Abstract

We present a detailed analytical model to describe optical coherence tomography (OCT) systems, which considers the propagation of the optical field within a scattering medium in the framework of the extended Huygens–Fresnel principle. The model includes use of the discrete-particle model and the fractal approach in treating biological tissue as being packed with scattering particles with a power-law distribution. In contrast to previous models, an imaginary lens proximal to the tissue surface is introduced that approximates the real focusing lens in the sample arm of the OCT system. This treatment avoids the consideration of backscattering light as traveling in the free space between the focusing lens and the tissue surface before mixing with the reference beam. Experiments on tissue phantoms were carried out to verify the validity of this model.

© 2003 Optical Society of America

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2002 (5)

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463 (2002).
[CrossRef] [PubMed]

R. K. Wang, J. B. Elder, “Propylene glycol as a contrasting agent for optical coherence tomography to image gastrointestinal tissues,” Lasers Surg. Med. 30, 201–208 (2002).
[CrossRef] [PubMed]

R. K. Wang, J. B. Elder, “High resolution optical tomographic imaging of soft biological tissues,” Laser Phys. 12, 611–616 (2002).

V. V. Tuchin, X. Xu, R. K. Wang, “Dynamic optical coherence tomography in studies of optical clearing, sedimentation, and appregation of immersed blood,” Appl. Opt. 41, 258–271 (2002).
[CrossRef] [PubMed]

E. Bordenave, E. Abraham, G. Jonusauskas, N. Tsurumachi, J. Oberle, C. Rulliere, P. E. Minot, M. Lassegues, J. E. S. Bazeille, “Wide-field optical coherence tomography: imaging of biological tissues,” Appl. Opt. 41, 2059–2064 (2002).
[CrossRef] [PubMed]

2001 (2)

R. K. Wang, X. Xu, V. V. Tuchin, J. B. Elder, “Concurrent enhancement of imaging depth and contrast for optical coherence tomography by hyperosmotic agents,” J. Opt. Soc. Am. B 18, 948–953 (2001).
[CrossRef]

U. Seitz, J. Freund, S. Jaeckle, E. Feldchtein, S. Bohnacker, F. Thonke, N. Gladkova, B. Brand, S. Schroder, N. Soehendra, “First in vivo optical coherence tomography in the human bile duct,” Endoscopy 33, 1018–1021 (2001).
[CrossRef] [PubMed]

2000 (3)

C. Pitris, C. Jesser, S. A. Boppart, D. Stamper, M. E. Brezinski, J. G. Fujimoto, “Feasibility of optical coherence tomography for high resolution imaging of human gastrointestinal tract malignancies,” J. Gastroenterol. 35, 87–92 (2000).
[CrossRef]

R. K. Wang, “Modelling optical properties of soft tissue by fractal distribution of scatterers,” J. Mod. Opt. 47, 103–120 (2000).
[CrossRef]

L. Thrane, H. T. Yura, P. E. Andersen, “Analysis of optical coherence tomography systems based on the extended Huygens–Fresenel principle,” J. Opt. Soc. Am. A 17, 484–490 (2000).
[CrossRef]

1999 (1)

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

1998 (5)

J. G. Fujimoto, B. Bouma, G. J. Tearney, S. A. Boppart, C. Pitris, J. F. Southern, M. E. Brezinski, “New technology for high speed and high resolution optical coherence tomography,” Ann. N.Y. Acad. Sci. 838, 95–107 (1998).
[CrossRef]

K. Kobayashi, J. A. Izzat, M. D. Kulkarni, J. Willis, M. V. Sivak, “High resolution cross sectional imaging of the gastrointestinal tract using optical coherence tomography: preliminary results,” Gastrointest. Endosc. 47, 515–523 (1998).
[CrossRef] [PubMed]

B. W. Colston, M. J. Everett, L. B. Da Silva, L. L. Otis, P. Stroeve, H. Nathel, “Imaging of hard- and soft-tissue structure in the oral cavity by optical coherence tomography,” Appl. Opt. 37, 3582–3585 (1998).
[CrossRef]

J. M. Schmitt, S. H. Xiang, “Cross-polarized backscatter in optical coherence tomography of biological tissue,” Opt. Lett. 23, 1060–1062 (1998).
[CrossRef]

J. M. Schmitt, G. Kumar, “Optical scattering properties of soft tissue: a discrete particle model,” Appl. Opt. 37, 2788–2797 (1998).
[CrossRef]

1997 (3)

1996 (2)

B. Gélébart, E. Tinet, J. M. Tualle, S. Avrillier, “Phase function simulation in tissue phantoms: a fractal approach,” Pure Appl. Opt. 5, 377–388 (1996).
[CrossRef]

H. T. Yura, S. G. Hanson, “Effects of receiver optics contamination on the performance of laser velocimeter systems,” J. Opt. Soc. Am. A 13, 1891–1902 (1996).
[CrossRef]

1995 (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,” Ophthalmology 102, 217–229 (1995).
[CrossRef] [PubMed]

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

Y. Pan, R. Birngruber, J. Rosperich, R. Engelhardt, “Low-coherence optical tomography in turbid tissue: theoretical analysis,” Appl. Opt. 34, 6564–6574 (1995).
[CrossRef] [PubMed]

1991 (2)

H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 30, 4507–4514 (1991).
[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]

1979 (1)

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

1971 (1)

Abraham, E.

Andersen, P. E.

L. Thrane, H. T. Yura, P. E. Andersen, “Analysis of optical coherence tomography systems based on the extended Huygens–Fresenel principle,” J. Opt. Soc. Am. A 17, 484–490 (2000).
[CrossRef]

L. Thrane, H. T. Yura, P. E. Andersen, “Optical coherence tomography: new analytical model and shower curtain effect,” in Saratov Fall Meeting ’99: Optical Technologies in Biophysics and Medicine, V. V. Tuchin, D. A. Zimnyakov, A. B. Pravdin, eds., Proc. SPIE4001, 202–208 (2000).
[CrossRef]

P. E. Andersen, L. Thrane, H. T. Yura, A. Tycho, T. M. Jørgensen, “Modeling the optical coherence tomography geometry using the extended Huygens–Fresnel principle and Monto Carlo simulations,” in Optical Pulse and Beam Propagation II, Y. B. Band, ed., Proc. SPIE3927, 166–178 (2000).
[CrossRef]

Avrillier, S.

B. Gélébart, E. Tinet, J. M. Tualle, S. Avrillier, “Phase function simulation in tissue phantoms: a fractal approach,” Pure Appl. Opt. 5, 377–388 (1996).
[CrossRef]

Bajraszewski, T.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463 (2002).
[CrossRef] [PubMed]

Bazeille, J. E. S.

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]

Bohnacker, S.

U. Seitz, J. Freund, S. Jaeckle, E. Feldchtein, S. Bohnacker, F. Thonke, N. Gladkova, B. Brand, S. Schroder, N. Soehendra, “First in vivo optical coherence tomography in the human bile duct,” Endoscopy 33, 1018–1021 (2001).
[CrossRef] [PubMed]

Bonner, R. F.

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

J. M. Schmitt, A. Knüttle, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissue 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.

C. Pitris, C. Jesser, S. A. Boppart, D. Stamper, M. E. Brezinski, J. G. Fujimoto, “Feasibility of optical coherence tomography for high resolution imaging of human gastrointestinal tract malignancies,” J. Gastroenterol. 35, 87–92 (2000).
[CrossRef]

J. G. Fujimoto, B. Bouma, G. J. Tearney, S. A. Boppart, C. Pitris, J. F. Southern, M. E. Brezinski, “New technology for high speed and high resolution optical coherence tomography,” Ann. N.Y. Acad. Sci. 838, 95–107 (1998).
[CrossRef]

Bordenave, E.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964), Chap. 8.

Bouma, B.

J. G. Fujimoto, B. Bouma, G. J. Tearney, S. A. Boppart, C. Pitris, J. F. Southern, M. E. Brezinski, “New technology for high speed and high resolution optical coherence tomography,” Ann. N.Y. Acad. Sci. 838, 95–107 (1998).
[CrossRef]

Brand, B.

U. Seitz, J. Freund, S. Jaeckle, E. Feldchtein, S. Bohnacker, F. Thonke, N. Gladkova, B. Brand, S. Schroder, N. Soehendra, “First in vivo optical coherence tomography in the human bile duct,” Endoscopy 33, 1018–1021 (2001).
[CrossRef] [PubMed]

Brezinski, M. E.

C. Pitris, C. Jesser, S. A. Boppart, D. Stamper, M. E. Brezinski, J. G. Fujimoto, “Feasibility of optical coherence tomography for high resolution imaging of human gastrointestinal tract malignancies,” J. Gastroenterol. 35, 87–92 (2000).
[CrossRef]

J. G. Fujimoto, B. Bouma, G. J. Tearney, S. A. Boppart, C. Pitris, J. F. Southern, M. E. Brezinski, “New technology for high speed and high resolution optical coherence tomography,” Ann. N.Y. Acad. Sci. 838, 95–107 (1998).
[CrossRef]

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]

Colston, B. W.

Da Silva, L. B.

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,” Ophthalmology 102, 217–229 (1995).
[CrossRef] [PubMed]

Elder, J. B.

R. K. Wang, J. B. Elder, “Propylene glycol as a contrasting agent for optical coherence tomography to image gastrointestinal tissues,” Lasers Surg. Med. 30, 201–208 (2002).
[CrossRef] [PubMed]

R. K. Wang, J. B. Elder, “High resolution optical tomographic imaging of soft biological tissues,” Laser Phys. 12, 611–616 (2002).

R. K. Wang, X. Xu, V. V. Tuchin, J. B. Elder, “Concurrent enhancement of imaging depth and contrast for optical coherence tomography by hyperosmotic agents,” J. Opt. Soc. Am. B 18, 948–953 (2001).
[CrossRef]

Engehardt, R.

J. Welzel, E. Lankenau, R. Birngruber, R. Engehardt, “Optical coherence tomography of the human skin,” J. Am. Acad. Dermatol. 37, 958–963 (1997).
[CrossRef]

Engelhardt, R.

Everett, M. J.

Feldchtein, E.

U. Seitz, J. Freund, S. Jaeckle, E. Feldchtein, S. Bohnacker, F. Thonke, N. Gladkova, B. Brand, S. Schroder, N. Soehendra, “First in vivo optical coherence tomography in the human bile duct,” Endoscopy 33, 1018–1021 (2001).
[CrossRef] [PubMed]

Fercher, A. F.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463 (2002).
[CrossRef] [PubMed]

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]

Freund, J.

U. Seitz, J. Freund, S. Jaeckle, E. Feldchtein, S. Bohnacker, F. Thonke, N. Gladkova, B. Brand, S. Schroder, N. Soehendra, “First in vivo optical coherence tomography in the human bile duct,” Endoscopy 33, 1018–1021 (2001).
[CrossRef] [PubMed]

Fujimoto, J. G.

C. Pitris, C. Jesser, S. A. Boppart, D. Stamper, M. E. Brezinski, J. G. Fujimoto, “Feasibility of optical coherence tomography for high resolution imaging of human gastrointestinal tract malignancies,” J. Gastroenterol. 35, 87–92 (2000).
[CrossRef]

J. G. Fujimoto, B. Bouma, G. J. Tearney, S. A. Boppart, C. Pitris, J. F. Southern, M. E. Brezinski, “New technology for high speed and high resolution optical coherence tomography,” Ann. N.Y. Acad. Sci. 838, 95–107 (1998).
[CrossRef]

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,” Ophthalmology 102, 217–229 (1995).
[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. Knüttle, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissue using low-coherence interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine III, H. Podbielska , ed., Proc. SPIE1889, 197–211 (1993).
[CrossRef]

Gélébart, B.

B. Gélébart, E. Tinet, J. M. Tualle, S. Avrillier, “Phase function simulation in tissue phantoms: a fractal approach,” Pure Appl. Opt. 5, 377–388 (1996).
[CrossRef]

Gladkova, N.

U. Seitz, J. Freund, S. Jaeckle, E. Feldchtein, S. Bohnacker, F. Thonke, N. Gladkova, B. Brand, S. Schroder, N. Soehendra, “First in vivo optical coherence tomography in the human bile duct,” Endoscopy 33, 1018–1021 (2001).
[CrossRef] [PubMed]

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]

Hanson, S. G.

Hee, M. R.

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,” Ophthalmology 102, 217–229 (1995).
[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]

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]

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,” Ophthalmology 102, 217–229 (1995).
[CrossRef] [PubMed]

Izzat, J. A.

K. Kobayashi, J. A. Izzat, M. D. Kulkarni, J. Willis, M. V. Sivak, “High resolution cross sectional imaging of the gastrointestinal tract using optical coherence tomography: preliminary results,” Gastrointest. Endosc. 47, 515–523 (1998).
[CrossRef] [PubMed]

Jaeckle, S.

U. Seitz, J. Freund, S. Jaeckle, E. Feldchtein, S. Bohnacker, F. Thonke, N. Gladkova, B. Brand, S. Schroder, N. Soehendra, “First in vivo optical coherence tomography in the human bile duct,” Endoscopy 33, 1018–1021 (2001).
[CrossRef] [PubMed]

Jesser, C.

C. Pitris, C. Jesser, S. A. Boppart, D. Stamper, M. E. Brezinski, J. G. Fujimoto, “Feasibility of optical coherence tomography for high resolution imaging of human gastrointestinal tract malignancies,” J. Gastroenterol. 35, 87–92 (2000).
[CrossRef]

Jonusauskas, G.

Jørgensen, T. M.

P. E. Andersen, L. Thrane, H. T. Yura, A. Tycho, T. M. Jørgensen, “Modeling the optical coherence tomography geometry using the extended Huygens–Fresnel principle and Monto Carlo simulations,” in Optical Pulse and Beam Propagation II, Y. B. Band, ed., Proc. SPIE3927, 166–178 (2000).
[CrossRef]

Knüttel, A.

J. M. Schmitt, A. Knüttel, “Model of optical coherence tomography of heterogeneous tissue,” J. Opt. Soc. Am. A 14, 1231–1242 (1997).
[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]

Knüttle, A.

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

Kobayashi, K.

K. Kobayashi, J. A. Izzat, M. D. Kulkarni, J. Willis, M. V. Sivak, “High resolution cross sectional imaging of the gastrointestinal tract using optical coherence tomography: preliminary results,” Gastrointest. Endosc. 47, 515–523 (1998).
[CrossRef] [PubMed]

Kowalczyk, A.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463 (2002).
[CrossRef] [PubMed]

Kulkarni, M. D.

K. Kobayashi, J. A. Izzat, M. D. Kulkarni, J. Willis, M. V. Sivak, “High resolution cross sectional imaging of the gastrointestinal tract using optical coherence tomography: preliminary results,” Gastrointest. Endosc. 47, 515–523 (1998).
[CrossRef] [PubMed]

Kumar, G.

Lankenau, E.

J. Welzel, E. Lankenau, R. Birngruber, R. Engehardt, “Optical coherence tomography of the human skin,” J. Am. Acad. Dermatol. 37, 958–963 (1997).
[CrossRef]

Lassegues, M.

Leitgeb, R.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463 (2002).
[CrossRef] [PubMed]

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,” Ophthalmology 102, 217–229 (1995).
[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]

Lutomirski, R. F.

Minot, P. E.

Moes, C. J. M.

Nathel, H.

Oberle, J.

Otis, L. L.

Pan, Y.

Pitris, C.

C. Pitris, C. Jesser, S. A. Boppart, D. Stamper, M. E. Brezinski, J. G. Fujimoto, “Feasibility of optical coherence tomography for high resolution imaging of human gastrointestinal tract malignancies,” J. Gastroenterol. 35, 87–92 (2000).
[CrossRef]

J. G. Fujimoto, B. Bouma, G. J. Tearney, S. A. Boppart, C. Pitris, J. F. Southern, M. E. Brezinski, “New technology for high speed and high resolution optical coherence tomography,” Ann. N.Y. Acad. Sci. 838, 95–107 (1998).
[CrossRef]

Prahl, S. A.

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,” Ophthalmology 102, 217–229 (1995).
[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]

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,” Ophthalmology 102, 217–229 (1995).
[CrossRef] [PubMed]

Rosperich, J.

Rulliere, C.

Schmitt, J. M.

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

J. M. Schmitt, S. H. Xiang, “Cross-polarized backscatter in optical coherence tomography of biological tissue,” Opt. Lett. 23, 1060–1062 (1998).
[CrossRef]

J. M. Schmitt, G. Kumar, “Optical scattering properties of soft tissue: a discrete particle model,” Appl. Opt. 37, 2788–2797 (1998).
[CrossRef]

J. M. Schmitt, A. Knüttel, “Model of optical coherence tomography of heterogeneous tissue,” J. Opt. Soc. Am. A 14, 1231–1242 (1997).
[CrossRef]

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

J. M. Schmitt, A. Knüttle, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissue 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]

Schroder, S.

U. Seitz, J. Freund, S. Jaeckle, E. Feldchtein, S. Bohnacker, F. Thonke, N. Gladkova, B. Brand, S. Schroder, N. Soehendra, “First in vivo optical coherence tomography in the human bile duct,” Endoscopy 33, 1018–1021 (2001).
[CrossRef] [PubMed]

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,” Ophthalmology 102, 217–229 (1995).
[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]

Seitz, U.

U. Seitz, J. Freund, S. Jaeckle, E. Feldchtein, S. Bohnacker, F. Thonke, N. Gladkova, B. Brand, S. Schroder, N. Soehendra, “First in vivo optical coherence tomography in the human bile duct,” Endoscopy 33, 1018–1021 (2001).
[CrossRef] [PubMed]

Sivak, M. V.

K. Kobayashi, J. A. Izzat, M. D. Kulkarni, J. Willis, M. V. Sivak, “High resolution cross sectional imaging of the gastrointestinal tract using optical coherence tomography: preliminary results,” Gastrointest. Endosc. 47, 515–523 (1998).
[CrossRef] [PubMed]

Soehendra, N.

U. Seitz, J. Freund, S. Jaeckle, E. Feldchtein, S. Bohnacker, F. Thonke, N. Gladkova, B. Brand, S. Schroder, N. Soehendra, “First in vivo optical coherence tomography in the human bile duct,” Endoscopy 33, 1018–1021 (2001).
[CrossRef] [PubMed]

Southern, J. F.

J. G. Fujimoto, B. Bouma, G. J. Tearney, S. A. Boppart, C. Pitris, J. F. Southern, M. E. Brezinski, “New technology for high speed and high resolution optical coherence tomography,” Ann. N.Y. Acad. Sci. 838, 95–107 (1998).
[CrossRef]

Stamper, D.

C. Pitris, C. Jesser, S. A. Boppart, D. Stamper, M. E. Brezinski, J. G. Fujimoto, “Feasibility of optical coherence tomography for high resolution imaging of human gastrointestinal tract malignancies,” J. Gastroenterol. 35, 87–92 (2000).
[CrossRef]

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]

Stroeve, P.

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,” Ophthalmology 102, 217–229 (1995).
[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.

J. G. Fujimoto, B. Bouma, G. J. Tearney, S. A. Boppart, C. Pitris, J. F. Southern, M. E. Brezinski, “New technology for high speed and high resolution optical coherence tomography,” Ann. N.Y. Acad. Sci. 838, 95–107 (1998).
[CrossRef]

Thonke, F.

U. Seitz, J. Freund, S. Jaeckle, E. Feldchtein, S. Bohnacker, F. Thonke, N. Gladkova, B. Brand, S. Schroder, N. Soehendra, “First in vivo optical coherence tomography in the human bile duct,” Endoscopy 33, 1018–1021 (2001).
[CrossRef] [PubMed]

Thrane, L.

L. Thrane, H. T. Yura, P. E. Andersen, “Analysis of optical coherence tomography systems based on the extended Huygens–Fresenel principle,” J. Opt. Soc. Am. A 17, 484–490 (2000).
[CrossRef]

P. E. Andersen, L. Thrane, H. T. Yura, A. Tycho, T. M. Jørgensen, “Modeling the optical coherence tomography geometry using the extended Huygens–Fresnel principle and Monto Carlo simulations,” in Optical Pulse and Beam Propagation II, Y. B. Band, ed., Proc. SPIE3927, 166–178 (2000).
[CrossRef]

L. Thrane, H. T. Yura, P. E. Andersen, “Optical coherence tomography: new analytical model and shower curtain effect,” in Saratov Fall Meeting ’99: Optical Technologies in Biophysics and Medicine, V. V. Tuchin, D. A. Zimnyakov, A. B. Pravdin, eds., Proc. SPIE4001, 202–208 (2000).
[CrossRef]

Tinet, E.

B. Gélébart, E. Tinet, J. M. Tualle, S. Avrillier, “Phase function simulation in tissue phantoms: a fractal approach,” Pure Appl. Opt. 5, 377–388 (1996).
[CrossRef]

Tsurumachi, N.

Tualle, J. M.

B. Gélébart, E. Tinet, J. M. Tualle, S. Avrillier, “Phase function simulation in tissue phantoms: a fractal approach,” Pure Appl. Opt. 5, 377–388 (1996).
[CrossRef]

Tuchin, V. V.

Tuchin, Valery

Valery Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis (SPIE Press, Bellingham, Washington, 2000).

Tycho, A.

P. E. Andersen, L. Thrane, H. T. Yura, A. Tycho, T. M. Jørgensen, “Modeling the optical coherence tomography geometry using the extended Huygens–Fresnel principle and Monto Carlo simulations,” in Optical Pulse and Beam Propagation II, Y. B. Band, ed., Proc. SPIE3927, 166–178 (2000).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1981).

van Gemert, M. J. C.

van Marle, J.

van Staveren, H. J.

Wang, R. K.

V. V. Tuchin, X. Xu, R. K. Wang, “Dynamic optical coherence tomography in studies of optical clearing, sedimentation, and appregation of immersed blood,” Appl. Opt. 41, 258–271 (2002).
[CrossRef] [PubMed]

R. K. Wang, J. B. Elder, “High resolution optical tomographic imaging of soft biological tissues,” Laser Phys. 12, 611–616 (2002).

R. K. Wang, J. B. Elder, “Propylene glycol as a contrasting agent for optical coherence tomography to image gastrointestinal tissues,” Lasers Surg. Med. 30, 201–208 (2002).
[CrossRef] [PubMed]

R. K. Wang, X. Xu, V. V. Tuchin, J. B. Elder, “Concurrent enhancement of imaging depth and contrast for optical coherence tomography by hyperosmotic agents,” J. Opt. Soc. Am. B 18, 948–953 (2001).
[CrossRef]

R. K. Wang, “Modelling optical properties of soft tissue by fractal distribution of scatterers,” J. Mod. Opt. 47, 103–120 (2000).
[CrossRef]

Welzel, J.

J. Welzel, E. Lankenau, R. Birngruber, R. Engehardt, “Optical coherence tomography of the human skin,” J. Am. Acad. Dermatol. 37, 958–963 (1997).
[CrossRef]

Willis, J.

K. Kobayashi, J. A. Izzat, M. D. Kulkarni, J. Willis, M. V. Sivak, “High resolution cross sectional imaging of the gastrointestinal tract using optical coherence tomography: preliminary results,” Gastrointest. Endosc. 47, 515–523 (1998).
[CrossRef] [PubMed]

Wojtkowski, M.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463 (2002).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964), Chap. 8.

Xiang, S. H.

Xu, X.

Yadlowsky, M.

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

Yura, H. T.

L. Thrane, H. T. Yura, P. E. Andersen, “Analysis of optical coherence tomography systems based on the extended Huygens–Fresenel principle,” J. Opt. Soc. Am. A 17, 484–490 (2000).
[CrossRef]

H. T. Yura, S. G. Hanson, “Effects of receiver optics contamination on the performance of laser velocimeter systems,” J. Opt. Soc. Am. A 13, 1891–1902 (1996).
[CrossRef]

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

R. F. Lutomirski, H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
[CrossRef] [PubMed]

P. E. Andersen, L. Thrane, H. T. Yura, A. Tycho, T. M. Jørgensen, “Modeling the optical coherence tomography geometry using the extended Huygens–Fresnel principle and Monto Carlo simulations,” in Optical Pulse and Beam Propagation II, Y. B. Band, ed., Proc. SPIE3927, 166–178 (2000).
[CrossRef]

L. Thrane, H. T. Yura, P. E. Andersen, “Optical coherence tomography: new analytical model and shower curtain effect,” in Saratov Fall Meeting ’99: Optical Technologies in Biophysics and Medicine, V. V. Tuchin, D. A. Zimnyakov, A. B. Pravdin, eds., Proc. SPIE4001, 202–208 (2000).
[CrossRef]

Ann. N.Y. Acad. Sci. (1)

J. G. Fujimoto, B. Bouma, G. J. Tearney, S. A. Boppart, C. Pitris, J. F. Southern, M. E. Brezinski, “New technology for high speed and high resolution optical coherence tomography,” Ann. N.Y. Acad. Sci. 838, 95–107 (1998).
[CrossRef]

Appl. Opt. (8)

H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 30, 4507–4514 (1991).
[CrossRef] [PubMed]

Y. Pan, R. Birngruber, R. Engelhardt, “Contrast limits of coherence-gated imaging in scattering media,” Appl. Opt. 36, 2979–2983 (1997).
[CrossRef] [PubMed]

J. M. Schmitt, G. Kumar, “Optical scattering properties of soft tissue: a discrete particle model,” Appl. Opt. 37, 2788–2797 (1998).
[CrossRef]

Y. Pan, R. Birngruber, J. Rosperich, R. Engelhardt, “Low-coherence optical tomography in turbid tissue: theoretical analysis,” Appl. Opt. 34, 6564–6574 (1995).
[CrossRef] [PubMed]

B. W. Colston, M. J. Everett, L. B. Da Silva, L. L. Otis, P. Stroeve, H. Nathel, “Imaging of hard- and soft-tissue structure in the oral cavity by optical coherence tomography,” Appl. Opt. 37, 3582–3585 (1998).
[CrossRef]

V. V. Tuchin, X. Xu, R. K. Wang, “Dynamic optical coherence tomography in studies of optical clearing, sedimentation, and appregation of immersed blood,” Appl. Opt. 41, 258–271 (2002).
[CrossRef] [PubMed]

E. Bordenave, E. Abraham, G. Jonusauskas, N. Tsurumachi, J. Oberle, C. Rulliere, P. E. Minot, M. Lassegues, J. E. S. Bazeille, “Wide-field optical coherence tomography: imaging of biological tissues,” Appl. Opt. 41, 2059–2064 (2002).
[CrossRef] [PubMed]

R. F. Lutomirski, H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
[CrossRef] [PubMed]

Dermatology (1)

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

Endoscopy (1)

U. Seitz, J. Freund, S. Jaeckle, E. Feldchtein, S. Bohnacker, F. Thonke, N. Gladkova, B. Brand, S. Schroder, N. Soehendra, “First in vivo optical coherence tomography in the human bile duct,” Endoscopy 33, 1018–1021 (2001).
[CrossRef] [PubMed]

Gastrointest. Endosc. (1)

K. Kobayashi, J. A. Izzat, M. D. Kulkarni, J. Willis, M. V. Sivak, “High resolution cross sectional imaging of the gastrointestinal tract using optical coherence tomography: preliminary results,” Gastrointest. Endosc. 47, 515–523 (1998).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

J. Am. Acad. Dermatol. (1)

J. Welzel, E. Lankenau, R. Birngruber, R. Engehardt, “Optical coherence tomography of the human skin,” J. Am. Acad. Dermatol. 37, 958–963 (1997).
[CrossRef]

J. Biomed. Opt. (1)

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463 (2002).
[CrossRef] [PubMed]

J. Gastroenterol. (1)

C. Pitris, C. Jesser, S. A. Boppart, D. Stamper, M. E. Brezinski, J. G. Fujimoto, “Feasibility of optical coherence tomography for high resolution imaging of human gastrointestinal tract malignancies,” J. Gastroenterol. 35, 87–92 (2000).
[CrossRef]

J. Mod. Opt. (1)

R. K. Wang, “Modelling optical properties of soft tissue by fractal distribution of scatterers,” J. Mod. Opt. 47, 103–120 (2000).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Laser Phys. (1)

R. K. Wang, J. B. Elder, “High resolution optical tomographic imaging of soft biological tissues,” Laser Phys. 12, 611–616 (2002).

Lasers Surg. Med. (1)

R. K. Wang, J. B. Elder, “Propylene glycol as a contrasting agent for optical coherence tomography to image gastrointestinal tissues,” Lasers Surg. Med. 30, 201–208 (2002).
[CrossRef] [PubMed]

Ophthalmology (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,” Ophthalmology 102, 217–229 (1995).
[CrossRef] [PubMed]

Opt. Acta (1)

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

Opt. Lett. (1)

Pure Appl. Opt. (1)

B. Gélébart, E. Tinet, J. M. Tualle, S. Avrillier, “Phase function simulation in tissue phantoms: a fractal approach,” Pure Appl. Opt. 5, 377–388 (1996).
[CrossRef]

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)

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964), Chap. 8.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1981).

J. M. Schmitt, A. Knüttle, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissue 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]

L. Thrane, H. T. Yura, P. E. Andersen, “Optical coherence tomography: new analytical model and shower curtain effect,” in Saratov Fall Meeting ’99: Optical Technologies in Biophysics and Medicine, V. V. Tuchin, D. A. Zimnyakov, A. B. Pravdin, eds., Proc. SPIE4001, 202–208 (2000).
[CrossRef]

P. E. Andersen, L. Thrane, H. T. Yura, A. Tycho, T. M. Jørgensen, “Modeling the optical coherence tomography geometry using the extended Huygens–Fresnel principle and Monto Carlo simulations,” in Optical Pulse and Beam Propagation II, Y. B. Band, ed., Proc. SPIE3927, 166–178 (2000).
[CrossRef]

Valery Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis (SPIE Press, Bellingham, Washington, 2000).

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Figures (10)

Fig. 1
Fig. 1

OCT system scanning a tissue medium.

Fig. 2
Fig. 2

Simplified sample arm in the OCT system.

Fig. 3
Fig. 3

Optical fields in the sample arm.

Fig. 4
Fig. 4

Heterodyne signal power I as a function of depth z of the tissue with all scatterings (solid curves) and with only single backscattering (dashed curves). Model parameters: F=5 mm, A0=0.5 mm, n¯=1.4. Concentration of the microspheres by weight is 0.9525%.

Fig. 5
Fig. 5

Heterodyne signal power I as a function of beam radius A0 of the light source and the numerical aperture NA with wavelengths of 820 and 1300 nm. Model parameters: focal length F=5 mm, n¯=1.4. Concentration of the microspheres by weight is 0.9525%. Probing depth (a) z=0.5 mm, (b) z=1.0 mm.

Fig. 6
Fig. 6

Heterodyne signal power I as a function of the scattering coefficient μs when the depth z=0.5 mm and z=1.0 mm. Model parameters: F=5 mm, A0=0.5 mm, n¯=1.4, λ=820 nm, g=0.95 (θrms=0.3).

Fig. 7
Fig. 7

Heterodyne signal power I as a function of the focal length F and the numerical aperture NA. Model parameters: A0=0.5 mm, n¯=1.4, λ=820 nm, probing depth z=0.5.

Fig. 8
Fig. 8

Heterodyne signal power I as a function of the sphere diameter d when the probing depth z=0.5 mm and the wavelength is 820 and 1300 nm. Model parameters: F=5 mm, A0=0.5 mm, n¯=1.4.

Fig. 9
Fig. 9

Theoretical curves from the current model (solid curves) and Thrane’s model (dashed curves) and experimental measurements (symbols) of the heterodyne signal power I in the polystyrene solutions as a function of the scattering coefficient μs when the depth z=0.5 mm and z=1.0 mm at the wavelength λ=1300 nm. Model parameters: A0=2.5 mm, n¯=1.4, focal length F=50 mm.

Fig. 10
Fig. 10

Theoretical curves from current model (solid curves) and experimental measurements (symbols) of the heterodyne signal power I in the intralipid solutions as a function of the scattering coefficient μs when the depth z=0.6 mm (circles) and z=1.2 mm (squares) at the wavelength λ=820 nm. Model parameters: A0=2.5 mm, n¯=1.4, focal length F=70 mm.

Equations (56)

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U(r)=AG(p, r)U(p)d2p,
G(p, r)=K(χ) exp[iks+ψ(p, r)]s=G0(p, r)UM(p, r),
UM(p, r)=exp[ψ(p, r)]
G0(p, r)=K(χ) exp(iks)s
U(r)=AG0(p, r)UM(p, r)U(p)d2p,
G0(p, r)=-ik2πzexp(ikz)expik2z (p-r)2.
a0=fF A0.
USi(p)=US0exp-p221a02+iknf,
Uf(r, z)=G0(p, r)UM(p, r)USi(p)d2p.
Ib(r, z)=σbIf (r, z).
If(r, z)=|Uf(r, z)|2=G0(p1, r)G0*(p2, r)×UM(p1, r)UM*(p2, r)×USi(p1)USi*(p2)d2p1d2p2
Γpt(p1, p2, r)=UM(p1, r)UM*(p2, r)=exp[ψ(p1, r)+ψ*(p2, r)]
Γpt(|p1-p2|, z)exp(-μsz)+[1-exp(-μsz)]×exp-|p1-p2|2ρ02,
ρ0=2μsz2kθrms,
If(r, z)=G0(p1, r)G0*(p2, r)Γpt(|p1-p2|, z)×ΓSi(p1, p2)d2p1d2p2,
If(r, z)=Ps0πexp(-μsz)ah21-πrahexp-r2ah2L1+1+exp(-μsz)as2×1-πrasexp-r2as2L2
ah=zka0,as2=ah2+2zkρ02,
ρ0=2μsz2kθrms;
L1=2πn=0rah2n+1n!(2n+1),
L2=2πn=0ras2n+1n!(2n+1),
Ub(r, z)=Ib(r, z)exp[iϕ(r, t)],
exp[iϕ(r1, t)]exp[iϕ(r2, t)]=4πk2 δ(r1-r2).
US(p, z)=rG0(r, p)UM(r, p)Ub(r, z)=rUS(p, r, z),
US(p, z)=rUS(p, r, z)=rIb(r, z)G0(r, p)UM(r, p)×exp[iϕ(r, t)].
i2(z)=2α2|g(τ)|2RΓS(p1, p2; z)×ΓR(p1, p2)d2p1d2p2,
ΓS(p1, p2; z)=US(p1, z)US*(p2, z),
ΓR(p1, p2)=UR(p1)UR(p2),
ΓS(p1, p2; z)=r1r2US(p1, r1, z)US*(p2, r2, z)=4πk2rIb(r, z)G0(r, p1)G0(r, p2)×Γpt(|p1-p2|, z).
i2(z)=2α2|g(τ)|2R4πk2rσbIf(r, z)IR(r, z),
IR(r, z)=G0(r, p1)G0(r, p2)Γpt(|p1-p2|, z)×ΓR(p1, p2)d2p1d2p2.
UR(p)=UR0exp-p221a02+ikf.
i2(z)=2α2|g(τ)|2PS0PR04πk2rσbexp(-μsz) 1ah2×1-πrahexp-r2ah2L1+[1+exp(-μsz)] 1as2×1-πrasexp-r2as2L22,
θrms2=0πsin2(θ)σ(θ)sin(θ)dθ0πσ(θ)sin(θ)dθ,
g=cos θ¯=0πcos(θ)σ(θ)sin(θ)dθ0πσ(θ)sin(θ)dθ=1-2θrms22,
θrms=2(1-g).
μs=Nσsca=NQscaG,
η(di)=η0di3-Df,
η0=Fvi=1mdi3-Df.
μ¯s=i=1mη(di)vi σsca(di)=6π η0i=1mdi-Dfσsca(di),
g¯=i=1mμs(di)g(di)i=1mμs(di)=i=1mdi-Dfσsca(di)g(di)i=1mdi-Dfσsca(di),
ρ¯0=2μsz2kθ¯rms,
G0(p1, r)G0*(p2, r)=k2πz2expikz [Pρ-rρ].
If(r, z)=k2πz2MSi(ρ)Γpt(ρ, z)×exp-ikzrρd2ρ,
MSi(ρ)=USiP+12 ρUSi*P-12 ρ×expikzPρd2P.
MSi(ρ)=US02exp-k2z2ah2ρ2S1,
ah2=a021-znf2+zka02,
S1=exp-1a02P+ika022z1-znfρ2d2P.
MSi(ρ)=US02πa02exp-k2z2ah2ρ2.
If(r, z)=2US02ka02z2exp(-μsz)exp-r2ah2S2+[1-exp(-μsz)]exp-r2as2S3,
as2=ah2+2zkρ02,
exp-r2ah2S2=exp-r2ah2exp-kah2z2×ρ+2izkah2 r2ρdρ=exp-kah2z2ρ2-ikzrρρdρ=122zkah21+iπrahexp-r2ah2×erfi rah-1,
exp-r2as2S3=exp-r2as2exp-kas2z2×ρ+2izkas2 r2ρdρ=exp-kas2z2ρ2-ikzrρρdρ=122zkas21+iπrasexp-r2as2×erfi ras-1,
erf(z)=2πn=0(-1)nz2n+1n!(2n+1).
If(r, z)=PS0πexp(-μsz)ah21-πrahexp-r2ah2L1+1+exp(-μsz)as2×1-πrasexp-r2as2L2,
L1=2πn=0(r/ah)2n+1n!(2n+1),
L2=2πn=0(r/as)2n+1n!(2n+1).

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