Abstract

Photon propagation in a turbid medium characterized by highly forward-directed light scattering can be described by a generalized Fermat principle, a counterpart of the conventional Fermat principle for transparent media. An exact analytic solution for the most favorable photon paths (Fermat paths) has been found and is investigated. An analysis intended to encourage and facilitate experimental observation of Fermat photons is presented. The feasibility of using the theory in time-resolved optical tomography for the recognition of objects hidden in turbid media is discussed.

© 1995 Optical Society of America

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References

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  1. V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, Wave Propagation in Random Media (Scintillation) (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993); V. I. Tatarskii, V. U. Zavorotnyi, Prog. Opt. 18, 205 (1980); R. Dashen, J. Math Phys, 20, 894 (1979); V. I. Tatarskii, The Effect of Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  2. J. Tassendorf, Phys. Rev. A 35, 872 (1987).
    [Crossref]
  3. I. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  4. K. M. Yoo, R. R. Alfano, Opt. Lett. 15, 320 (1990).
    [Crossref] [PubMed]
  5. L. M. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
    [Crossref] [PubMed]
  6. S. Feng, F. Zeng, B. Chance, Proc. Soc. Photo-Opt. Instrum. Eng. 1888, 78 (1993).
  7. L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, Phys. Rev. Lett. 72, 1341 (1994).
    [Crossref] [PubMed]
  8. J. A. Moon, J. Reintjes, Opt. Lett. 19, 521 (1994).
    [Crossref] [PubMed]
  9. J. S. Shotland, J. C. Haselgrove, J. S. Leigh, Appl. Opt. 32, 448 (1993).
    [Crossref]
  10. M. S. Patterson, S. Andersson-Engels, B. S. Wilson, E. K. Osei, Appl. Opt. 34, 22 (1995).
    [Crossref] [PubMed]
  11. V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, Phys. Med. Biol. 35, 1317 (1990).
    [Crossref] [PubMed]
  12. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
    [Crossref]
  13. R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), Chap. 12.
  14. A. Ya Polishchuk, R. R. Alfano, “Fermat photons in random media: diffusion on the velocity sphere and principle of the least mean-square curvature,” submitted toPhys. Rev. Lett.
  15. K. M. Yoo, F. Liu, R. R. Alfano, Phys. Rev. Lett. 64, 2647 (1990); errata K. M. Yoo, F. Liu, R. R. Alfano, Phys. Rev. Lett. 65, 2210 (1990).
    [Crossref] [PubMed]
  16. M. R. Hee, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, Opt. Lett. 18, 1107 (1993).
    [Crossref] [PubMed]

1995 (1)

1994 (2)

J. A. Moon, J. Reintjes, Opt. Lett. 19, 521 (1994).
[Crossref] [PubMed]

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, Phys. Rev. Lett. 72, 1341 (1994).
[Crossref] [PubMed]

1993 (3)

1992 (1)

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[Crossref]

1991 (1)

L. M. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
[Crossref] [PubMed]

1990 (3)

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, Phys. Med. Biol. 35, 1317 (1990).
[Crossref] [PubMed]

K. M. Yoo, F. Liu, R. R. Alfano, Phys. Rev. Lett. 64, 2647 (1990); errata K. M. Yoo, F. Liu, R. R. Alfano, Phys. Rev. Lett. 65, 2210 (1990).
[Crossref] [PubMed]

K. M. Yoo, R. R. Alfano, Opt. Lett. 15, 320 (1990).
[Crossref] [PubMed]

1987 (1)

J. Tassendorf, Phys. Rev. A 35, 872 (1987).
[Crossref]

Alfano, R. R.

L. M. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
[Crossref] [PubMed]

K. M. Yoo, R. R. Alfano, Opt. Lett. 15, 320 (1990).
[Crossref] [PubMed]

K. M. Yoo, F. Liu, R. R. Alfano, Phys. Rev. Lett. 64, 2647 (1990); errata K. M. Yoo, F. Liu, R. R. Alfano, Phys. Rev. Lett. 65, 2210 (1990).
[Crossref] [PubMed]

A. Ya Polishchuk, R. R. Alfano, “Fermat photons in random media: diffusion on the velocity sphere and principle of the least mean-square curvature,” submitted toPhys. Rev. Lett.

Andersson-Engels, S.

Boas, D. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[Crossref]

Chance, B.

S. Feng, F. Zeng, B. Chance, Proc. Soc. Photo-Opt. Instrum. Eng. 1888, 78 (1993).

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[Crossref]

Feld, M. S.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, Phys. Rev. Lett. 72, 1341 (1994).
[Crossref] [PubMed]

Feng, S.

S. Feng, F. Zeng, B. Chance, Proc. Soc. Photo-Opt. Instrum. Eng. 1888, 78 (1993).

Feynman, R. P.

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

Frank, G. L.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, Phys. Med. Biol. 35, 1317 (1990).
[Crossref] [PubMed]

Fujimoto, J. G.

Haselgrove, J. C.

Hee, M. R.

Hibbs, A. R.

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

Ho, P. P.

L. M. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
[Crossref] [PubMed]

Ishimaru, A.

V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, Wave Propagation in Random Media (Scintillation) (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993); V. I. Tatarskii, V. U. Zavorotnyi, Prog. Opt. 18, 205 (1980); R. Dashen, J. Math Phys, 20, 894 (1979); V. I. Tatarskii, The Effect of Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Ishimaru, I.

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

Itzkan, I.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, Phys. Rev. Lett. 72, 1341 (1994).
[Crossref] [PubMed]

Izatt, J. A.

Leigh, J. S.

Liu, C.

L. M. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
[Crossref] [PubMed]

Liu, F.

K. M. Yoo, F. Liu, R. R. Alfano, Phys. Rev. Lett. 64, 2647 (1990); errata K. M. Yoo, F. Liu, R. R. Alfano, Phys. Rev. Lett. 65, 2210 (1990).
[Crossref] [PubMed]

Moon, J. A.

O’Leary, M. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[Crossref]

Osei, E. K.

Patterson, M. S.

M. S. Patterson, S. Andersson-Engels, B. S. Wilson, E. K. Osei, Appl. Opt. 34, 22 (1995).
[Crossref] [PubMed]

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, Phys. Med. Biol. 35, 1317 (1990).
[Crossref] [PubMed]

Perelman, L. T.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, Phys. Rev. Lett. 72, 1341 (1994).
[Crossref] [PubMed]

Peters, V. G.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, Phys. Med. Biol. 35, 1317 (1990).
[Crossref] [PubMed]

Reintjes, J.

Shotland, J. S.

Swanson, E. A.

Tassendorf, J.

J. Tassendorf, Phys. Rev. A 35, 872 (1987).
[Crossref]

Tatarskii, V. I.

V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, Wave Propagation in Random Media (Scintillation) (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993); V. I. Tatarskii, V. U. Zavorotnyi, Prog. Opt. 18, 205 (1980); R. Dashen, J. Math Phys, 20, 894 (1979); V. I. Tatarskii, The Effect of Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Wang, L. M.

L. M. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
[Crossref] [PubMed]

Wilson, B. S.

Wu, J.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, Phys. Rev. Lett. 72, 1341 (1994).
[Crossref] [PubMed]

Wyman, D. R.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, Phys. Med. Biol. 35, 1317 (1990).
[Crossref] [PubMed]

Ya Polishchuk, A.

A. Ya Polishchuk, R. R. Alfano, “Fermat photons in random media: diffusion on the velocity sphere and principle of the least mean-square curvature,” submitted toPhys. Rev. Lett.

Yodh, A. G.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[Crossref]

Yoo, K. M.

K. M. Yoo, R. R. Alfano, Opt. Lett. 15, 320 (1990).
[Crossref] [PubMed]

K. M. Yoo, F. Liu, R. R. Alfano, Phys. Rev. Lett. 64, 2647 (1990); errata K. M. Yoo, F. Liu, R. R. Alfano, Phys. Rev. Lett. 65, 2210 (1990).
[Crossref] [PubMed]

Zavorotny, V. U.

V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, Wave Propagation in Random Media (Scintillation) (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993); V. I. Tatarskii, V. U. Zavorotnyi, Prog. Opt. 18, 205 (1980); R. Dashen, J. Math Phys, 20, 894 (1979); V. I. Tatarskii, The Effect of Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Zeng, F.

S. Feng, F. Zeng, B. Chance, Proc. Soc. Photo-Opt. Instrum. Eng. 1888, 78 (1993).

Zhang, G.

L. M. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
[Crossref] [PubMed]

Appl. Opt. (2)

Opt. Lett. (3)

Phys. Med. Biol. (1)

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, Phys. Med. Biol. 35, 1317 (1990).
[Crossref] [PubMed]

Phys. Rev. A (1)

J. Tassendorf, Phys. Rev. A 35, 872 (1987).
[Crossref]

Phys. Rev. Lett. (3)

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, Phys. Rev. Lett. 72, 1341 (1994).
[Crossref] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[Crossref]

K. M. Yoo, F. Liu, R. R. Alfano, Phys. Rev. Lett. 64, 2647 (1990); errata K. M. Yoo, F. Liu, R. R. Alfano, Phys. Rev. Lett. 65, 2210 (1990).
[Crossref] [PubMed]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

S. Feng, F. Zeng, B. Chance, Proc. Soc. Photo-Opt. Instrum. Eng. 1888, 78 (1993).

Science (1)

L. M. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
[Crossref] [PubMed]

Other (4)

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

V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, Wave Propagation in Random Media (Scintillation) (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993); V. I. Tatarskii, V. U. Zavorotnyi, Prog. Opt. 18, 205 (1980); R. Dashen, J. Math Phys, 20, 894 (1979); V. I. Tatarskii, The Effect of Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

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

A. Ya Polishchuk, R. R. Alfano, “Fermat photons in random media: diffusion on the velocity sphere and principle of the least mean-square curvature,” submitted toPhys. Rev. Lett.

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Figures (1)

Fig. 1
Fig. 1

Calculated Fermat paths for fixed source and detector positions and time of detection T = 500 ps (index of refraction is taken to be 1.33). The angles between the directions of emission and detection are 180 deg (curve 1) and 130 deg (curve 2) (α = 50 deg). The inset is a schematic diagram of the experimental setup for observing Fermat photons’ paths. A particular source–detector configuration is depicted; the corresponding most favorable path ∑ and other less probable paths σ possessing the same path lengths are also shown.

Equations (4)

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P [ σ ] = exp ( - c 4 D s σ d s d l · d s ) .
( δ P [ σ ] δ σ ) σ = Σ = 0 ,
d 2 d l 2 S ( l ) - λ ( l ) S ( l ) = C ,
x = τ γ 0 γ cos ( γ - φ ) d γ ( ± 1 + δ sin γ ) 1 / 2 , y = τ γ 0 γ sin ( γ - φ ) d γ ( ± 1 + δ sin γ ) 1 / 2 , l = τ γ 0 γ d γ ( ± 1 + δ sin γ ) 1 / 2 ,

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