Abstract

We examine interference effects resulting from the superposition of photon-density waves produced by coherently modulated light incident upon a turbid medium. Photon-diffusion theory is used to derive expressions for the ac magnitude and phase of the aggregate diffusive wave produced in full- and half-space volumes by two sources. Using a frequency-domain spectrometer operating at 410 MHz, we verify interference patterns predicted by the model in scattering samples having optical properties similar to those of skin tissue. Potential imaging applications of interfering diffusive waves are discussed in the context of the theoretical and experimental results.

© 1992 Optical Society of America

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  1. R. L. Longini, R. Zdrojkowski, “A note on the theory of backscattering of light by living tissue,”IEEE Trans. Biomed. Eng. BME-15, 4–10 (1968).
    [CrossRef]
  2. A. Ishimaru, “Diffusion of light in turbid material,” Appl. Opt. 28, 2210–2222 (1989).
    [CrossRef] [PubMed]
  3. L. F. Gate, “The determination of light absorption in diffusing materials by a photon diffusion model,”J. Phys. D 4, 1049–1056 (1971).
    [CrossRef]
  4. B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscles,” Anal. Biochem. 174, 698–707 (1988).
    [CrossRef] [PubMed]
  5. 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]
  6. J. R. Lakowicz, K. W. Berndt, “Frequency-domain measurements of photon migration in tissues,” Chem. Phys. Lett. 166, 246–252 (1990).
    [CrossRef]
  7. B. Chance, ed., Photon Migration in Tissues (Plenum, New York, 1989).
    [CrossRef]
  8. J. Fishkin, E. Gratton, M. J. vandeVen, W. W. Mantulin, “Diffusion of intensity-modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 122–135 (1991).
    [CrossRef]
  9. A. Knüttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-density waves,” Appl. Opt. (to be published).
  10. A. Knüttel, J. R. Knutson, “Methods and apparatus for imaging a physical parameter in turbid media using diffusive waves,” U.S. patent application 07/722,823 (May7, 1991).
  11. E. Gratton, W. W. Mantulin, M. J. vandeVen, J. B. Fishkin, M. B. Maris, B. Chance, “The possibility of a near-infrared optical imaging system using frequency domain methods,” in Proceedings of the Third International Conference on Peace through Mind/Brain Science (Hamamatsu Corp., Hamamatsu, Japan, 1990), pp. 183–188.
  12. 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]
  13. H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, New York, 1980), Vol. 1, Chap. 14, pp. 477–492.
  14. R. J. Hirko, R. J. Fretterd, R. L. Longini, “Diffusion dipole source,”J. Opt. Soc. Am. 63, 336–337 (1973).
    [CrossRef]
  15. J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1970), p. 245.
  16. M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering methods to photodynamic dosimetry,” in Photodynamic Therapy: Mechanisms II, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1203, 62–75 (1990).
    [CrossRef]
  17. J. M. Schmitt, G. X. Zhou, E. C. Walker, R. T. Wall, “Multilayer model of photon diffusion in skin,” J. Opt. Soc. Am. A 7, 2141–2153 (1990).
    [CrossRef] [PubMed]
  18. D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1989).
  19. G. Yoon, S. A. Prahl, A. J. Welch, “Accuracies of the diffusion approximation and its similarity relations for laser irradiated biological media,” Appl. Opt. 28, 2250–2255 (1989).
    [CrossRef] [PubMed]
  20. K. M. Yoo, L. Feng, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
    [CrossRef] [PubMed]
  21. 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. (to be published).
  22. K. Furutsu, “Diffusion equation derived from space-time transport equation,”J. Opt. Soc. Am. 70, 360–366 (1980).
    [CrossRef]
  23. J. R. Lakowicz, K. W. Berndt, “Lifetime-selective fluorescence imaging using an rf phase-sensitive camera,” Rev. Sci. Instrum. 62, 1727–1734 (1991).
    [CrossRef]

1991

J. R. Lakowicz, K. W. Berndt, “Lifetime-selective fluorescence imaging using an rf phase-sensitive camera,” Rev. Sci. Instrum. 62, 1727–1734 (1991).
[CrossRef]

1990

J. M. Schmitt, G. X. Zhou, E. C. Walker, R. T. Wall, “Multilayer model of photon diffusion in skin,” J. Opt. Soc. Am. A 7, 2141–2153 (1990).
[CrossRef] [PubMed]

J. R. Lakowicz, K. W. Berndt, “Frequency-domain measurements of photon migration in tissues,” Chem. Phys. Lett. 166, 246–252 (1990).
[CrossRef]

K. M. Yoo, L. Feng, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

1989

1988

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscles,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

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]

1980

1973

1971

L. F. Gate, “The determination of light absorption in diffusing materials by a photon diffusion model,”J. Phys. D 4, 1049–1056 (1971).
[CrossRef]

1968

R. L. Longini, R. Zdrojkowski, “A note on the theory of backscattering of light by living tissue,”IEEE Trans. Biomed. Eng. BME-15, 4–10 (1968).
[CrossRef]

Alfano, R. R.

K. M. Yoo, L. Feng, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

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]

Berndt, K. W.

J. R. Lakowicz, K. W. Berndt, “Lifetime-selective fluorescence imaging using an rf phase-sensitive camera,” Rev. Sci. Instrum. 62, 1727–1734 (1991).
[CrossRef]

J. R. Lakowicz, K. W. Berndt, “Frequency-domain measurements of photon migration in tissues,” Chem. Phys. Lett. 166, 246–252 (1990).
[CrossRef]

Bonner, R. F.

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. (to be published).

Chaikin, P. M.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1989).

Chance, B.

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]

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscles,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

E. Gratton, W. W. Mantulin, M. J. vandeVen, J. B. Fishkin, M. B. Maris, B. Chance, “The possibility of a near-infrared optical imaging system using frequency domain methods,” in Proceedings of the Third International Conference on Peace through Mind/Brain Science (Hamamatsu Corp., Hamamatsu, Japan, 1990), pp. 183–188.

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering methods to photodynamic dosimetry,” in Photodynamic Therapy: Mechanisms II, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1203, 62–75 (1990).
[CrossRef]

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]

Feng, L.

K. M. Yoo, L. Feng, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Fishkin, J.

J. Fishkin, E. Gratton, M. J. vandeVen, W. W. Mantulin, “Diffusion of intensity-modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 122–135 (1991).
[CrossRef]

Fishkin, J. B.

E. Gratton, W. W. Mantulin, M. J. vandeVen, J. B. Fishkin, M. B. Maris, B. Chance, “The possibility of a near-infrared optical imaging system using frequency domain methods,” in Proceedings of the Third International Conference on Peace through Mind/Brain Science (Hamamatsu Corp., Hamamatsu, Japan, 1990), pp. 183–188.

Fountain, M.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscles,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Fretterd, R. J.

Furutsu, K.

Gandjbakhche, A. H.

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. (to be published).

Gate, L. F.

L. F. Gate, “The determination of light absorption in diffusing materials by a photon diffusion model,”J. Phys. D 4, 1049–1056 (1971).
[CrossRef]

Gratton, E.

J. Fishkin, E. Gratton, M. J. vandeVen, W. W. Mantulin, “Diffusion of intensity-modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 122–135 (1991).
[CrossRef]

E. Gratton, W. W. Mantulin, M. J. vandeVen, J. B. Fishkin, M. B. Maris, B. Chance, “The possibility of a near-infrared optical imaging system using frequency domain methods,” in Proceedings of the Third International Conference on Peace through Mind/Brain Science (Hamamatsu Corp., Hamamatsu, Japan, 1990), pp. 183–188.

Greenfeld, R.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscles,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1989).

Hirko, R. J.

Holtom, G.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscles,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Ishimaru, A.

Kent, J.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscles,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Knutson, J. R.

A. Knüttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-density waves,” Appl. Opt. (to be published).

A. Knüttel, J. R. Knutson, “Methods and apparatus for imaging a physical parameter in turbid media using diffusive waves,” U.S. patent application 07/722,823 (May7, 1991).

Knüttel, A.

A. Knüttel, J. R. Knutson, “Methods and apparatus for imaging a physical parameter in turbid media using diffusive waves,” U.S. patent application 07/722,823 (May7, 1991).

A. Knüttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-density waves,” Appl. Opt. (to be published).

Lakowicz, J. R.

J. R. Lakowicz, K. W. Berndt, “Lifetime-selective fluorescence imaging using an rf phase-sensitive camera,” Rev. Sci. Instrum. 62, 1727–1734 (1991).
[CrossRef]

J. R. Lakowicz, K. W. Berndt, “Frequency-domain measurements of photon migration in tissues,” Chem. Phys. Lett. 166, 246–252 (1990).
[CrossRef]

Longini, R. L.

R. J. Hirko, R. J. Fretterd, R. L. Longini, “Diffusion dipole source,”J. Opt. Soc. Am. 63, 336–337 (1973).
[CrossRef]

R. L. Longini, R. Zdrojkowski, “A note on the theory of backscattering of light by living tissue,”IEEE Trans. Biomed. Eng. BME-15, 4–10 (1968).
[CrossRef]

Mantulin, W. W.

J. Fishkin, E. Gratton, M. J. vandeVen, W. W. Mantulin, “Diffusion of intensity-modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 122–135 (1991).
[CrossRef]

E. Gratton, W. W. Mantulin, M. J. vandeVen, J. B. Fishkin, M. B. Maris, B. Chance, “The possibility of a near-infrared optical imaging system using frequency domain methods,” in Proceedings of the Third International Conference on Peace through Mind/Brain Science (Hamamatsu Corp., Hamamatsu, Japan, 1990), pp. 183–188.

Maret, G.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1989).

Maris, M. B.

E. Gratton, W. W. Mantulin, M. J. vandeVen, J. B. Fishkin, M. B. Maris, B. Chance, “The possibility of a near-infrared optical imaging system using frequency domain methods,” in Proceedings of the Third International Conference on Peace through Mind/Brain Science (Hamamatsu Corp., Hamamatsu, Japan, 1990), pp. 183–188.

Mathews, J.

J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1970), p. 245.

McCully, K.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscles,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Moulton, J. D.

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering methods to photodynamic dosimetry,” in Photodynamic Therapy: Mechanisms II, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1203, 62–75 (1990).
[CrossRef]

Nioka, S.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscles,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Patterson, M. S.

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]

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering methods to photodynamic dosimetry,” in Photodynamic Therapy: Mechanisms II, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1203, 62–75 (1990).
[CrossRef]

Pine, D. J.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1989).

Prahl, S. A.

Schmitt, J. M.

J. M. Schmitt, G. X. Zhou, E. C. Walker, R. T. Wall, “Multilayer model of photon diffusion in skin,” J. Opt. Soc. Am. A 7, 2141–2153 (1990).
[CrossRef] [PubMed]

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. (to be published).

A. Knüttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-density waves,” Appl. Opt. (to be published).

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, New York, 1980), Vol. 1, Chap. 14, pp. 477–492.

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]

vandeVen, M. J.

E. Gratton, W. W. Mantulin, M. J. vandeVen, J. B. Fishkin, M. B. Maris, B. Chance, “The possibility of a near-infrared optical imaging system using frequency domain methods,” in Proceedings of the Third International Conference on Peace through Mind/Brain Science (Hamamatsu Corp., Hamamatsu, Japan, 1990), pp. 183–188.

J. Fishkin, E. Gratton, M. J. vandeVen, W. W. Mantulin, “Diffusion of intensity-modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 122–135 (1991).
[CrossRef]

Walker, E. C.

Walker, R. L.

J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1970), p. 245.

Wall, R. T.

Weitz, D. A.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1989).

Welch, A. J.

Wilson, B. C.

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]

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering methods to photodynamic dosimetry,” in Photodynamic Therapy: Mechanisms II, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1203, 62–75 (1990).
[CrossRef]

Wolf, P. E.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1989).

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]

Yoo, K. M.

K. M. Yoo, L. Feng, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Yoon, G.

Zdrojkowski, R.

R. L. Longini, R. Zdrojkowski, “A note on the theory of backscattering of light by living tissue,”IEEE Trans. Biomed. Eng. BME-15, 4–10 (1968).
[CrossRef]

Zhou, G. X.

Anal. Biochem.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscles,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Appl. Opt.

Chem. Phys. Lett.

J. R. Lakowicz, K. W. Berndt, “Frequency-domain measurements of photon migration in tissues,” Chem. Phys. Lett. 166, 246–252 (1990).
[CrossRef]

IEEE Trans. Biomed. Eng.

R. L. Longini, R. Zdrojkowski, “A note on the theory of backscattering of light by living tissue,”IEEE Trans. Biomed. Eng. BME-15, 4–10 (1968).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D

L. F. Gate, “The determination of light absorption in diffusing materials by a photon diffusion model,”J. Phys. D 4, 1049–1056 (1971).
[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. Lett.

K. M. Yoo, L. Feng, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

J. R. Lakowicz, K. W. Berndt, “Lifetime-selective fluorescence imaging using an rf phase-sensitive camera,” Rev. Sci. Instrum. 62, 1727–1734 (1991).
[CrossRef]

Other

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. (to be published).

H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, New York, 1980), Vol. 1, Chap. 14, pp. 477–492.

J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1970), p. 245.

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering methods to photodynamic dosimetry,” in Photodynamic Therapy: Mechanisms II, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1203, 62–75 (1990).
[CrossRef]

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1989).

B. Chance, ed., Photon Migration in Tissues (Plenum, New York, 1989).
[CrossRef]

J. Fishkin, E. Gratton, M. J. vandeVen, W. W. Mantulin, “Diffusion of intensity-modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 122–135 (1991).
[CrossRef]

A. Knüttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-density waves,” Appl. Opt. (to be published).

A. Knüttel, J. R. Knutson, “Methods and apparatus for imaging a physical parameter in turbid media using diffusive waves,” U.S. patent application 07/722,823 (May7, 1991).

E. Gratton, W. W. Mantulin, M. J. vandeVen, J. B. Fishkin, M. B. Maris, B. Chance, “The possibility of a near-infrared optical imaging system using frequency domain methods,” in Proceedings of the Third International Conference on Peace through Mind/Brain Science (Hamamatsu Corp., Hamamatsu, Japan, 1990), pp. 183–188.

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Figures (10)

Fig. 1
Fig. 1

Coordinate systems and source arrangements for calculating the photon-density wave produced by one or more point sources in a volume of random scatterers. (a) A single source in a medium of infinite extent. (b) Two sources separated by a distance R in a medium of infinite extent. One plane of the cylindrically symmetrical volume is shown. (c) Two collimated beams separated by a distance R incident upon a semi-infinite medium. One plane of the cylindrically symmetrical volume is shown.

Fig. 2
Fig. 2

Block diagram of the experimental setup for measuring interference patterns of diffusive photon-density waves. Component names are abbreviated as follows: M’s, mirrors; BS’s, beam splitters; PD, photodiode; ND, neutral-density filter; Pr, retroreflecting prism; V ND, variable neutral-density filter; PMT, photomultiplier tube.

Fig. 3
Fig. 3

Magnitude and phase of the photon-density wave produced by a 410-MHz modulated beam incident upon the experimental scattering medium. The data points shown are measurements made throughout the medium at a radial distance greater than 20 mm from the detector at depths greater than 10 mm. (a) Loge of the product of the ac magnitude and the radial distance between the source and the detector. (b) Measured phase in degrees. All measurements are shown normalized to the value measured at the shortest source–detector separation distance.

Fig. 4
Fig. 4

Calculated phase and magnitude of a 410-MHz photon-diffusion wave generated by a point source in a half-space medium (solid curves) and in the forward hemisphere of a full-space medium (dashed curves). Note the distortion of the spherical wave front caused by the absorbing boundary: (a) isomagnitude contours [loge(mac)], (b) isophase contours. The model parameters, which were chosen to represent experimental conditions, were as follows: I0 = 1, Cn = 0.222 mm/ps, ω = 2.58 × 109 rad, μs′ = 1.1 mm−1, μa = 0.008 mm−1.

Fig. 5
Fig. 5

Contour plot showing the phase differences Δϕdes (dotted curves) and source-intensity ratios (I2/I1)des (solid curves) required simultaneously for complete destructive interference at points in the plane of the sources. I1 is the intensity of source 1, located at r = −6 mm; I2 is the intensity of source 2, located at r = +6 mm. The arrows indicate the locations of the sources. The model parameters used were the same as those used to generate Fig. 4 (see the caption to Fig. 4). (a) The semi-infinite medium [from Eq. (13)]. (b) The infinite medium [from Eqs. (19) and (20)]. Only the forward hemisphere is shown.

Fig. 6
Fig. 6

Surface plots of the magnitude of a 410-MHz wave formed by two interfering diffusive waves. Measurements were made with the tip of the detector fiber placed at different positions in the source plane. The interfering waves were produced by equal-intensity beams incident upon the medium at lateral positions r = −12 mm and r = +12 mm: (a) destructively interfering, (b) constructively interfering.

Fig. 7
Fig. 7

Surface plots of the phase of a 410-MHz wave formed by two constructively interfering diffusive waves. Measurements were made with the tip of the detector fiber placed at different positions in the source plane. The interfering waves were produced by equal-intensity beams incident upon the medium at lateral positions r = −12 mm and r = +12 mm: (a) destructively interfering, (b) constructively interfering.

Fig. 8
Fig. 8

Comparison of predicted magnitudes and phases with those measured at a fixed depth (z = 12.5 mm) under constructive- and destructive-interference conditions. The plotted values have been normalized to the average of the values measured at r = −5 mm and r = 5 mm under constructive-interference conditions. Model parameters for constructive interference: I1 = I2 = 1, Δϕ = 1; for destructive interference: I1 = I2 = 1, Δϕ = 180°. The remaining parameters were the same as those listed in the caption to Fig. 4. (a) Ac magnitude for the constructive-interference (upper curve) and destructive-interference cases (lower curve). (b) Phases for the constructive-interference (upper curve) and destructive-interference cases (lower curve).

Fig. 9
Fig. 9

Constructive- and destructive-interference patterns produced by two 410-MHz sources having different magnitudes and phases, derived by using the half-space model [Eq. (17)]. The contour lines, which connect the points having equal loge ac magnitudes, are spaced to show an incremental decrease in photon density equal to e1.5. Sources 1 and 2 are located at r = −6 mm and r = +6 mm, respectively (the arrows indicate the points of entry of the photons), and the model parameters are the same as those listed in the caption to Fig. 4. (a) Destructive conditions (solid curves): I2/I1 = 0.453, Δϕ = −169.1°; constructive conditions (dashed curves): I2/I1 = 0.453, Δϕ = +10.9°. (b) Loge ratio of the constructive and destructive magnitudes shown in Fig. 10(a). (c) Destructive conditions (solid curves): I2/I1 = 0.0598, Δϕ = −135.8°; constructive conditions (dashed curves): I2/I1 = 0.0598, Δϕ = +44.2°. (d) Loge ratio of the constructive and destructive magnitudes shown in Fig. 10(c). (e) Destructive conditions (solid curves): I2/I1 = 0.0278, Δϕ = −114.3°; constructive conditions (dashed curves): I2/I1 = 0.0278, Δϕ = +65.7°. (f) Loge ratio of the constructive and destructive magnitudes shown in Fig. 10(e). (g) Sum of loge constructive/destructive ratios produced by two source pairs located symmetrically about the origin (one source pair is centered around r = +10 mm and the other around r = −10 mm, both with a source separation of 12 mm). Initial intensities and phases for the source pair centered around r = −10 mm are the same as those used to generate the plot in Fig. 10(c); the phase and the magnitude of the sources constituting the other pair are interchanged.

Fig. 10
Fig. 10

Ac magnitude and phase of a 410-MHz wave formed by two interfering diffusive waves plotted as a function of depth into the medium. The two beams, which were incident upon the medium at lateral positions r = 0 mm and r = +12 mm, were adjusted to; produce a null at r (lateral position), z (depth) coordinates (−30 mm, 2 mm). The data measured at two lateral positions, r = 10 mm and r = 15 mm, are compared with theory. The plotted values are shown normalized to the phase and magnitude measured at the depth z = 5 mm. Model parameters for constrictive interference: I2/I1 = 0.0428, Δϕ = +67.8°; for destructive interference: I2/I1 = 0.0428, Δϕ = −112.2°; the remaining parameters were the same as those listed in the caption to Fig. 4. (a) Ac magnitude for the constructive-interference and destructive-interference cases. (b) Phases for the constructive-interference and destructive-interference cases. The filled and open squares are experimental values recorded under constructive-interference conditions, and the filled and open squares are experimental values recorded under destructive-interference conditions.

Equations (24)

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1 c n t ψ ( r , z , t ) - D 2 ψ ( r , z , t ) + μ a ψ ( r , z , t ) = S ( r , z , t ) ,
D = [ 3 ( μ a + μ s ) ] - 1 .
1 c n t ψ ( r , z , t ) - D 2 ψ ( r , z , t ) + μ a ψ ( r , z , t ) = I 0 δ ( t = 0 ) δ ( 0 , 0 ) .
- D 2 ψ ( r , z , ω ) + [ μ a + ( i ω / c n ) ] ψ ( r , z , ω ) = I 0 δ ( 0 , 0 ) ,
ψ ( r , z , ω ) = I 0 4 π D ρ exp [ - ρ ( μ a D + i ω D c n ) 1 / 2 ] ,
ψ ( ρ , ω ) = I 0 4 π D ρ exp { - a ρ [ cos ( θ 2 ) + i sin ( θ 2 ) ] } ,
a = [ ( μ a D ) 2 + ( ω D c n ) 2 ] 1 / 4 , θ = tan - 1 [ ω / ( μ a c n ) ] .
m ac ( ρ ) = 1 4 π D ρ exp { - ρ [ ( μ a D ) 2 + ( ω D c n ) 2 ] 1 / 4 × cos [ 1 2 tan - 1 ( ω μ a c n ) ] } ,
ϕ ( ρ ) = ρ [ ( μ a D ) 2 + ( ω D c n ) 2 ] 1 / 4 sin [ 1 2 tan - 1 ( ω μ a c n ) ] ,
m dc ( ρ ) = 1 4 π D ρ exp { - ρ [ ( μ a D ) 1 / 2 ] } .
ψ fs ( ρ , ω ) = 1 4 π D ( I 1 ρ 1 exp { - a ρ 1 [ cos ( θ 2 ) + i sin ( θ 2 ) ] } + I 2 ρ 2 exp { - a ρ 2 [ cos ( θ 2 ) + i sin ( θ 2 ) + i Δ ϕ 0 ] } ) ,
M fs ( ρ , ω ) = { I 1 2 m ac 2 ( ρ 1 ) + I 2 2 m ac 2 ( ρ 2 ) + 2 I 1 I 2 m ac ( ρ 1 ) m ac ( ρ 2 ) × cos [ ϕ ( ρ 2 ) - ϕ ( ρ 1 ) + Δ ϕ 0 ] } 1 / 2 ,
Φ fs ( ρ , ω ) = tan - 1 [ ( { I 1 m ac ( ρ 1 ) sin [ ϕ ( ρ 1 ) ] + I 2 m ac ( ρ 2 ) sin [ ϕ ( ρ 2 ) ] + Δ ϕ 0 } ) / ( { I 1 m ac ( ρ 1 ) × cos [ ϕ ( ρ 1 ) ] + I 2 m ac ( ρ 2 ) cos [ ϕ ( ρ 2 ) + Δ ϕ 0 } ) ] ,
( I 2 / I 1 ) des = ρ 2 ρ 1 exp ( - ( ρ 1 - ρ 2 ) [ ( μ a D ) 2 + ( ω D c n ) 2 ] 1 / 4 × { cos [ 1 2 tan - 1 ( ω μ a c n ) ] } ) ,
Δ ϕ 0 des = π - ( ρ 1 - ρ 2 ) [ ( μ a D ) 2 + ( ω D c n ) 2 ] 1 / 4 × sin [ 1 2 tan - 1 ( ω μ a c n ) ] .
ψ hs ( ρ , ω ) = 1 4 π D ( I 1 ρ 1 exp { - a ρ 1 [ cos ( θ 2 ) + i sin ( θ 2 ) ] } - I 1 ρ 1 exp { - a ρ 1 [ cos ( θ 2 ) + i sin ( θ 2 ) ] } + I 2 ρ 2 exp { - a ρ 2 [ cos ( θ 2 ) + i sin ( θ 2 ) + i Δ ϕ 0 ] } - I 2 ρ 2 exp { - a ρ 2 [ cos ( θ 2 ) + i sin ( θ 2 ) + i Δ ϕ 0 ] } ) ,
ρ 1 = [ ( r + R / 2 ) 2 + ( z - z s ) 2 ] 1 / 2
ρ 1 = [ ( r + R / 2 ) 2 + ( z + z s ) 2 ] 1 / 2
ρ 2 = [ ( r - R / 2 ) 2 + ( z - z s ) 2 ] 1 / 2
ρ 2 = [ ( r - R / 2 ) 2 + ( z + z s ) 2 ] 1 / 2 .
M hs ( ρ , ω ) = ( { I 1 m ac ( ρ 1 ) cos [ ϕ ( ρ 1 ) ] - I 1 m ac ( ρ 1 ) cos [ ϕ ( ρ 1 ) ] + I 2 m ac ( ρ 2 ) cos [ ϕ ( ρ 2 ) + Δ ϕ 0 ] - I 2 m ac ( ρ 2 ) cos [ ϕ ( ρ 2 ) + Δ ϕ 0 ] } 2 + { I 1 m ac ( ρ 1 ) sin [ ϕ ( ρ 1 ) ] - I 1 m ac ( ρ 1 ) sin [ ϕ ( ρ 1 ) ] + I 2 m ac ( ρ 2 ) sin [ ϕ ( ρ 2 ) + Δ ϕ 0 ] - I 2 m ac ( ρ 2 ) sin [ ϕ ( ρ 2 ) + Δ ϕ 0 ] } 2 ) 1 / 2 ,
Φ hs ( ρ , ω ) = tan - 1 ( { I 1 m ac ( ρ 1 ) sin [ ϕ ( ρ 1 ) ] - I 1 m ac ( ρ 1 ) sin [ ϕ ( ρ 1 ) + I 2 m ac ( ρ 2 ) sin [ ϕ ( ρ 2 ) + Δ ϕ 0 ] - I 2 m ac ( ρ 2 ) sin [ ϕ ( ρ 2 ) + Δ ϕ 0 ] } / { I 1 m ac ( ρ 1 ) cos [ ϕ ( ρ 1 ) ] - I 1 m ac ( ρ 1 ) cos [ ϕ ( ρ 1 ) ] + I 2 m ac ( ρ 2 ) cos [ ϕ ( ρ 2 ) + Δ ϕ 0 ] - I 2 m ac ( ρ 2 ) cos [ ϕ ( ρ 2 ) + Δ ϕ 0 ] } ) .
( I 2 / I 1 ) des = ( { m ac 2 ( ρ 1 ) + m ac 2 ( ρ 1 ) - m ac ( ρ 1 ) m ac ( ρ 1 ) cos [ ϕ ( ρ 1 ) - ϕ ( ρ 1 ) ] } / { m ac 2 ( ρ 2 ) + m ac 2 ( ρ 2 ) - m ac ( ρ 2 ) m ac ( ρ 2 ) × cos [ ϕ ( ρ 2 ) - ϕ ( ρ 2 ) ] } ) 1 / 2 ,
Δ ϕ 0 des = π - tan - 1 ( { m ac ( ρ 1 ) sin [ ϕ ( ρ 1 ) ] - m ac ( ρ 1 ) sin [ ρ ( ρ 1 ) ] } / { m ac ( ρ ) cos [ ϕ ( ρ 1 ) - m ac ( ρ 1 ) cos [ ϕ ( ρ 1 ) ] } ) - tan - 1 ( { m ac ( ρ 2 ) × sin [ ϕ ( ρ 2 ) ] - m ac ( ρ 2 ) sin [ ϕ ( ρ 2 ) ] } / { m ac ( ρ 2 ) × cos [ ϕ ( ρ 2 ) ] - m ac ( ρ 2 ) cos [ ϕ ( ρ 2 ) ] } ) .

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