Abstract

Laser Doppler flow measurements and Monte Carlo simulations on small blood perfusion flow models at 780 nm are presented and compared. The dimensions of the optical sample volume are investigated as functions of the distance of the laser to the detector and as functions of the angle of penetration of the laser into the sample. The effects of homodyne and heterodyne scattering are investigated.

© 1995 Optical Society of America

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    [CrossRef]
  2. R. F. Bonner, R. Nossal, “Model for laser Doppler measurements of blood flow in tissue,” Appl. Opt. 20, 2097–2107 (1981).
    [CrossRef] [PubMed]
  3. G. E. Nilsson, T. Tenland, P. A. Oberg, “A new instrument for continuous measurement of blood flow by light beating spectroscopy,” IEEE Trans. Biomed. Eng. BME-27, 12–19 (1980).
    [CrossRef]
  4. R. J. Gush, T. A. King, M. I. V. Jayson, “Aspects of laser light scattering from skin tissue with application to laser Doppler blood flow measurements,” Phys. Med. Biol. 29, 1463–1476 (1984).
    [CrossRef] [PubMed]
  5. A. N. Obeid, D. M. Boggett, N. J. Barnett, G. Dougherty, P. Rolfe, “Depth discrimination in laser Doppler skin blood flow measurement using different lasers,” Med. Biol. Eng. Comput. 26, 415–419 (1988).
    [CrossRef] [PubMed]
  6. H. W. Jentink, F. F. M. de Mul, R. G. A. M. Hermsen, R. Graaff, J. Greve, “Monte Carlo simulations of laser Doppler blood flow measurements in tissue,” Appl. Opt. 29, 2371–2381 (1990).
    [CrossRef] [PubMed]
  7. M. H. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Monte Carlo simulations and measurements of signals in laser Doppler flowmetry on human skin,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 63–72 (1991).
  8. H. W. Jentink, F. F. M. de Mul, R. Graaff, H. E. Suichies, J. G. Aarnoudse, J. Greve, “Laser Doppler flowmetry: measurements in a layered perfusion model and Monte Carlo simulations of measurements,” Appl. Opt. 30, 2592–2597 (1991).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  20. R. Graaff, A. C. M. Dassel, M. H. Koelink, F. F. M. de Mul, J. G. Aarnoudse, W. G. Zijlstra, “Optical properties of human dermis in vitro and in vivo,” Appl. Opt. 32, 435–447 (1993).
    [CrossRef] [PubMed]
  21. R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, H. W. Jentink, “Light propagation parameters for anisotropic scattering media based on a rigorous solution of the transport equation,” Appl. Opt. 26, 2273–2279 (1989).
    [CrossRef]
  22. M. H. Koelink, F. F. M. de Mul, B. Leerkotte, J. Greve, H. W. Jentink, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Signal processing for a laser Doppler perfusion meter,” Signal Process. 38, 239–252 (1994).
    [CrossRef]
  23. H. W. Jentink, J. A. J. van Beurden, M. A. Helsdingen, F. F. M. de Mul, H. E. Suichies, J. G. Aarnoudse, J. Greve, “A compact differential laser Doppler velocimeter using a semiconductor laser,” J. Phys. E 20, 129–131 (1987).
    [CrossRef]
  24. B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), Chaps. 4 and 5.
  25. M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin Optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
    [CrossRef] [PubMed]
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  29. S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–498 (1981).
    [PubMed]
  30. R. R. Anderson, J. A. Parrish, “The optics of human skin,” J. Invest. Dermatol. 77, 13–20 (1981).
    [CrossRef] [PubMed]

1994 (2)

M. H. Koelink, F. F. M. de Mul, B. Leerkotte, J. Greve, H. W. Jentink, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Signal processing for a laser Doppler perfusion meter,” Signal Process. 38, 239–252 (1994).
[CrossRef]

M. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Laser Doppler blood flowmetry using two wavelengths: Monte-Carlo simulations and measurements,” Appl. Opt. 33, 3549–3558 (1994).
[CrossRef] [PubMed]

1993 (2)

1991 (1)

1990 (3)

1989 (3)

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, H. W. Jentink, “Light propagation parameters for anisotropic scattering media based on a rigorous solution of the transport equation,” Appl. Opt. 26, 2273–2279 (1989).
[CrossRef]

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin Optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

R. Nossal, R. F. Bonner, G. H. Weiss, “Influence of path length on remote optical sensing of properties of biological tissue,” Appl. Opt. 28, 2238–2244 (1989).
[CrossRef] [PubMed]

1988 (1)

A. N. Obeid, D. M. Boggett, N. J. Barnett, G. Dougherty, P. Rolfe, “Depth discrimination in laser Doppler skin blood flow measurement using different lasers,” Med. Biol. Eng. Comput. 26, 415–419 (1988).
[CrossRef] [PubMed]

1987 (2)

H. W. Jentink, J. A. J. van Beurden, M. A. Helsdingen, F. F. M. de Mul, H. E. Suichies, J. G. Aarnoudse, J. Greve, “A compact differential laser Doppler velocimeter using a semiconductor laser,” J. Phys. E 20, 129–131 (1987).
[CrossRef]

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

1985 (1)

L. Duteil, J. C. Bernego, W. Schalla, “A double wavelength laser Doppler system to investigate skin microcirculation,” IEEE Trans. Biomed. Eng. BME-32, 439–447 (1985).
[CrossRef]

1984 (2)

R. J. Gush, T. A. King, M. I. V. Jayson, “Aspects of laser light scattering from skin tissue with application to laser Doppler blood flow measurements,” Phys. Med. Biol. 29, 1463–1476 (1984).
[CrossRef] [PubMed]

J. M. Maarek, G. Jarry, B. de Losnac, A. Lansiart, B. M. Hung, “A simulation method for the study of laser transillumination of biological tissues,” Ann. Biomed. Eng. 12, 281–284 (1984).
[CrossRef] [PubMed]

1983 (1)

1981 (3)

R. F. Bonner, R. Nossal, “Model for laser Doppler measurements of blood flow in tissue,” Appl. Opt. 20, 2097–2107 (1981).
[CrossRef] [PubMed]

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–498 (1981).
[PubMed]

R. R. Anderson, J. A. Parrish, “The optics of human skin,” J. Invest. Dermatol. 77, 13–20 (1981).
[CrossRef] [PubMed]

1980 (1)

G. E. Nilsson, T. Tenland, P. A. Oberg, “A new instrument for continuous measurement of blood flow by light beating spectroscopy,” IEEE Trans. Biomed. Eng. BME-27, 12–19 (1980).
[CrossRef]

1979 (1)

S. Takatani, M. D. Graham, “Analysis of diffuse reflectance from a two layer tissue model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
[CrossRef]

1975 (1)

M. D. Stern, “In vivo evaluation of microcirculation by coherent light scattering,” Nature (London) 254, 56–58 (1975).
[CrossRef]

1962 (1)

1951 (1)

Aarnoudse, J. G.

M. H. Koelink, F. F. M. de Mul, B. Leerkotte, J. Greve, H. W. Jentink, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Signal processing for a laser Doppler perfusion meter,” Signal Process. 38, 239–252 (1994).
[CrossRef]

M. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Laser Doppler blood flowmetry using two wavelengths: Monte-Carlo simulations and measurements,” Appl. Opt. 33, 3549–3558 (1994).
[CrossRef] [PubMed]

R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, J. G. Aarnoudse, “Condensed Monte Carlo simulation for the description of light transport,” Appl. Opt. 32, 426–434 (1993).
[CrossRef] [PubMed]

R. Graaff, A. C. M. Dassel, M. H. Koelink, F. F. M. de Mul, J. G. Aarnoudse, W. G. Zijlstra, “Optical properties of human dermis in vitro and in vivo,” Appl. Opt. 32, 435–447 (1993).
[CrossRef] [PubMed]

H. W. Jentink, F. F. M. de Mul, R. Graaff, H. E. Suichies, J. G. Aarnoudse, J. Greve, “Laser Doppler flowmetry: measurements in a layered perfusion model and Monte Carlo simulations of measurements,” Appl. Opt. 30, 2592–2597 (1991).
[CrossRef] [PubMed]

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, H. W. Jentink, “Light propagation parameters for anisotropic scattering media based on a rigorous solution of the transport equation,” Appl. Opt. 26, 2273–2279 (1989).
[CrossRef]

H. W. Jentink, J. A. J. van Beurden, M. A. Helsdingen, F. F. M. de Mul, H. E. Suichies, J. G. Aarnoudse, J. Greve, “A compact differential laser Doppler velocimeter using a semiconductor laser,” J. Phys. E 20, 129–131 (1987).
[CrossRef]

M. H. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Monte Carlo simulations and measurements of signals in laser Doppler flowmetry on human skin,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 63–72 (1991).

Anderson, R. R.

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–498 (1981).
[PubMed]

R. R. Anderson, J. A. Parrish, “The optics of human skin,” J. Invest. Dermatol. 77, 13–20 (1981).
[CrossRef] [PubMed]

Barnett, N. J.

A. N. Obeid, D. M. Boggett, N. J. Barnett, G. Dougherty, P. Rolfe, “Depth discrimination in laser Doppler skin blood flow measurement using different lasers,” Med. Biol. Eng. Comput. 26, 415–419 (1988).
[CrossRef] [PubMed]

Berne, B. J.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), Chaps. 4 and 5.

Bernego, J. C.

L. Duteil, J. C. Bernego, W. Schalla, “A double wavelength laser Doppler system to investigate skin microcirculation,” IEEE Trans. Biomed. Eng. BME-32, 439–447 (1985).
[CrossRef]

Boggett, D. M.

A. N. Obeid, D. M. Boggett, N. J. Barnett, G. Dougherty, P. Rolfe, “Depth discrimination in laser Doppler skin blood flow measurement using different lasers,” Med. Biol. Eng. Comput. 26, 415–419 (1988).
[CrossRef] [PubMed]

Bonner, R. F.

Cheong, W. F.

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

Crucio, J. A.

Dassel, A. C. M.

M. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Laser Doppler blood flowmetry using two wavelengths: Monte-Carlo simulations and measurements,” Appl. Opt. 33, 3549–3558 (1994).
[CrossRef] [PubMed]

M. H. Koelink, F. F. M. de Mul, B. Leerkotte, J. Greve, H. W. Jentink, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Signal processing for a laser Doppler perfusion meter,” Signal Process. 38, 239–252 (1994).
[CrossRef]

R. Graaff, A. C. M. Dassel, M. H. Koelink, F. F. M. de Mul, J. G. Aarnoudse, W. G. Zijlstra, “Optical properties of human dermis in vitro and in vivo,” Appl. Opt. 32, 435–447 (1993).
[CrossRef] [PubMed]

R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, J. G. Aarnoudse, “Condensed Monte Carlo simulation for the description of light transport,” Appl. Opt. 32, 426–434 (1993).
[CrossRef] [PubMed]

M. H. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Monte Carlo simulations and measurements of signals in laser Doppler flowmetry on human skin,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 63–72 (1991).

de Losnac, B.

J. M. Maarek, G. Jarry, B. de Losnac, A. Lansiart, B. M. Hung, “A simulation method for the study of laser transillumination of biological tissues,” Ann. Biomed. Eng. 12, 281–284 (1984).
[CrossRef] [PubMed]

de Mul, F. F. M.

M. H. Koelink, F. F. M. de Mul, B. Leerkotte, J. Greve, H. W. Jentink, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Signal processing for a laser Doppler perfusion meter,” Signal Process. 38, 239–252 (1994).
[CrossRef]

M. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Laser Doppler blood flowmetry using two wavelengths: Monte-Carlo simulations and measurements,” Appl. Opt. 33, 3549–3558 (1994).
[CrossRef] [PubMed]

R. Graaff, A. C. M. Dassel, M. H. Koelink, F. F. M. de Mul, J. G. Aarnoudse, W. G. Zijlstra, “Optical properties of human dermis in vitro and in vivo,” Appl. Opt. 32, 435–447 (1993).
[CrossRef] [PubMed]

R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, J. G. Aarnoudse, “Condensed Monte Carlo simulation for the description of light transport,” Appl. Opt. 32, 426–434 (1993).
[CrossRef] [PubMed]

H. W. Jentink, F. F. M. de Mul, R. Graaff, H. E. Suichies, J. G. Aarnoudse, J. Greve, “Laser Doppler flowmetry: measurements in a layered perfusion model and Monte Carlo simulations of measurements,” Appl. Opt. 30, 2592–2597 (1991).
[CrossRef] [PubMed]

H. W. Jentink, F. F. M. de Mul, R. G. A. M. Hermsen, R. Graaff, J. Greve, “Monte Carlo simulations of laser Doppler blood flow measurements in tissue,” Appl. Opt. 29, 2371–2381 (1990).
[CrossRef] [PubMed]

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, H. W. Jentink, “Light propagation parameters for anisotropic scattering media based on a rigorous solution of the transport equation,” Appl. Opt. 26, 2273–2279 (1989).
[CrossRef]

H. W. Jentink, J. A. J. van Beurden, M. A. Helsdingen, F. F. M. de Mul, H. E. Suichies, J. G. Aarnoudse, J. Greve, “A compact differential laser Doppler velocimeter using a semiconductor laser,” J. Phys. E 20, 129–131 (1987).
[CrossRef]

M. H. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Monte Carlo simulations and measurements of signals in laser Doppler flowmetry on human skin,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 63–72 (1991).

Dougherty, G.

A. N. Obeid, D. M. Boggett, N. J. Barnett, G. Dougherty, P. Rolfe, “Depth discrimination in laser Doppler skin blood flow measurement using different lasers,” Med. Biol. Eng. Comput. 26, 415–419 (1988).
[CrossRef] [PubMed]

Duteil, L.

L. Duteil, J. C. Bernego, W. Schalla, “A double wavelength laser Doppler system to investigate skin microcirculation,” IEEE Trans. Biomed. Eng. BME-32, 439–447 (1985).
[CrossRef]

Ferwerda, H. A.

Flock, S. T.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo-diffusion theory modelling of light distributions in tissue,” in Laser Interaction with Tissue, M. W. Berns, ed. Proc. Soc. Photo-Opt. Instrum. Eng. 908, 20–28 (1988).

Graaff, R.

M. H. Koelink, F. F. M. de Mul, B. Leerkotte, J. Greve, H. W. Jentink, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Signal processing for a laser Doppler perfusion meter,” Signal Process. 38, 239–252 (1994).
[CrossRef]

M. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Laser Doppler blood flowmetry using two wavelengths: Monte-Carlo simulations and measurements,” Appl. Opt. 33, 3549–3558 (1994).
[CrossRef] [PubMed]

R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, J. G. Aarnoudse, “Condensed Monte Carlo simulation for the description of light transport,” Appl. Opt. 32, 426–434 (1993).
[CrossRef] [PubMed]

R. Graaff, A. C. M. Dassel, M. H. Koelink, F. F. M. de Mul, J. G. Aarnoudse, W. G. Zijlstra, “Optical properties of human dermis in vitro and in vivo,” Appl. Opt. 32, 435–447 (1993).
[CrossRef] [PubMed]

H. W. Jentink, F. F. M. de Mul, R. Graaff, H. E. Suichies, J. G. Aarnoudse, J. Greve, “Laser Doppler flowmetry: measurements in a layered perfusion model and Monte Carlo simulations of measurements,” Appl. Opt. 30, 2592–2597 (1991).
[CrossRef] [PubMed]

H. W. Jentink, F. F. M. de Mul, R. G. A. M. Hermsen, R. Graaff, J. Greve, “Monte Carlo simulations of laser Doppler blood flow measurements in tissue,” Appl. Opt. 29, 2371–2381 (1990).
[CrossRef] [PubMed]

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, H. W. Jentink, “Light propagation parameters for anisotropic scattering media based on a rigorous solution of the transport equation,” Appl. Opt. 26, 2273–2279 (1989).
[CrossRef]

M. H. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Monte Carlo simulations and measurements of signals in laser Doppler flowmetry on human skin,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 63–72 (1991).

Graham, M. D.

S. Takatani, M. D. Graham, “Analysis of diffuse reflectance from a two layer tissue model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
[CrossRef]

Greve, J.

M. H. Koelink, F. F. M. de Mul, B. Leerkotte, J. Greve, H. W. Jentink, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Signal processing for a laser Doppler perfusion meter,” Signal Process. 38, 239–252 (1994).
[CrossRef]

M. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Laser Doppler blood flowmetry using two wavelengths: Monte-Carlo simulations and measurements,” Appl. Opt. 33, 3549–3558 (1994).
[CrossRef] [PubMed]

H. W. Jentink, F. F. M. de Mul, R. Graaff, H. E. Suichies, J. G. Aarnoudse, J. Greve, “Laser Doppler flowmetry: measurements in a layered perfusion model and Monte Carlo simulations of measurements,” Appl. Opt. 30, 2592–2597 (1991).
[CrossRef] [PubMed]

H. W. Jentink, F. F. M. de Mul, R. G. A. M. Hermsen, R. Graaff, J. Greve, “Monte Carlo simulations of laser Doppler blood flow measurements in tissue,” Appl. Opt. 29, 2371–2381 (1990).
[CrossRef] [PubMed]

H. W. Jentink, J. A. J. van Beurden, M. A. Helsdingen, F. F. M. de Mul, H. E. Suichies, J. G. Aarnoudse, J. Greve, “A compact differential laser Doppler velocimeter using a semiconductor laser,” J. Phys. E 20, 129–131 (1987).
[CrossRef]

M. H. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Monte Carlo simulations and measurements of signals in laser Doppler flowmetry on human skin,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 63–72 (1991).

Groenhuis, R. A. J.

Gush, R. J.

R. J. Gush, T. A. King, M. I. V. Jayson, “Aspects of laser light scattering from skin tissue with application to laser Doppler blood flow measurements,” Phys. Med. Biol. 29, 1463–1476 (1984).
[CrossRef] [PubMed]

Havlin, S.

Helsdingen, M. A.

H. W. Jentink, J. A. J. van Beurden, M. A. Helsdingen, F. F. M. de Mul, H. E. Suichies, J. G. Aarnoudse, J. Greve, “A compact differential laser Doppler velocimeter using a semiconductor laser,” J. Phys. E 20, 129–131 (1987).
[CrossRef]

Hermsen, R. G. A. M.

Hung, B. M.

J. M. Maarek, G. Jarry, B. de Losnac, A. Lansiart, B. M. Hung, “A simulation method for the study of laser transillumination of biological tissues,” Ann. Biomed. Eng. 12, 281–284 (1984).
[CrossRef] [PubMed]

Jacques, S. L.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin Optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Jarry, G.

J. M. Maarek, G. Jarry, B. de Losnac, A. Lansiart, B. M. Hung, “A simulation method for the study of laser transillumination of biological tissues,” Ann. Biomed. Eng. 12, 281–284 (1984).
[CrossRef] [PubMed]

Jayson, M. I. V.

R. J. Gush, T. A. King, M. I. V. Jayson, “Aspects of laser light scattering from skin tissue with application to laser Doppler blood flow measurements,” Phys. Med. Biol. 29, 1463–1476 (1984).
[CrossRef] [PubMed]

Jentink, H. W.

M. H. Koelink, F. F. M. de Mul, B. Leerkotte, J. Greve, H. W. Jentink, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Signal processing for a laser Doppler perfusion meter,” Signal Process. 38, 239–252 (1994).
[CrossRef]

H. W. Jentink, F. F. M. de Mul, R. Graaff, H. E. Suichies, J. G. Aarnoudse, J. Greve, “Laser Doppler flowmetry: measurements in a layered perfusion model and Monte Carlo simulations of measurements,” Appl. Opt. 30, 2592–2597 (1991).
[CrossRef] [PubMed]

H. W. Jentink, F. F. M. de Mul, R. G. A. M. Hermsen, R. Graaff, J. Greve, “Monte Carlo simulations of laser Doppler blood flow measurements in tissue,” Appl. Opt. 29, 2371–2381 (1990).
[CrossRef] [PubMed]

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, H. W. Jentink, “Light propagation parameters for anisotropic scattering media based on a rigorous solution of the transport equation,” Appl. Opt. 26, 2273–2279 (1989).
[CrossRef]

H. W. Jentink, J. A. J. van Beurden, M. A. Helsdingen, F. F. M. de Mul, H. E. Suichies, J. G. Aarnoudse, J. Greve, “A compact differential laser Doppler velocimeter using a semiconductor laser,” J. Phys. E 20, 129–131 (1987).
[CrossRef]

King, T. A.

R. J. Gush, T. A. King, M. I. V. Jayson, “Aspects of laser light scattering from skin tissue with application to laser Doppler blood flow measurements,” Phys. Med. Biol. 29, 1463–1476 (1984).
[CrossRef] [PubMed]

Koelink, M.

Koelink, M. H.

M. H. Koelink, F. F. M. de Mul, B. Leerkotte, J. Greve, H. W. Jentink, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Signal processing for a laser Doppler perfusion meter,” Signal Process. 38, 239–252 (1994).
[CrossRef]

R. Graaff, A. C. M. Dassel, M. H. Koelink, F. F. M. de Mul, J. G. Aarnoudse, W. G. Zijlstra, “Optical properties of human dermis in vitro and in vivo,” Appl. Opt. 32, 435–447 (1993).
[CrossRef] [PubMed]

R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, J. G. Aarnoudse, “Condensed Monte Carlo simulation for the description of light transport,” Appl. Opt. 32, 426–434 (1993).
[CrossRef] [PubMed]

M. H. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Monte Carlo simulations and measurements of signals in laser Doppler flowmetry on human skin,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 63–72 (1991).

Lansiart, A.

J. M. Maarek, G. Jarry, B. de Losnac, A. Lansiart, B. M. Hung, “A simulation method for the study of laser transillumination of biological tissues,” Ann. Biomed. Eng. 12, 281–284 (1984).
[CrossRef] [PubMed]

Leerkotte, B.

M. H. Koelink, F. F. M. de Mul, B. Leerkotte, J. Greve, H. W. Jentink, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Signal processing for a laser Doppler perfusion meter,” Signal Process. 38, 239–252 (1994).
[CrossRef]

Maarek, J. M.

J. M. Maarek, G. Jarry, B. de Losnac, A. Lansiart, B. M. Hung, “A simulation method for the study of laser transillumination of biological tissues,” Ann. Biomed. Eng. 12, 281–284 (1984).
[CrossRef] [PubMed]

Nilsson, G. E.

G. E. Nilsson, T. Tenland, P. A. Oberg, “A new instrument for continuous measurement of blood flow by light beating spectroscopy,” IEEE Trans. Biomed. Eng. BME-27, 12–19 (1980).
[CrossRef]

Nossal, R.

Obeid, A. N.

A. N. Obeid, D. M. Boggett, N. J. Barnett, G. Dougherty, P. Rolfe, “Depth discrimination in laser Doppler skin blood flow measurement using different lasers,” Med. Biol. Eng. Comput. 26, 415–419 (1988).
[CrossRef] [PubMed]

Oberg, P. A.

G. E. Nilsson, T. Tenland, P. A. Oberg, “A new instrument for continuous measurement of blood flow by light beating spectroscopy,” IEEE Trans. Biomed. Eng. BME-27, 12–19 (1980).
[CrossRef]

Parrish, J. A.

R. R. Anderson, J. A. Parrish, “The optics of human skin,” J. Invest. Dermatol. 77, 13–20 (1981).
[CrossRef] [PubMed]

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–498 (1981).
[PubMed]

Patterson, M. S.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo-diffusion theory modelling of light distributions in tissue,” in Laser Interaction with Tissue, M. W. Berns, ed. Proc. Soc. Photo-Opt. Instrum. Eng. 908, 20–28 (1988).

Pecora, R.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), Chaps. 4 and 5.

Petty, C. C.

Prahl, S. A.

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

Rolfe, P.

A. N. Obeid, D. M. Boggett, N. J. Barnett, G. Dougherty, P. Rolfe, “Depth discrimination in laser Doppler skin blood flow measurement using different lasers,” Med. Biol. Eng. Comput. 26, 415–419 (1988).
[CrossRef] [PubMed]

Schalla, W.

L. Duteil, J. C. Bernego, W. Schalla, “A double wavelength laser Doppler system to investigate skin microcirculation,” IEEE Trans. Biomed. Eng. BME-32, 439–447 (1985).
[CrossRef]

Schmitt, J. M.

Star, W. M.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin Optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Sterenborg, H. J. C. M.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin Optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Stern, M. D.

M. D. Stern, “In vivo evaluation of microcirculation by coherent light scattering,” Nature (London) 254, 56–58 (1975).
[CrossRef]

Suichies, H. E.

H. W. Jentink, F. F. M. de Mul, R. Graaff, H. E. Suichies, J. G. Aarnoudse, J. Greve, “Laser Doppler flowmetry: measurements in a layered perfusion model and Monte Carlo simulations of measurements,” Appl. Opt. 30, 2592–2597 (1991).
[CrossRef] [PubMed]

H. W. Jentink, J. A. J. van Beurden, M. A. Helsdingen, F. F. M. de Mul, H. E. Suichies, J. G. Aarnoudse, J. Greve, “A compact differential laser Doppler velocimeter using a semiconductor laser,” J. Phys. E 20, 129–131 (1987).
[CrossRef]

Takatani, S.

S. Takatani, M. D. Graham, “Analysis of diffuse reflectance from a two layer tissue model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
[CrossRef]

ten Bosch, J. J.

Tenland, T.

G. E. Nilsson, T. Tenland, P. A. Oberg, “A new instrument for continuous measurement of blood flow by light beating spectroscopy,” IEEE Trans. Biomed. Eng. BME-27, 12–19 (1980).
[CrossRef]

Twersky, V.

van Beurden, J. A. J.

H. W. Jentink, J. A. J. van Beurden, M. A. Helsdingen, F. F. M. de Mul, H. E. Suichies, J. G. Aarnoudse, J. Greve, “A compact differential laser Doppler velocimeter using a semiconductor laser,” J. Phys. E 20, 129–131 (1987).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

van Gemert, M. J. C.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin Optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Walker, E. C.

Wan, S.

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–498 (1981).
[PubMed]

Weiss, G. H.

Welch, A. J.

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

Wilson, B. C.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo-diffusion theory modelling of light distributions in tissue,” in Laser Interaction with Tissue, M. W. Berns, ed. Proc. Soc. Photo-Opt. Instrum. Eng. 908, 20–28 (1988).

Zhou, G. X.

Zijlstra, W. G.

Ann. Biomed. Eng. (1)

J. M. Maarek, G. Jarry, B. de Losnac, A. Lansiart, B. M. Hung, “A simulation method for the study of laser transillumination of biological tissues,” Ann. Biomed. Eng. 12, 281–284 (1984).
[CrossRef] [PubMed]

Appl. Opt. (9)

R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, J. G. Aarnoudse, “Condensed Monte Carlo simulation for the description of light transport,” Appl. Opt. 32, 426–434 (1993).
[CrossRef] [PubMed]

R. F. Bonner, R. Nossal, “Model for laser Doppler measurements of blood flow in tissue,” Appl. Opt. 20, 2097–2107 (1981).
[CrossRef] [PubMed]

H. W. Jentink, F. F. M. de Mul, R. G. A. M. Hermsen, R. Graaff, J. Greve, “Monte Carlo simulations of laser Doppler blood flow measurements in tissue,” Appl. Opt. 29, 2371–2381 (1990).
[CrossRef] [PubMed]

H. W. Jentink, F. F. M. de Mul, R. Graaff, H. E. Suichies, J. G. Aarnoudse, J. Greve, “Laser Doppler flowmetry: measurements in a layered perfusion model and Monte Carlo simulations of measurements,” Appl. Opt. 30, 2592–2597 (1991).
[CrossRef] [PubMed]

R. Nossal, R. F. Bonner, G. H. Weiss, “Influence of path length on remote optical sensing of properties of biological tissue,” Appl. Opt. 28, 2238–2244 (1989).
[CrossRef] [PubMed]

M. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Laser Doppler blood flowmetry using two wavelengths: Monte-Carlo simulations and measurements,” Appl. Opt. 33, 3549–3558 (1994).
[CrossRef] [PubMed]

R. Graaff, A. C. M. Dassel, M. H. Koelink, F. F. M. de Mul, J. G. Aarnoudse, W. G. Zijlstra, “Optical properties of human dermis in vitro and in vivo,” Appl. Opt. 32, 435–447 (1993).
[CrossRef] [PubMed]

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, H. W. Jentink, “Light propagation parameters for anisotropic scattering media based on a rigorous solution of the transport equation,” Appl. Opt. 26, 2273–2279 (1989).
[CrossRef]

R. A. J. Groenhuis, H. A. Ferwerda, J. J. ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. I. Theory,” Appl. Opt. 22, 2456–2462 (1983).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

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

IEEE Trans. Biomed. Eng. (4)

L. Duteil, J. C. Bernego, W. Schalla, “A double wavelength laser Doppler system to investigate skin microcirculation,” IEEE Trans. Biomed. Eng. BME-32, 439–447 (1985).
[CrossRef]

S. Takatani, M. D. Graham, “Analysis of diffuse reflectance from a two layer tissue model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
[CrossRef]

G. E. Nilsson, T. Tenland, P. A. Oberg, “A new instrument for continuous measurement of blood flow by light beating spectroscopy,” IEEE Trans. Biomed. Eng. BME-27, 12–19 (1980).
[CrossRef]

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin Optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

J. Invest. Dermatol. (1)

R. R. Anderson, J. A. Parrish, “The optics of human skin,” J. Invest. Dermatol. 77, 13–20 (1981).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (2)

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

J. Phys. E (1)

H. W. Jentink, J. A. J. van Beurden, M. A. Helsdingen, F. F. M. de Mul, H. E. Suichies, J. G. Aarnoudse, J. Greve, “A compact differential laser Doppler velocimeter using a semiconductor laser,” J. Phys. E 20, 129–131 (1987).
[CrossRef]

Med. Biol. Eng. Comput. (1)

A. N. Obeid, D. M. Boggett, N. J. Barnett, G. Dougherty, P. Rolfe, “Depth discrimination in laser Doppler skin blood flow measurement using different lasers,” Med. Biol. Eng. Comput. 26, 415–419 (1988).
[CrossRef] [PubMed]

Nature (London) (1)

M. D. Stern, “In vivo evaluation of microcirculation by coherent light scattering,” Nature (London) 254, 56–58 (1975).
[CrossRef]

Photochem. Photobiol. (1)

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–498 (1981).
[PubMed]

Phys. Med. Biol. (1)

R. J. Gush, T. A. King, M. I. V. Jayson, “Aspects of laser light scattering from skin tissue with application to laser Doppler blood flow measurements,” Phys. Med. Biol. 29, 1463–1476 (1984).
[CrossRef] [PubMed]

Signal Process. (1)

M. H. Koelink, F. F. M. de Mul, B. Leerkotte, J. Greve, H. W. Jentink, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Signal processing for a laser Doppler perfusion meter,” Signal Process. 38, 239–252 (1994).
[CrossRef]

Other (4)

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), Chaps. 4 and 5.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

M. H. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M. Dassel, J. G. Aarnoudse, “Monte Carlo simulations and measurements of signals in laser Doppler flowmetry on human skin,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 63–72 (1991).

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo-diffusion theory modelling of light distributions in tissue,” in Laser Interaction with Tissue, M. W. Berns, ed. Proc. Soc. Photo-Opt. Instrum. Eng. 908, 20–28 (1988).

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Figures (13)

Fig. 1
Fig. 1

Optical probe A: dimensions, ϕ 5.0 cm × 2.0 cm; laser, Hitachi HL7806MG, 3 mW, 780 nm; gradient-index (grin) lens, Melles-Griot 06LGE214; Photodiode array, UDT A4V-4, 4 elements, each 2.0 mm × 1.5 mm. Source–detector distances: A, 2.8 mm; B, 4.8 mm; C, 6.8 mm; D, 8.8 mm (see text for locations of A–D).

Fig. 2
Fig. 2

Experimental flow model with liquid layers. Thicknesses: layers, 3.0 mm; glass plates, 2.0 mm. Overall dimensions: 10 cm × 5 cm.

Fig. 3
Fig. 3

Experimental flow model with gelatin layers: 1, support table; 2, marble bar; 3, platform; 4, rubber strip; 5, roller axis; 6–8, platforms; 9, gelatin sample; 10, probe; 11, 12, supports; 13, Hinge; 14, counterweight; 15, strip gear; 16, motor. For details about the gelatin layers and the polystyrene scatterers, see text.

Fig. 4
Fig. 4

Flow model with liquid layers: measured (○) and simulated (△) intensity versus the source–detector distance; (a) one layer; (b) two layers; mirror at backside; intensity normalization on the first detector (A).

Fig. 5
Fig. 5

Flow model with liquid layers: measured (△) and simulated (○, ▲) first weighted moment of the spectral power density of the detector signal versus the source–detector distance; (a) one layer; (b), (c) two layers, with flow in the (b) first layer; (c) second layer.

Fig. 6
Fig. 6

Flow model with gelatin layers: (●) measured and (□, ▽) simulated intensities; particle concentration, 1.9 × 106 mm−3; absorption, (□) 0.0253 mm−1; (▽) 0.0299 mm−1.

Fig. 7
Fig. 7

Flow model with gelatin layers. Relative intensity versus the source-detector distance for different thicknesses of the sample: +, 3 cm; △, 1 cm; ○, 0.5 cm. Curves are drawn to guide the eye.

Fig. 8
Fig. 8

Flow model with gelatin layers. Variation in the relative intensities (see Fig. 10) as a function of the angle of incidence of the laser beam. Detectors: +, B; △, C; ○, D. Lines are drawn to guide the eye.

Fig. 9
Fig. 9

Angular distribution function P(ϑ, ϕ), for detection at the X axis, where ϕ = 0 or π: Dependence on ϑ (a), (b) and ϕ (c), (d), either close to the source (x = 0–1 mm) (a), (c), or far away (x = 3–5 mm) (b), (d). Dependence on ϑ for different values of ϕ (e) along the +X axis, ϕ = −22.5°–22.5°, (f) along the −X axis, or ϕ = 157.5°–202.5°. In (a), (c) 110,000 photons, in (b), (d) 5611 photons, in (e) 40,100 photons, in (f) 7756 photons were analyzed.

Fig. 10
Fig. 10

(○) Number of detected photons, (□) averaged Doppler shift, and (△) averaged absolute Doppler shift as functions of the numerical aperture of the detector. The curves indicate linear and quadratic fits.

Fig. 11
Fig. 11

(a) Homodyne and (b) heterodyne power spectra of a moving layer with scatterers (see text for details). Detector B, velocity, 3.941 mm/s, was used. (c) Weighted first moment of the (○) frequency (average) and (△) zeroth moment as functions of the velocity.

Fig. 12
Fig. 12

Comparison between (——) measurements and (---) simulations for (a) homodyne and (b) heterodyne cases. Detector B, velocity 3.941 mm/s was used. In (b), no scaling was performed; in (a), scaling of the horizontal coordinate of the simulated spectrum with a factor of 3.0 (the spectrum shown corresponds in fact to that at 1.3 mm/s) was performed.

Fig. 13
Fig. 13

Simulations with the gelatin model: (a), (c) effects of the numerical aperture of the detector on the averaged Doppler shift, (b), (d) averaged absolute Doppler shift. The curves are linear and quadratic fits. In (c) and (d) a coherence correction factor cos2 ϑ u u is the polar angle) is applied. Curves (from top to bottom): +, q i ) = (cos2 ϑ i + 1)/2, n = 2; △, q i ) = (cos2 ϑ i + 1)/2, n = 1; ○, q i ) = (cos2 ϑ i + 1)/2, n = 1/2; ●, q i ) = cos2 ϑ i ; n = 1/2; ▲, q i ) = 1, n = 1/2 [Eqs. (16a)(16c)].

Equations (45)

Equations on this page are rendered with MathJax. Learn more.

Δ f = ( k v / π ) sin ½ ϑ cos α ,
v ~ ω = M 1 / M 0 ,             with M n - ω n S ( ω ) d ω .
g = 1 2 π ( 2 π ) f ( ϑ ) cos ϑ d ( cos ϑ ) ,
N ( i Δ ω ) = a i ,
a i = n s b - b + D ( ω ) d ω ,
S k ( k Δ ω ) = C β 2 Δ ω i = 0 n - k a i a i + k
Δ f = 1 2 π [ ( k 1 - k 0 ) · v + + ( k f - k f - 1 ) · v ] = 1 2 π ( k f - k 0 ) · v .
I ( r ) = exp ( - b r ) r + c r 2
z ( r ) = C 1 + C 2 r 2 / 3 ,
P ( ϑ ) = ½ sin ϑ .
P ( ϑ ) = ½ Q ( h ) d h sin ϑ exp - ( s + a ) h cos ϑ .
P ( ϑ ) = sin 2 ϑ .
Δ f = ( v / λ ) sin ϑ cos ϕ .
N = P ( ϑ , ϕ ) Ω , Δ f = v N λ P ( ϑ , ϕ ) sin ϑ cos ϕ Ω , Δ f = v N λ P ( ϑ , ϕ ) sin ϑ cos ϕ Ω ,
Δ f max = v / λ ,
Δ ω = d r 3 f A ( r ) f L ( r , r u ) k n · v d r 3 f A ( r ) f L ( r , r u ) ,
f A = f A ( r , ϑ i ) = exp ( - b r ) r - n q ( ϑ i ) ,
f L = exp [ - s + a ) r - r u ] ,
L eff , j = ( a , j + v = 1 N v C j v σ s v ) - 1 .
L abs , j = ( a , j + v = 1 N v C j v σ a v ) - 1 .
P = 1 - exp ( - x L eff , j - 1 ) .
x = - L eff , j ln ( 1 - R 1 ) .
f j v = C j v σ s v v C j v σ s v .
R 3 < σ abs , j σ s v + σ abs , j
R 4 < 1 - c v .
σ abs , j = abs , j C tot , j - 1 = ( L abs , j C tot , j ) - 1 .
0 2 π d ϕ s 0 π S ( ϑ s ) sin ϑ s d ϑ s = 1 ,
Q v k = m = 1 k ½ ( sin ϑ m - 1 S v , m - 1 + sin ϑ m S v m ) ( ϑ m - ϑ m - 1 ) ,
ϑ s = ϑ m ( R 5 - Q v , m - 1 ) + ϑ m - 1 ( Q v , m - R 5 ) Q v m - Q v , m - 1 .
ϕ s = R 6 2 π ,
E ( t ) = j = 0 N a j exp [ i ( j δ t + ϕ j ) ] ,
i ( n Δ t ) = c v = 0 N w = 0 N β v w ( a v a w ) 1 / 2 × exp { i [ δ ( w - v ) n Δ t + ϕ w - ϕ v ] } ,
i ( n Δ t ) = c β v = 0 N w = v N ( a v a w ) 1 / 2 × exp { i [ δ ( w - v ) n Δ t + ϕ w - ϕ v ] } + c . c . ,
f p k = c β ( a p a p + k ) 1 / 2 exp [ i ( ϕ p + k - ϕ p ) ] ,
i ( n Δ t ) = p = 0 N k = 0 N - p f p k exp ( i δ k n Δ t ) + c . c . ,
S ( t ) = i * ( 0 ) i ( t ) ,
S ( m Δ t ) = 1 N - m + 1 n = 0 N - m i ( n Δ t ) i [ ( n + m ) Δ t ] .
S ( j δ ) = m = 0 N S ( m Δ t ) exp ( - i j δ m Δ t ) .
S ( j δ ) = m = 0 N n = 0 N - m p = 0 N k = 0 N - p p = 0 N k = 0 N - p Δ t N - m + 1 × ( f p k f p k exp { i δ Δ t [ k n + k ( n + m ) - m j ] } + f p k * f p k * exp { - i δ Δ t [ k n + k ( n + m ) + m j ] } + f p k f p k * exp { i δ Δ t [ k n - k ( n + m ) - m j ] } + f p k * f p k * exp { - i δ Δ t [ k n - k ( n + m ) + n j ] } ) .
S ( j δ ) = m = 0 N n = 0 N - m p = 0 N k = 0 N - p p = 0 N k = 0 N - p Δ t N - m + 1 × f p k * f p k δ ( k - k ) δ ( k - j ) .
S ( j δ ) = m = 0 N n = 0 N - m p = 0 N - j p = 0 N - j Δ t N - m + 1 × f p j * f p j = p = 0 N - j p = 0 N - j f p j * f p j / Δ f ,
S ( j δ ) = c 2 β 2 Δ f p = 0 N - j p = 0 N - j ( a p a p a p + j a p + j ) 1 / 2 × exp [ i ( ϕ p + j - ϕ p + j + ϕ p - ϕ p ) ] .
S ( j δ ) = c 2 β 2 Δ f [ p = 0 N - j ( a p a p ) 1 / 2 ] 2 .
S ( j δ ) = c 2 β 2 Δ f p = 0 N - j ( a p a p + j a p a p + j ) 1 / 2 = c 2 β 2 Δ f p = 0 N - j a p a p + j .
S ( j δ ) het = c 2 β 2 Δ f a 0 a j .

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