Abstract

We describe experiments to measure the spatial and the temporal distribution of photons traversing a turbid medium in the early-arriving regime in which the photons are multiply scattered but are not completely randomized. The photon paths are resolved temporally by a streak camera and spatially by an adjustable absorbing screen with a small aperture. The results are compared with predictions of a theory based on path integrals (PIs) and with the standard diffusion approximation. The PI theory agrees with the data for both long and short times of flight; this agreement is in contrast to the diffusion approximation, which fails for short times. An alternative PI calculation, based on the use of an effective Lagrangian, also agrees with the experiments. PI theory succeeds because it preserves causality. The implications for optical tomography are discussed.

© 1998 Optical Society of America

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  1. A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today, 48(3), 34–40 (1995).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978).
  3. L. Wang, P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-d imaging through scattering wall using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
    [CrossRef] [PubMed]
  4. S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 180–213 (1997).
    [CrossRef] [PubMed]
  5. 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]
  6. R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), Chap. 12.
  7. J. Tessendorf, “Radiative-transfer as a sum over paths,” Phys. Rev. A 35, 872–878 (1987).
    [CrossRef] [PubMed]
  8. R. Bonner, R. Nossal, S. Halvin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
    [CrossRef] [PubMed]
  9. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), p. 12.
  10. J. A. Moon, J. Reintjes, “Image resolution by use of multiply scattering light,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceeding Series (Optical Society of America, Washington, DC1994), pp. 149–152.
  11. A. Ya. Polishchuk, R. R. Alfano, “Fermat photons in turbid media,” Opt. Lett. 20, 1937–1939 (1995).
    [CrossRef] [PubMed]
  12. L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path-integrals,” Phys. Rev. E 51, 6134–6141 (1995).
    [CrossRef]
  13. L. T. Perelman, J. N. Winn, J. Wu, R. R. Dasari, M. S. Feld, “Photon migration of near-diffusive photons in turbid media: a Lagrangian-based approach,” J. Opt. Soc. Am. A 14, 224–229 (1997).
    [CrossRef]
  14. I. S. Gradstein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, London1980).
  15. M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  16. See, for example, J. M. Blackledge, Quantitative Coherent Imaging (Academic, London, 1989).
  17. To incorporate the effects of absorption into the PI expression, each probability P(t, R1, R2) is multiplied by the factor exp(-μat), where μa is the absorption coefficient.
  18. A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply imbedded in tissues,” Med. Phys. 21, 185–191 (1994).
    [CrossRef] [PubMed]
  19. A. H. Gandjbakhche, National Institutes of Health, Bethesda, Md. 20892 (personal communication).

1997 (2)

1995 (3)

A. Ya. Polishchuk, R. R. Alfano, “Fermat photons in turbid media,” Opt. Lett. 20, 1937–1939 (1995).
[CrossRef] [PubMed]

L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path-integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[CrossRef]

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

1994 (2)

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]

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply imbedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

1991 (1)

L. Wang, P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-d imaging through scattering wall using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

1989 (1)

1987 (2)

’t Hooft, G. W.

Alfano, R. R.

A. Ya. Polishchuk, R. R. Alfano, “Fermat photons in turbid media,” Opt. Lett. 20, 1937–1939 (1995).
[CrossRef] [PubMed]

L. Wang, P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-d imaging through scattering wall using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Blackledge, J. M.

See, for example, J. M. Blackledge, Quantitative Coherent Imaging (Academic, London, 1989).

Bonner, R.

Bonner, R. F.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply imbedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

Chance, B.

Colak, S. B.

Dasari, R. R.

L. T. Perelman, J. N. Winn, J. Wu, R. R. Dasari, M. S. Feld, “Photon migration of near-diffusive photons in turbid media: a Lagrangian-based approach,” J. Opt. Soc. Am. A 14, 224–229 (1997).
[CrossRef]

L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path-integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[CrossRef]

Feld, M. S.

L. T. Perelman, J. N. Winn, J. Wu, R. R. Dasari, M. S. Feld, “Photon migration of near-diffusive photons in turbid media: a Lagrangian-based approach,” J. Opt. Soc. Am. A 14, 224–229 (1997).
[CrossRef]

L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path-integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[CrossRef]

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]

Feynman, R. P.

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), Chap. 12.

Gandjbakhche, A. H.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply imbedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

A. H. Gandjbakhche, National Institutes of Health, Bethesda, Md. 20892 (personal communication).

Gradstein, I. S.

I. S. Gradstein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, London1980).

Halvin, S.

Hibbs, A. R.

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), Chap. 12.

Ho, P.

L. Wang, P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-d imaging through scattering wall using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978).

Itzkan, I.

L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path-integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[CrossRef]

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]

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), p. 12.

Katsev, I. L.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), p. 12.

Liu, C.

L. Wang, P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-d imaging through scattering wall using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Melissen, J. B. M.

Moon, J. A.

J. A. Moon, J. Reintjes, “Image resolution by use of multiply scattering light,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceeding Series (Optical Society of America, Washington, DC1994), pp. 149–152.

Nossal, R.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply imbedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

R. Bonner, R. Nossal, S. Halvin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
[CrossRef] [PubMed]

Paasschens, J. C. J.

Papaioannou, D. G.

Patterson, M. S.

Perelman, L. T.

L. T. Perelman, J. N. Winn, J. Wu, R. R. Dasari, M. S. Feld, “Photon migration of near-diffusive photons in turbid media: a Lagrangian-based approach,” J. Opt. Soc. Am. A 14, 224–229 (1997).
[CrossRef]

L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path-integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[CrossRef]

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]

Polishchuk, A. Ya.

Reintjes, J.

J. A. Moon, J. Reintjes, “Image resolution by use of multiply scattering light,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceeding Series (Optical Society of America, Washington, DC1994), pp. 149–152.

Ryzhik, I. M.

I. S. Gradstein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, London1980).

Schomberg, H.

Tessendorf, J.

J. Tessendorf, “Radiative-transfer as a sum over paths,” Phys. Rev. A 35, 872–878 (1987).
[CrossRef] [PubMed]

van Asten, N. A. A. J.

van der Mark, M. B.

Wang, L.

L. Wang, P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-d imaging through scattering wall using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Wang, Y.

L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path-integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[CrossRef]

Weiss, G. H.

Wilson, B. C.

Winn, J. N.

Wu, J.

L. T. Perelman, J. N. Winn, J. Wu, R. R. Dasari, M. S. Feld, “Photon migration of near-diffusive photons in turbid media: a Lagrangian-based approach,” J. Opt. Soc. Am. A 14, 224–229 (1997).
[CrossRef]

L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path-integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[CrossRef]

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]

Yodh, A. G.

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

Zege, E. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), p. 12.

Zhang, G.

L. Wang, P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-d imaging through scattering wall using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Appl. Opt. (2)

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

Med. Phys. (1)

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply imbedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. A (1)

J. Tessendorf, “Radiative-transfer as a sum over paths,” Phys. Rev. A 35, 872–878 (1987).
[CrossRef] [PubMed]

Phys. Rev. E (1)

L. T. Perelman, J. Wu, I. Itzkan, Y. Wang, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path-integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[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).
[CrossRef] [PubMed]

Phys. Today (1)

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

Science (1)

L. Wang, P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-d imaging through scattering wall using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Other (8)

A. H. Gandjbakhche, National Institutes of Health, Bethesda, Md. 20892 (personal communication).

I. S. Gradstein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, London1980).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978).

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), Chap. 12.

See, for example, J. M. Blackledge, Quantitative Coherent Imaging (Academic, London, 1989).

To incorporate the effects of absorption into the PI expression, each probability P(t, R1, R2) is multiplied by the factor exp(-μat), where μa is the absorption coefficient.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), p. 12.

J. A. Moon, J. Reintjes, “Image resolution by use of multiply scattering light,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceeding Series (Optical Society of America, Washington, DC1994), pp. 149–152.

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Figures (7)

Fig. 1
Fig. 1

Schematic view of the tank that contains the scattering medium, from above. To be detected, photons must travel from the source to the aperture in the screen and then to the collection fibers. The source and detector are separated by Δx = 10 cm, Δy = 12.7 cm. If ever y(t) > y screen, the photon is absorbed (a fact not accounted for in the theory, but negligible at small t).

Fig. 2
Fig. 2

Distribution of photon paths that pass through (x = 5 cm, y = 6.35 cm) as a function of time, for the case μ s ′ = 0.60 cm-1. The dots are the data, and the theoretical predictions are labeled.

Fig. 3
Fig. 3

Distribution of photon paths that pass through (x = 5 cm and y = 6.35 cm) as a function of time, for the case μ s ′ = 0.19 cm-1. The dots are the data, and the theoretical predictions are labeled.

Fig. 4
Fig. 4

Distribution of photon paths arriving at time t = 900 ps, as a function of position, for the case μ s ′ = 0.19 cm-1. Each shaded square represents the measured intensity (normalized along each horizontal line) when the aperture was placed at that position. Black squares are the sites of highest intensity. The open circles represent the most probable path as predicted by use of the PI theory, Eq. (4), and the open squares represent the most probable path as predicted by use of the diffusion theory, Eq. (3).

Fig. 5
Fig. 5

Distribution of photon paths arriving at time t = 2500 ps, as a function of position, for the case μ s ′ = 0.60 cm-1. The shading scheme and the symbols are the same as those in Fig. 4.

Fig. 6
Fig. 6

Distribution of photon paths arriving at time t = 900 ps, as a function of position, for the case μ s ′ = 0.19 cm-1. The data and the shading scheme are the same as in Fig. 4. This time the open circles represent the most probable path, as predicted by use of the Lagrange equation of motion, Eq. (5).

Fig. 7
Fig. 7

Computed half-width at half-maximum of the distribution of photon paths for a photon that begins at (x = 0, y = 0) and is detected at (0, 6.35 cm), as a function of Δt = t - t 0, where t 0 is the flight time in the absence of scattering. The medium has μ s ′ = 7.4 cm-1 and μ a = 0.04 cm-1. PI prediction, solid curve; diffusion approximation), dotted curve; random-walk lattice model (and corrected diffusion approximation), dashed curve.

Equations (6)

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P t ,   r 1 ,   r 2 =   D r t × exp - 1 2 μ s 0 t r ¨ t d t × J r t .
P t ,   r 2 - r 1 = 0 2 t 3 Β 3 / 2 ,   3 μ s t / 4 + 1 × 1 - r 2 - r 1 2 t 2 3 μ s t / 4   | r 2 - r 1 | > t | r 2 - r 1 | < t ,
P PI t ,   R 1 ,   R 2 = 0 3 μ s 4 π 3 Y 1 Y 2 r 1 t - r 2 d τ   1 τ 5 / 2 1 - R 1 2 τ 2 3 μ s τ / 4 - 1 × 1 t - τ 5 / 2 1 - R 2 2 t - τ 2 3 μ s t - τ / 4 - 1 | R 2 + R 1 | > t | R 2 + R 1 | < t ,
P DA t ,   R 1 ,   R 2 = 3 μ s 3 / 2 8 π Y 1 Y 2 R 1 + R 2 4 π 2 R 1 3 R 2 3 t 5 / 2 × exp - R 1 + R 2 2 t 2 2 R 1 - R 2 2 + 2 R 1 R 2 3 μ s R 1 + R 2 2 2 t + 1 .
L r t = 3 μ s 2 r ¨ t 2 + r ˙ t 2 .
δ PSF = 0.406 c Δ t μ s 1 / 2 ,

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