Abstract

We describe a method for discriminating short- and long-path photons transmitted through a multiply scattering medium that is based on the relationship between the polarization states of the incident and forward-scattered light. Results of Monte Carlo simulations and experiments show that if the scattering anisotropy of the scatterers is sufficiently small, absorbing barriers embedded in optically dense suspensions of polystyrene spheres can be resolved with good contrast by selectively detecting a component of the scattered-light intensity that has preserved its incident circular polarization state.

The principles of operation of a polarization-modulation system capable of measuring small polarization fractions are explained. Using this system we were able to measure polarized light in a depolarized background over 1000 times as large.

© 1992 Optical Society of America

Full Article  |  PDF Article

Errata

J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, "Use of polarized light to discriminate short-path photons in a multiply scattering medium: erratum," Appl. Opt. 32, 2186-2186 (1993)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-32-12-2186

References

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    [CrossRef]
  2. K. M. Yoo, R. R. Alfano, “Time-resolved depolarization of multiple backscattered light from random media,” Phys. Lett. A 142, 531–536 (1989).
    [CrossRef]
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    [CrossRef]
  4. D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
    [CrossRef]
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    [CrossRef] [PubMed]
  7. B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  20. L. Henyey, J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  21. S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
    [CrossRef] [PubMed]
  22. W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).
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    [CrossRef]
  24. W. F. Cheong, S. A. Prahl, A. J. Welsh, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]
  25. Y. Kuga, A. Ishimaru, A. Bruckner, “Experiments on picosecond pulse propagation in a diffuse medium,” J. Opt. Soc. Am. 73, 1812–1815 (1983).
    [CrossRef]

1991

1990

W. F. Cheong, S. A. Prahl, A. J. Welsh, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

M. J. McCormick, “Particle-size-distribution retrieval from backscattered polarized radiation measurements: a proposed method,” J. Opt. Soc. Am. A 7, 1811–1816 (1990).
[CrossRef]

S. Andersson-Engels, R. Berg, S. Svanberg, “Time-resolved transillumination for medical diagnostics,” Opt. Lett. 15, 1179–1181 (1990).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

1989

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

K. M. Yoo, R. R. Alfano, “Time-resolved depolarization of multiple backscattered light from random media,” Phys. Lett. A 142, 531–536 (1989).
[CrossRef]

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

1988

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

1987

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

1983

1982

1981

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
[CrossRef]

1979

1973

1959

I. Kuscer, M. Ribaric, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

1941

L. Henyey, J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Alfano, R. R.

Andersson-Engels, S.

Arridge, S.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Berg, R.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–129.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 46–53.

Boretsky, R.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Bruckner, A.

Bruckner, A. P.

Bucher, E. A.

Chance, B.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Chandresekhar, S.

S. Chandresekhar, Radiative Transfer (Oxford U. Press, New York, 1950).

Cheong, W. F.

W. F. Cheong, S. A. Prahl, A. J. Welsh, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Cheung, R. L. T.

Cohen, P.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Cope, M.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Delpy, D. T.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Finlander, M.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Flock, S. T.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Greenfield, R.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Greenstein, J.

L. Henyey, J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hebden, J. C.

Henyey, L.

L. Henyey, J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 46–53.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–129.

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.

John, S.

F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

Kaufmann, K.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

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]

Kruger, R. A.

Kuga, Y.

Kuscer, I.

I. Kuscer, M. Ribaric, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

Leigh, J.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Levy, W.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

MacKintosh, F. C.

F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

McCormick, M. J.

Miyake, H.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Nioka, S.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Patterson, M. S.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Pine, D. J.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Prahl, S. A.

W. F. Cheong, S. A. Prahl, A. J. Welsh, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Reynolds, L.

Ribaric, M.

I. Kuscer, M. Ribaric, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

Shimuzu, K.

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).

Siewert, C. E.

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
[CrossRef]

Smith, D.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Svanberg, S.

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 Zee, P.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Weitz, D. A.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Welsh, A. J.

W. F. Cheong, S. A. Prahl, A. J. Welsh, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Wilson, B. C.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Wong, K. S.

Wray, S.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Wyatt, J.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Wyman, D. R.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

Xing, Q. R.

Yoo, K. M.

Yoshioka, H.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Young, M.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Zhu, J. X.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Appl. Opt.

Appl. Phys. B

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]

Astrophys. J.

L. Henyey, J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
[CrossRef]

IEEE J. Quantum Electron.

W. F. Cheong, S. A. Prahl, A. J. Welsh, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

IEEE Trans. Biomed. Eng.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. (Paris)

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

Med. Phys.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Opt. Acta

I. Kuscer, M. Ribaric, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

Opt. Lett.

Phys. Lett. A

K. M. Yoo, R. R. Alfano, “Time-resolved depolarization of multiple backscattered light from random media,” Phys. Lett. A 142, 531–536 (1989).
[CrossRef]

Phys. Med. Biol.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Phys. Rev. B

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. U.S.A. 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Other

S. Chandresekhar, Radiative Transfer (Oxford U. Press, New York, 1950).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 46–53.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–129.

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).

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Figures (12)

Fig. 1
Fig. 1

Geometrical relationship between a circularly polarized plane wave incident on a single scatterer and the scattered wave. The directions of propagation of the incident and scattered waves define the scattering plane, which is aligned parallel to the orientation of one of the orthogonal electric-field vectors that characterize the polarization of the incident wave.

Fig. 2
Fig. 2

Circular polarization fraction Fc of light scattered from a polystyrene sphere in water as a function of the polar scattering angle θ: (a) 0.22-μm-diameter sphere (solid curve) and Rayleigh scatterer (dotted curve), (b) 0.48-μm-diameter sphere, (c) 1.03-μm-diameter sphere, and (d) 2.0-μm-diameter sphere. Curves are symmetrical about the direction of propagation of the incident wave (0°); Fc is displayed over only one hemisphere. Curves pertain to right circularly polarized light (633 nm) incident on a nonabsorbing sphere.

Fig. 3
Fig. 3

Geometry of the scattering volume employed in the Monte Carlo simulations.

Fig. 4
Fig. 4

Radial intensity profiles at the exit aperture of a multiply scattering medium composed of 0.22-μm-diameter polystyrene spheres at different optical thicknesses determined by the Monte Carlo method: (a) total average intensity and (b) circularly polarized component of total intensity. Each curve is shown normalized to the intensity value at the center of the aperture.

Fig. 5
Fig. 5

Radial intensity profiles at the exit aperture of a multiply scattering medium composed of 1.03-μm-diameter polystyrene spheres at different optical thicknesses determined by the Monte Carlo method: (a) total average intensity and (b) circularly polarized component of total intensity. Each curve is shown normalized to the intensity value at the center of the aperture.

Fig. 6
Fig. 6

Beam width at half-maximum determined by the Monte Carlo method at the exit aperture of a multiply scattering medium composed of polystyrene spheres of different sizes. The optical thickness, τ = ∑sd, has not been scaled to reflect the effect of the anisotropy of scattering on the optical density of the simulated medium.

Fig. 7
Fig. 7

Fraction of circular polarization of the total flux of photons collected through the aperture of the simulated scattering medium versus optical thickness τ, for three different sizes of scattering particles.

Fig. 8
Fig. 8

Schematic of experimental apparatus for measuring unpolarized and polarized light transmitted through multiply scattering samples containing absorbers: P1, polarization-splitting cube; R, λ/4 retarder (Polaroid); P2, linear polarizer (Polaroid); F, interference filter (633 nm, 10-nm passband); LPF, dc 1-kHz low-pass filter; BPF, 35- to 50-kHz bandpass filter. In some of the experiments (see text in Subsection 3A) R was placed before the PEM in the beam path.

Fig. 9
Fig. 9

Drawing showing the orientations of the polarizing elements in the experimental setup of Fig. 8.

Fig. 10
Fig. 10

Drawing of the sample container showing the positions of the embedded absorbing strips. The container walls were made of optical-quality glass and the absorbers were constructed from black plastic.

Fig. 11
Fig. 11

Scan profiles measured across the width of the sample container, for a suspension of (a) 0.1-μm-diameter spheres, (b) 0.22-μm-diameter spheres, and (c) 1.03-μm-diameter spheres. To generate the profiles, we recorded the voltages Vpol (solid curves) and Vtot (dashed curves) each time the sample was translated 1 mm; an interpolating curve was drawn through the measured points. In this series of experiments Vpol was a measure of the circularly polarized fraction of the total intensity. Each curve is shown normalized to the voltage measured at the 12-mm scan position (midway between two of the absorbing strips). The fraction of polarization, F = Vpol/Vtot, determined at this point is displayed beside each profile.

Fig. 12
Fig. 12

Scan profiles measured across the width of the sample container containing a suspension of 1.03-μm-diameter spheres. The scan profiles shown here were made in the same way as those in Fig. 11, except in this series of experiments Vpol was a measure of the linearly polarized fraction of the total intensity.

Equations (18)

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I i = [ I i Q i U i V i ] ,
I i = E x E x * + E y E y * , Q i = E x E x * - E y E y * , U i = E x E y * - E y E x * , V i = E x E y * - E y E x * .
I s = [ S 11 ( θ ) S 12 ( θ ) 0 0 S 12 ( θ ) S 11 ( θ ) 0 0 0 0 S 33 ( θ ) S 44 ( θ ) 0 0 - S 44 ( θ ) S 33 ( θ ) ] I i ,
S 11 ( θ ) = ½ ( S 2 2 + S 1 2 ) , S 12 ( θ ) = ½ ( S 2 2 - S 1 2 ) , S 33 ( θ ) = ½ ( S 2 * S 1 + S 2 S 1 * ) , S 34 ( θ ) = ½ ( S 2 * S 1 - S 2 S 1 * ) ,
[ E x s E y s ] = exp [ i k ( r - z ) ] - i k r [ S 2 0 0 S 1 ] [ E x i E y i ] .
I s = [ S 11 ( θ ) S 12 ( θ ) 0 0 S 12 ( θ ) S 11 ( θ ) 0 0 0 0 S 33 ( θ ) S 44 ( θ ) 0 0 - S 44 ( θ ) S 33 ( θ ) ] [ 1 0 0 1 ] I 0 = [ S 11 ( θ ) S 12 ( θ ) S 44 ( θ ) S 33 ( θ ) ] I 0 .
I R = 1 2 [ 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 ] I s = 1 2 [ S 11 ( θ ) + S 33 ( θ ) 0 0 S 11 ( θ ) + S 33 ( θ ) ] I 0 ,
I L = 1 2 [ 1 0 0 - 1 0 0 0 0 0 0 0 0 - 1 0 0 1 ] I s = 1 2 [ S 11 ( θ ) - S 33 ( θ ) 0 0 S 11 ( θ ) - S 33 ( θ ) ] I 0 ,
F c ( θ ) = 1 2 S 11 ( θ ) + S 33 ( θ ) - [ S 11 ( θ ) - S 33 ( θ ) ] S 11 ( θ ) = S 33 ( θ ) S 11 ( θ ) .
cos ( θ ) = 1 + g 2 - ( 1 - g 2 ) 2 2 g ( 1 - g + 2 g R ) 2 ,             0 < θ < π ,
Σ s = σ s v i V f .
S 1 = I 0 [ 1 1 0 0 ] .
S 2 = I 0 [ 1 0 0 0 0 cos δ 0 - sin δ 0 0 1 1 0 sin δ 0 cos δ ] S 1 = I 0 [ 1 cos δ 0 sin δ ] .
δ = δ 0 cos ( ω t ) ,
cos δ = cos ( δ 0 cos ω t ) = J 0 ( δ 0 ) - 2 J 2 ( δ 0 ) cos ( 2 ω t ) + 2 J 4 ( δ 0 ) cos ( 4 ω t ) + ,
sin δ = sin ( δ 0 cos ω t ) = 2 J 1 ( δ 0 ) cos ( ω t ) - 2 J 3 ( δ 0 ) cos ( 3 ω t ) + ,
S 3 = [ 1 - 1 0 0 - 1 1 0 0 0 0 0 0 0 0 0 0 ] [ 1 0 0 0 0 0 0 - 1 0 0 1 0 0 1 0 0 ] S 2 = 1 2 [ 1 + sin δ - ( 1 + sin δ ) 0 0 ] I 0 .
I t = I 0 2 ( 1 + sin δ ) = I 0 2 ( 1 + 2 J 1 ( δ 0 ) cos ( ω t ) - 2 J 3 ( δ 0 ) cos ( 3 ω t ) + )

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