Abstract

A comprehensive model of optical coherence tomography (OCT) is described that includes the interference effects that produce speckle in images of dense heterogeneous tissue. It is based on the extended Huygens–Fresnel formulation of beam propagation in a turbulent atmosphere, adapted to the analysis of OCT. Incorporated in the model is a fractal description of the size distribution of scatterers in tissue. We demonstrate its application in the simulation of images of tissue volumes containing high-contrast targets embedded in a mixture of two sizes of particles. The simulated images show the degradation of image quality caused by speckle noise, along with the benefits of employing a light source with a short coherence time and an objective lens with a high numerical aperture. Based on model results, an estimate of the maximum probing depth is given in terms of the design variables of an OCT scanner and the optical properties of the tissue.

© 1997 Optical Society of America

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  1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [CrossRef] [PubMed]
  2. A. F. Fercher, K. Mengedoht, W. Werner, “Eye length measurement by interferometry with partially coherent light,” Opt. Lett. 13, 186–188 (1988).
    [CrossRef] [PubMed]
  3. C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).
  4. B. Bouma, G. J. Tearney, S. A. Boppart, M. R. Hee, M. E. Brezinski, J. G. Fujimoto, “High resolution optical coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source,” Opt. Lett. 20, 1486–1488 (1995).
    [CrossRef] [PubMed]
  5. J. M. Schmitt, M. Yadlowsky, R. F. Bonner, “Subsurface imaging of living skin with optical coherence microscopy,” Dermatology 191, 93–98 (1995).
  6. J. M. Schmitt, A. Knüttel, M. Yadlowsky, M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
    [CrossRef] [PubMed]
  7. J. M. Schmitt, A. Knüttel, R. F. Bonner, “Measurement of optical properties of biological tissues by low-coherence reflectometry,” Appl. Opt. 32, 6032–6042 (1993).
    [CrossRef] [PubMed]
  8. M. J. Yadlowsky, J. M. Schmitt, R. F. Bonner, “Multiple scattering in optical coherence microscopy,” Appl. Opt. 34, 5699–5707 (1995).
    [CrossRef] [PubMed]
  9. Y. T. Pan, R. Birngruber, J. Rosperich, R. Engelhardt, “Low-coherence optical tomography in turbid tissue: theoretical analysis,” Appl. Opt. 34, 6564–6574 (1995).
    [CrossRef] [PubMed]
  10. J. M. Schmitt, A. Knüttel, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 197–211 (1993).
    [CrossRef]
  11. J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
    [CrossRef] [PubMed]
  12. A. Knüttel, R. Schork, D. Böcker, “Analytical modeling of spatial resolution curves in turbid media acquired with optical coherence tomography (OCT),” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 258–270 (1996).
    [CrossRef]
  13. H. T. Yura, “Mutual coherence function of a finite cross section beam propagating in a turbulent medium,” Appl. Opt., 11, 1399–1406 (1972).
    [CrossRef] [PubMed]
  14. H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
    [CrossRef]
  15. C. M. Sonnenschein, F. A. Horrigan, “Signal-to-noise relationships for coaxial systems that heterodyne backscatter from the atmosphere,” Appl. Opt. 10, 1600–1604 (1971).
    [CrossRef] [PubMed]
  16. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2, pp. 448–454.
  17. J. M. Schmitt, G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. 21, 1310–1312 (1996).
    [CrossRef] [PubMed]
  18. G. Kumar, J. M. Schmitt, “Micro-optical properties of tissue,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases III: Optical Biopsy, R. R. Alfano, ed., Proc. SPIE2679, 106–116 (1996).
    [CrossRef]
  19. R. F. Lutomirski, “Atmospheric degradation of electrooptic system performance,” Appl. Opt. 17, 3915–3921 (1978).
    [CrossRef] [PubMed]
  20. Ref. 16, p. 318.
  21. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1977).
  22. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 121.
  23. A. Ishimaru, “Limitation on image resolution imposed by a random medium,” Appl. Opt. 17, 348–352 (1978).
    [CrossRef] [PubMed]

1996 (1)

1995 (5)

1994 (2)

J. M. Schmitt, A. Knüttel, M. Yadlowsky, M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

1993 (1)

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

1988 (1)

1979 (1)

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

1978 (2)

1972 (1)

1971 (1)

Birngruber, R.

Böcker, D.

A. Knüttel, R. Schork, D. Böcker, “Analytical modeling of spatial resolution curves in turbid media acquired with optical coherence tomography (OCT),” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 258–270 (1996).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 121.

Bonner, R. F.

J. M. Schmitt, M. Yadlowsky, R. F. Bonner, “Subsurface imaging of living skin with optical coherence microscopy,” Dermatology 191, 93–98 (1995).

M. J. Yadlowsky, J. M. Schmitt, R. F. Bonner, “Multiple scattering in optical coherence microscopy,” Appl. Opt. 34, 5699–5707 (1995).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, R. F. Bonner, “Measurement of optical properties of biological tissues by low-coherence reflectometry,” Appl. Opt. 32, 6032–6042 (1993).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 197–211 (1993).
[CrossRef]

Boppart, S. A.

Bouma, B.

Brezinski, M. E.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Duker, J. S.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

Eckhaus, M. A.

J. M. Schmitt, A. Knüttel, M. Yadlowsky, M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

Engelhardt, R.

Fercher, A. F.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

B. Bouma, G. J. Tearney, S. A. Boppart, M. R. Hee, M. E. Brezinski, J. G. Fujimoto, “High resolution optical coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source,” Opt. Lett. 20, 1486–1488 (1995).
[CrossRef] [PubMed]

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Gandjbakhche, A. S.

J. M. Schmitt, A. Knüttel, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 197–211 (1993).
[CrossRef]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hee, M. R.

B. Bouma, G. J. Tearney, S. A. Boppart, M. R. Hee, M. E. Brezinski, J. G. Fujimoto, “High resolution optical coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source,” Opt. Lett. 20, 1486–1488 (1995).
[CrossRef] [PubMed]

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Horrigan, F. A.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 121.

Ishimaru, A.

A. Ishimaru, “Limitation on image resolution imposed by a random medium,” Appl. Opt. 17, 348–352 (1978).
[CrossRef] [PubMed]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2, pp. 448–454.

Izatt, J. A.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

Knüttel, A.

J. M. Schmitt, A. Knüttel, M. Yadlowsky, M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, R. F. Bonner, “Measurement of optical properties of biological tissues by low-coherence reflectometry,” Appl. Opt. 32, 6032–6042 (1993).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 197–211 (1993).
[CrossRef]

A. Knüttel, R. Schork, D. Böcker, “Analytical modeling of spatial resolution curves in turbid media acquired with optical coherence tomography (OCT),” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 258–270 (1996).
[CrossRef]

Kumar, G.

J. M. Schmitt, G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. 21, 1310–1312 (1996).
[CrossRef] [PubMed]

G. Kumar, J. M. Schmitt, “Micro-optical properties of tissue,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases III: Optical Biopsy, R. R. Alfano, ed., Proc. SPIE2679, 106–116 (1996).
[CrossRef]

Lin, C. P.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Lutomirski, R. F.

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1977).

Mengedoht, K.

Owen, G. M.

Pan, Y. T.

Puliafito, C. A.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Reichel, E.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

Rosperich, J.

Schmitt, J. M.

J. M. Schmitt, G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. 21, 1310–1312 (1996).
[CrossRef] [PubMed]

M. J. Yadlowsky, J. M. Schmitt, R. F. Bonner, “Multiple scattering in optical coherence microscopy,” Appl. Opt. 34, 5699–5707 (1995).
[CrossRef] [PubMed]

J. M. Schmitt, M. Yadlowsky, R. F. Bonner, “Subsurface imaging of living skin with optical coherence microscopy,” Dermatology 191, 93–98 (1995).

J. M. Schmitt, A. Knüttel, M. Yadlowsky, M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, R. F. Bonner, “Measurement of optical properties of biological tissues by low-coherence reflectometry,” Appl. Opt. 32, 6032–6042 (1993).
[CrossRef] [PubMed]

G. Kumar, J. M. Schmitt, “Micro-optical properties of tissue,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases III: Optical Biopsy, R. R. Alfano, ed., Proc. SPIE2679, 106–116 (1996).
[CrossRef]

J. M. Schmitt, A. Knüttel, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 197–211 (1993).
[CrossRef]

Schork, R.

A. Knüttel, R. Schork, D. Böcker, “Analytical modeling of spatial resolution curves in turbid media acquired with optical coherence tomography (OCT),” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 258–270 (1996).
[CrossRef]

Schuman, J. S.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Sonnenschein, C. M.

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Tearney, G. J.

Werner, W.

Yadlowsky, M.

J. M. Schmitt, M. Yadlowsky, R. F. Bonner, “Subsurface imaging of living skin with optical coherence microscopy,” Dermatology 191, 93–98 (1995).

J. M. Schmitt, A. Knüttel, M. Yadlowsky, M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

Yadlowsky, M. J.

Yura, H. T.

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

H. T. Yura, “Mutual coherence function of a finite cross section beam propagating in a turbulent medium,” Appl. Opt., 11, 1399–1406 (1972).
[CrossRef] [PubMed]

Appl. Opt. (7)

Dermatology (1)

J. M. Schmitt, M. Yadlowsky, R. F. Bonner, “Subsurface imaging of living skin with optical coherence microscopy,” Dermatology 191, 93–98 (1995).

Opt. Acta (1)

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

Opt. Lett. (4)

Opthalmology (1)

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

Phys. Med. Biol. (1)

J. M. Schmitt, A. Knüttel, M. Yadlowsky, M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Other (7)

J. M. Schmitt, A. Knüttel, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 197–211 (1993).
[CrossRef]

A. Knüttel, R. Schork, D. Böcker, “Analytical modeling of spatial resolution curves in turbid media acquired with optical coherence tomography (OCT),” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 258–270 (1996).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2, pp. 448–454.

Ref. 16, p. 318.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1977).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 121.

G. Kumar, J. M. Schmitt, “Micro-optical properties of tissue,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases III: Optical Biopsy, R. R. Alfano, ed., Proc. SPIE2679, 106–116 (1996).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Optical coherence tomography system scanning a tissue medium. The lens is focused at a distance z below the surface. During the axial and lateral scans, the optical path in the reference arm is adjusted such that it always equals the optical distance to the focal plane, L=d+nz, where n is the mean index of the tissue.

Fig. 2
Fig. 2

Geometry for calculation of the mutual coherence and interference of the propagating beams. Interference is assumed to take place in the plane of the objective lens.

Fig. 3
Fig. 3

Simulated tissue volume. In the simulations reported in this paper, the voxel size was set at 3 μm×3 μm×3 μm, and the target slice in the center was 3 voxels (9 μm) wide. For the calculation of the scattering coefficient μs(ri, zi) and the mean scattering angle ρ0(ri, zi), the numbers of particles of each size of sphere within the cone of illumination (shaded area) were counted.

Fig. 4
Fig. 4

(a) Backscatter map and (b) attenuation map for a simulated tissue volume containing spheres with two different diameters (0.4 μm and 6.4 μm). The volume fractions of the small-size and large-size spheres were 0.05 and 0.022, respectively. The labels are in units of pixels, with each pixel equal to 3 μm. The brightness of each pixel in the backscatter map is linearly proportional to μb(ri, zi) in the target slice; in the attenuation map, the brightness of each pixel is proportional to the optical depth μs(ri, zi)z calculated over the cone of illumination of a NA=0.1 lens (see Fig. 3). The striations evident in the attenuation map result from shadowing by the large particles.

Fig. 5
Fig. 5

Simulated optical coherence images: (a) ideal speckle-free image of the target slice [same as Fig. 4(a)], (b) image with speckle (no ensemble averaging), (c) image with speckle removed, (d) ensemble average of four images. The labels are in units of pixels, with each pixel equal to 3 μm. Beside each of the images are profiles taken through the targets along the diagonals of the images. Model parameters: R=2 mm, f=20 mm, Fv,0.4 μm=0.05, Fv,6.4 μm=0.022, cτc=15 μm, λ=1300 nm, n=1.35.

Fig. 6
Fig. 6

Effect of source coherence time on simulated optical coherence images: (a) cτc=30 μm, (b) cτc=15 μm, and (c) cτc =5 μm. The labels are in units of pixels, with each pixel equal to 3 μm. Beside each of the images are profiles taken through the targets along the diagonals of the images. Model parameters: R=2 mm, f=20 mm, Fv,0.4 μm=0.05, Fv,6.4 μm=0.022, λ =1300 nm, n=1.35.

Fig. 7
Fig. 7

Effect of particle density on simulated optical coherence images: (a) Fv,6.4 μm=0.022, (b) Fv,6.4 μm=0.06, (c) Fv,6.4 μm=0.1. Beside each of the images are profiles taken through the targets along the diagonals of the images. Model parameters: R=2 mm, f=20 mm, Fv,0.4 μm=0.05, cτc=15 μm, λ=1300 nm, n=1.35.

Fig. 8
Fig. 8

Effect of lens NA on the simulated optical coherence images: (a) NA=0.1, (b) NA=0.2, (c) NA=0.4. Beside each of the images are profiles taken through the targets along the diagonals of the images. To simulate the change in NA, the focal length f was fixed and R was varied. Model parameters: f=20 mm, Fv,0.4 μm=0.05, Fv,6.4 μm=0.1, cτc=15 μm, λ=1300 nm, n=1.35.

Equations (39)

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is=2ηq ReAUr(ρ, t)Us*(ρ, t+τ)dρ.
is=2ηq ReAUr(ρ)Us*(ρ)dρ|g(τ)|fortτc,
Us(r, L)=Psπ k2LR |g(τ)|exp-μsz2×expi k2L (r-p)2-i k2nf p2-p2R2dp,
Us(p, 0)=Us(r, L)σbL exp(-μsz)×expi k2L (r-p)2-i k2nf p-ikL.
Ur(p, 0)=Pr(πR)1/2 exp-p22R2-i k2 f.
isb(r, z)=PsPrηq|g(τ)| σbkas2 exp(-2μsz)×exp-|r|2as2cos2kL+2k|r|2L ϕ+2 tan-12z0L 1-Lnf,
as2=R2Lz02+1-Lnf2,
ϕ=1-z02Lnf 1-Lnf1+z02L2 1-Lnf2,
z0=kR2,
is2=2η2q2|g(τ)|2Γs(r1, r2,z)Γr(r1, r2, z)dr1 dr2,
Γr(r1, r2, z)=Ur(r1)Ur*(r2),
Γs(r1, r2, z)=Us(r1)Us*(r2).
Γr(r, rd, z)=PrπR2 exp-(|r|2+0.25|rd|2)R2-i knf (rrd),
Γs(r, rd, L)=k2(2πL)2 Γ0 expi kL (rd-pd)(r-p)-Hdpdpd,
Γ0=PsπR2 exp-(|p|2+0.25|pd|2)R2-i knf (ppd).
H=1ρ02 [3rdpd+(rd-pd)2].
ρ0=3μsz2kθrms.
Γms(r, rd, L)=Iom(r0, z)σb4πL2 [1-exp(-μsz)]×exp(-rd2/ρ02)×expi kL rd(r-r0),
Iom(r, z)=Psπam2 [1-exp(-μsz)]exp(-|r|2/am2),
am2=R2Lz02+1-Lnf2+2LkRρ02,
z0=kR2.
ims2(r, z)=PsPrη2q2|g(τm)|2 σbk2am4 [1-exp(-μsz)]2×exp(-2|r|2/am2)cos2[2k(L+Φ)].
τm=2nc (L-f+Φ),
iT,sb(r, z)=i=1misb(ri, zi).
iT,ms2(r, z)=i=1mims2(ri, zi),
|is(r, z)|=|iT,sb|+iT,ms21/2.
N=(do/di)Df,
η(di)=(πdi3/6) Nπdo3/6=N0di3-Df,
Fv=N0 dmindmax di3-Df ddiN0=Fvdmindmax di3-Df ddi=Fv(4-Df)dmax4-Df-dmin4-Df.
Fv=N0i=1mdi3-DfN0=Fvi=1m di3-Df.
μs=dmindmaxη(di) σs(di)vi ddi,
μs=i=1mη(di) σs(di)vi=6π N0i=1mdi-Dfσs(di),
θ¯rms=i=1mdi-Dfσs(di)θrms(di)i=1mdi-Dfσs(di),
ρ¯0=3μsz2kθ¯rms,
σb(di)=σs(di)|Si(180°)|2,
μb=6π N0i=1mdi-Dfσs(di)Pi(180°).
isbims=am2as2 exp(-2μsz)[1-exp(-μsz)]2=(1+2R2/ρ02) exp(-2μsz)[1-exp(-μsz)]2
isb/ims(1+2R2/ρ02)exp(-2μsz).
zmax12μs ln1+2R2ρ02.

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