 210–212 (1976).

Chan, V. W. S.

This method has been used by G. L. Abbas, V. W. S. Chan, T. K. Yee, “A dual detector optical heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. 3, 1110–1112 (1985).
[CrossRef]

H. P. Yuen, V. W. S. Chan, “Noise in homodyne and heterodyne detection,” Opt. Lett. 8, 177–179 (1983).
[CrossRef] [PubMed]

Chance, B.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[CrossRef]

Cheng, C.

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, “Propagation of Wigner coherence functions in multiple scattering media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 236–238.

Cheung, R.

Clarke, L.

Cooper, J.

J. Cooper, P. Zoller, “Radiative transfer equations in broad-band, time-varying fields,” Astrophys. J. 277, 813–819 (1984).
[CrossRef]

Corcoran, V. J.

See, for example, V. J. Corcoran, “Directional characteristics in optical heterodyne detection processes,” J. Appl. Phys. 36, 1819–1825 (1965); A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Appl. Opt. 5, 1588–1594 (1966); S. Cohen, “Heterodyne detection: phase front alignment, beam spot size, and detector uniformity,” Appl. Opt. 14, 1953–1959 (1975); A. L. Migdall, B. Roop, Y. C. Zheng, J. E. Hardis, Gu Jun Xia, “Use of heterodyne detection to measure optical transmittance over a wide range,” Appl. Opt. 29, 5136–5144 (1990).
[CrossRef] [PubMed]

Corey, R.

Cummins, H. Z.

This method has been used in light beating spectroscopy. See H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” in Progress in Optics VIII, E. Wolf, ed. (North-Holland, New York, 1970), Chap. 3, pp. 133–200.

de Wolf, D. A.

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);L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Photon paths in turbid media: theory and experimental observation,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 153–155.
[CrossRef] [PubMed]

Fujimoto, J. G.

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]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Gerstl, S. A. W.

Glauber, R. J.

The idea of making measurements of Wigner distributions in this regime was suggested to us by R. J. Glauber, Department of Physics, Harvard University, Cambridge, Massachusetts 02138 (personal communication, May1996).

Hee, M. R.

Hillery, M.

M. Hillery, R. F. O’Connel, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Iaconis, C.

Ishimaru, A.

A. Ishimaru, Y. Kuga, R. Cheung, K. Shimizu, “Scattering and diffusion of a beam wave in randomly distributed scatterers,” J. Opt. Soc. Am. 73, 131–136 (1983).
[CrossRef]

See, for example, A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. I and II.

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);L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Photon paths in turbid media: theory and experimental observation,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 153–155.
[CrossRef] [PubMed]

Izatt, J. A.

J. A. Izatt, M. D. Kulkerni, K. Kobayashi, M. S. Sivak, J. K. Barton, A. J. Welch, “Optical coherence tomography for biodiagnostics,” Opt. Photon. News 8(5), 41–47, 65 (1997).
[CrossRef]

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]

John, S.

S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef] [PubMed]

Kobayashi, K.

J. A. Izatt, M. D. Kulkerni, K. Kobayashi, M. S. Sivak, J. K. Barton, A. J. Welch, “Optical coherence tomography for biodiagnostics,” Opt. Photon. News 8(5), 41–47, 65 (1997).
[CrossRef]

Kuga, Y.

Kulkerni, M. D.

J. A. Izatt, M. D. Kulkerni, K. Kobayashi, M. S. Sivak, J. K. Barton, A. J. Welch, “Optical coherence tomography for biodiagnostics,” Opt. Photon. News 8(5), 41–47, 65 (1997).
[CrossRef]

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Mayer, A.

McAlister, D. F.

O’Connel, R. F.

M. Hillery, R. F. O’Connel, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Owen, G. M.

Pack, J.-K.

Pang, G.

S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef] [PubMed]

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);L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Photon paths in turbid media: theory and experimental observation,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 153–155.
[CrossRef] [PubMed]

Polishchuk, A. Ya.

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Raymer, M.

M. Beck, M. E. Anderson, M. Raymer, “Imaging through scattering media using pulsed homodyne detection,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 257–260.

Raymer, M. G.

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
[CrossRef] [PubMed]

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, “Propagation of Wigner coherence functions in multiple scattering media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 236–238.

Saulnier, P.

Schmidt, A.

Schmidt, J. M.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Scully, M. O.

M. Hillery, R. F. O’Connel, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Shimizu, K.

Sivak, M. S.

J. A. Izatt, M. D. Kulkerni, K. Kobayashi, M. S. Sivak, J. K. Barton, A. J. Welch, “Optical coherence tomography for biodiagnostics,” Opt. Photon. News 8(5), 41–47, 65 (1997).
[CrossRef]

Swanson, E. A.

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]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swinney, H. L.

This method has been used in light beating spectroscopy. See H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” in Progress in Optics VIII, E. Wolf, ed. (North-Holland, New York, 1970), Chap. 3, pp. 133–200.

Thomas, J. E.

A. Wax, J. E. Thomas, “Optical heterodyne imaging and Wigner phase space distributions,” Opt. Lett. 21, 1427–1429 (1996).
[CrossRef] [PubMed]

A. Wax, J. E. Thomas, “Heterodyne measurement of Wigner phase space distributions in turbid media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 238–242.

Toloudis, D. M.

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, “Propagation of Wigner coherence functions in multiple scattering media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 236–238.

Walmsley, I. A.

Wax, A.

A. Wax, J. E. Thomas, “Optical heterodyne imaging and Wigner phase space distributions,” Opt. Lett. 21, 1427–1429 (1996).
[CrossRef] [PubMed]

A. Wax, J. E. Thomas, “Heterodyne measurement of Wigner phase space distributions in turbid media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 238–242.

Welch, A. J.

J. A. Izatt, M. D. Kulkerni, K. Kobayashi, M. S. Sivak, J. K. Barton, A. J. Welch, “Optical coherence tomography for biodiagnostics,” Opt. Photon. News 8(5), 41–47, 65 (1997).
[CrossRef]

Wigner, E. P.

M. Hillery, R. F. O’Connel, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

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);L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Photon paths in turbid media: theory and experimental observation,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 153–155.
[CrossRef] [PubMed]

Yadlowsky, M. J.

Yang, Y.

S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef] [PubMed]

Yee, T. K.

This method has been used by G. L. Abbas, V. W. S. Chan, T. K. Yee, “A dual detector optical heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. 3, 1110–1112 (1985).
[CrossRef]

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[CrossRef]

Yuen, H. P.

Zardecki, A.

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J. Cooper, P. Zoller, “Radiative transfer equations in broad-band, time-varying fields,” Astrophys. J. 277, 813–819 (1984).
[CrossRef]

Appl. Opt. (2)

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J. Cooper, P. Zoller, “Radiative transfer equations in broad-band, time-varying fields,” Astrophys. J. 277, 813–819 (1984).
[CrossRef]

J. Appl. Phys. (1)

See, for example, V. J. Corcoran, “Directional characteristics in optical heterodyne detection processes,” J. Appl. Phys. 36, 1819–1825 (1965); A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Appl. Opt. 5, 1588–1594 (1966); S. Cohen, “Heterodyne detection: phase front alignment, beam spot size, and detector uniformity,” Appl. Opt. 14, 1953–1959 (1975); A. L. Migdall, B. Roop, Y. C. Zheng, J. E. Hardis, Gu Jun Xia, “Use of heterodyne detection to measure optical transmittance over a wide range,” Appl. Opt. 29, 5136–5144 (1990).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef] [PubMed]

J. Lightwave Technol. (1)

This method has been used by G. L. Abbas, V. W. S. Chan, T. K. Yee, “A dual detector optical heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. 3, 1110–1112 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (7)

Opt. Photon. News (1)

J. A. Izatt, M. D. Kulkerni, K. Kobayashi, M. S. Sivak, J. K. Barton, A. J. Welch, “Optical coherence tomography for biodiagnostics,” Opt. Photon. News 8(5), 41–47, 65 (1997).
[CrossRef]

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[CrossRef]

Phys. Rev. (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. Lett. (1)

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);L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Photon paths in turbid media: theory and experimental observation,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 153–155.
[CrossRef] [PubMed]

Phys. Today (1)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[CrossRef]

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Science (1)

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[CrossRef] [PubMed]

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Yu. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969).

Other (7)

See, for example, A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. I and II.

This method has been used in light beating spectroscopy. See H. Z. Cummins, H. L. Swinney, “Light beating spectroscopy,” in Progress in Optics VIII, E. Wolf, ed. (North-Holland, New York, 1970), Chap. 3, pp. 133–200.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

The idea of making measurements of Wigner distributions in this regime was suggested to us by R. J. Glauber, Department of Physics, Harvard University, Cambridge, Massachusetts 02138 (personal communication, May1996).

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, “Propagation of Wigner coherence functions in multiple scattering media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 236–238.

A. Wax, J. E. Thomas, “Heterodyne measurement of Wigner phase space distributions in turbid media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 238–242.

M. Beck, M. E. Anderson, M. Raymer, “Imaging through scattering media using pulsed homodyne detection,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 257–260.

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Figures (9)

Fig. 1
Fig. 1

Heterodyne measurement of smoothed Wigner phase-space distributions.

Fig. 2
Fig. 2

Mie differential cross section for scattering from 11.4-µm polystyrene spheres, with parameters nrel=1.17, no=1.36, and λair=633 nm.

Fig. 3
Fig. 3

Smoothed Wigner distribution (log scale) for light transmitted though a turbid medium for ρ=2×106/cm3. x denotes the transverse position in millimeters, and p denotes the transverse wave vector (momentum) in units of the wave vector in air, ko. The central island is the ballistic contribution.

Fig. 4
Fig. 4

Smoothed Wigner distribution (log scale) for ρ=6×106/cm3. The central island is the ballistic contribution. Note the appearance of a narrow pedestal centered on the ballistic feature.

Fig. 5
Fig. 5

Smoothed Wigner distribution W(x=0, p) (linear scale, ballistic contribution not shown) for ρ=0.4×106/cm3. Note that here the input beam is large compared with the diameter of the LO beam. The theoretical prediction is shown as a solid curve.

Fig. 6
Fig. 6

Smoothed Wigner distribution W(x=0, p) for large input beam (linear scale, ballistic contribution not shown) for ρ=2×106/cm3. Note the narrow central pedestal arising from diffractive scattering. The solid curve shows the theoretical prediction.

Fig. 7
Fig. 7

Smoothed Wigner distribution W(x=0, p) (linear scale, ballistic contribution not shown) for ρ=6×106/cm3. Note the increase in the narrow central pedestal arising from diffractive scattering. The solid curve shows the theoretical prediction.

Fig. 8
Fig. 8

Amplitude of the narrow pedestal arising from multiple diffractive scattering as a function of scatterer concentration ρ. The solid curve shows the prediction with no free parameters. Note that the amplitude of the pedestal decays with an extinction coefficient that is 0.65 of the ballistic extinction coefficient of Fig. 9.

Fig. 9
Fig. 9

Amplitude of the ballistic component as a function of scatterer concentration ρ. The solid line shows the prediction for exponential decay with the extinction coefficient determined from the total scattering cross section.

Equations (70)

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W(x, p)=d2πexp(ip)E*(x+/2)E(x-/2),
|VB|2dx ELO*(x, zD)ES(x, zD)2
dx ELO*(x-dx, z=0)
×ES(x, z=0)expikodpfx2.
|VB(dx, dp)|2dxdp WLO(x-dx, p+kodp/f)WS(x, p),
S(xM, pM)=d2xd2p WLO(x-xM, p-pM)WS(x, p).
WS(x, p)=dpz WS(z=L, x, p).
W(x, p, t)=d3(2π)3exp(i·p)E*(x+/2, t)×E(x-/2, t),
c2pno2ω·xW(x, p)=-cnoμT(p)W(x, p)+d3p K(p, p)W(x, p).
K(p, p)=KN(p, p)+KB(p, p).
W=WBALL+WP+WB.
WBALL(z, x;p)
exp(-μTz)WBALL0(z=0, x-zp/k;p).
WBALL0(z=0, x;p)
δ(pz-k)1π2exp-x2+y2a2-a2(px2+py2).
WP(L, x;p)=δ(pz-k)WP(L, x, p).
WP(L, x, p)=exp(-x2/a2)π2θo2k2a2exp(-μTL)×0dη 2ηJ0η2pθok
×{exp[μNL exp(-η2)]-1}.
WLO(x-xM, p-pM)
=1π2exp-(x-xM)2+y2ao2
×exp{-ao2[(px-pM)2+py2]}.
WLO(x-xM, p-pM)δ(x-xM)δ(y)×δ(px-pM)δ(py).
SP(xM, pM)
=WP(L, x=xM, y=0; px=pM, py=0),
SBALL(xM, pM)=ao2π2a2exp-ao2pM2-xM2a2×exp(-μTL).
SMBALL=ηhet2(ao2/π2a2),
S(xM=0, pM)SMBALL=A(ρ)FP(p, ρ)+B exp[-p2/(ΔpB)2].
A(ρ)=exp(-μTL)ηhet2θo2k2ao20dη 2η×{exp[μNL exp(-η2)]-1}.
FP(p)=WP(x=0, p)WP(x=0, p=0),
WP(L, x, p)=exp(-x2/a2)πa2exp[-(μT-μN)L]×n=1 exp(-μNL)
×(μNL)nn!exp[-p2/(nθo2k2)]πnθo2k2.
exp[μNL exp(-η2)]-1exp[μNL(1-η2)].
WP(L, x, p)=exp(-x2/a2)πa2exp[-(μT-μN)L]×exp[-p2/(μNLθo2k2)]πμNLθo2k2.
c2pno2ω·xW(x, p)=-cnoμT(p)W(x, p)+d3p K(p, p)W(x, p).
K(p, p)=ρc2no2ωδp2-p22|f(p, p)|2,
d3p K(p, p)=cnoμs(p)
K(p, p)=KN(p, p)+KB(p, p).
W(x, p)=WBALL(x, p)+WP(x, p)+WB(x, p).
cpnok·xWBALL(x, p)=-cnoμTWBALL(x, p).
cpnok·xWP(x, p)=-cnoμTWP(x, p)+d3p KN(p, p)WP(x, p)+d3p KN(p, p)WBALL(x, p).
cpnok·xWB(x, p)=-cnoμTWB(x, p)+d3p[KN(p, p)+KB(p, p)]WB(x, p)+d3p KB(p, p)[WBALL(x, p)+WP(x, p)].
|f(p, p)|2=dσdΩ=σNπθo2exp-(Δp)2θo2p2.
dΩdσdΩ02πθ dθσNπθo2exp-θ2θo2=σN.
KN(p, p)KN(p-p)=cnoμNδ(pz-pz)1πθo2k2
×exp-(Δp)2θo2p2,
d3Δp KN(Δp)=ΓN=cnoμN,
cpnok·x+ΓTGp(x, x;p, p)-d3p K(p-p)
×Gp(x, x;p, p)=δ(x-x)δ(p-p).
ΓT=(c/no)μT.
Gp(x, x;p, p)
=d3q(2π)3d3r(2π)3×exp[iq·(x-x)+ir·(p-p)]×no/c0dl exp(-ilq·p/k)×exp-0ldl μ˜(r+ql/k),
cnoμ˜(r)cnoμT-K˜(r),
K˜(r)=d3Δp exp(-iΔp)K(Δp).
S(x, p)=d3p KN(p, p)WBALL(x, p).
WP(x, p)=d3xd3pGp(x, x;p, p)S(x, p).
μ˜(r)=μT-μN exp-θo2k24(rx2+ry2)μT-K˜(r).
S(x, p)=cnoμNexp(-μTz)πa2exp-x2+y2a2 δ(pz-k)πθo2k2×exp-px2+py2θo2k2.
WP(L, x;p)=δ(pz-k)exp(-μTL)×d2q(2π)2exp(iq·x)×d2r(2π)2exp(ir·p-a2q2/4)×0Ldz K˜[r+(L-z)q/k]×exp0L-zdl K˜(r+ql/k).
exp0Ldl K˜(r+ql/k)-1.
WP(L, x;p)=δ(pz-k)exp(-μTL)
×d2q(2π)2exp(iq·x-a2q2/4)×d2r(2π)2exp(ir·p)×exp0Ldl K˜(r+ql/k)-1.
WP(L, x; p)=δ(pz-k)WP(L, x, p).
d2xd2p W(L, x, p)
=exp(-μTL)[exp(μNL)-1]=exp(-μBL)-exp(-μTL).
0Ldz μN exp(-μTL)exp[-μB(L-z)]
=exp(-μBL)-exp(-μTL),
WP(L, x, p)=exp(-x2/a2)πa2exp(-μTL)×d2r(2π)2exp(ir·p)×expμNL exp-θo2k2r24-1.
02πdϕ2πexpi2pθokη cos ϕ=J0η2pθok.
WP(L, x, p)=exp(-x2/a2)π2θo2k2a2exp(-μTL)×0dη 2ηJ0η2pθok×{exp[μNL exp(-η2)]-1}.
d2p J0η2pθok=π2θo2k2 δ(η)η.

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