Abstract

A Monte Carlo (MC) method for modeling optical coherence tomography (OCT) measurements of a diffusely reflecting discontinuity embedded in a scattering medium is presented. For the first time to the authors’ knowledge it is shown analytically that the applicability of an MC approach to this optical geometry is firmly justified, because, as we show, in the conjugate image plane the field reflected from the sample is delta-correlated from which it follows that the heterodyne signal is calculated from the intensity distribution only. This is not a trivial result because, in general, the light from the sample will have a finite spatial coherence that cannot be accounted for by MC simulation. To estimate this intensity distribution adequately we have developed a novel method for modeling a focused Gaussian beam in MC simulation. This approach is valid for a softly as well as for a strongly focused beam, and it is shown that in free space the full three-dimensional intensity distribution of a Gaussian beam is obtained. The OCT signal and the intensity distribution in a scattering medium have been obtained for several geometries with the suggested MC method; when this model and a recently published analytical model based on the extended Huygens-Fresnel principle are compared, excellent agreement is found.

© 2002 Optical Society of America

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    [CrossRef] [PubMed]
  36. L. Thrane, Optical Coherence Tomography: Modeling and Applications, Ph.D. dissertation (Risø National Laboratory, Roskilde, Denmark, 2001).
  37. H. T. Yura, L. Thrane, P. E. Andersen, “Closed-form solution for the Wigner phase-space distribution function for diffuse reflection and small-angle scattering in a random medium,” J. Opt. Soc. Am. A 17, 2464–2474 (2000).
    [CrossRef]

2002 (1)

2000 (2)

1999 (4)

1998 (3)

1997 (1)

1996 (3)

C. Sturesson, S. Andersson-Engels, “Mathematical modeling of dynamic cooling and pre-heating, used to increase the depth of selective damage to blood vessels in laser treatment of port wine stains,” Phys. Med. Biol. 41, 413–428 (1996).
[CrossRef] [PubMed]

J. M. Schmitt, K. Ben-Letaief, “Efficient Monte Carlo simulation of confocal microscopy in biological tissue,” J. Opt. Soc. Am. A 13, 952–961 (1996).
[CrossRef]

A. K. Dunn, C. Smithpeter, A. J. Welch, R. Richards-Kortum, “Sources of contrast in confocal reflectance imaging,” Appl. Opt. 35, 3441–3446 (1996).
[CrossRef] [PubMed]

1995 (2)

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]

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

1994 (1)

J. Schmitt, A. Knüttel, M. Yadlowski, “Confocal microscopy in turbid media,” J. Opt. Soc. A 11, 2226–2235 (1994).
[CrossRef]

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]

1990 (2)

W. Denk, J. Strickler, W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

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

1989 (1)

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]

1988 (1)

J. Manyak, A. Russo, P. Smith, E. Glatstain, “Photodynamic therapy,” J. Clin. Oncol. 6, 380–391 (1988).
[PubMed]

1987 (1)

1983 (1)

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

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]

1974 (1)

N. Bloembergen, “Laser-induced electric breakdown in solids,” IEEE J. Quantum Electron. QE-10, 375–386 (1974).
[CrossRef]

1966 (1)

Adam, G.

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Andersen, P. E.

Andersson-Engels, S.

C. Sturesson, S. Andersson-Engels, “Mathematical modeling of dynamic cooling and pre-heating, used to increase the depth of selective damage to blood vessels in laser treatment of port wine stains,” Phys. Med. Biol. 41, 413–428 (1996).
[CrossRef] [PubMed]

Ben-Letaief, K.

Birngruber, R.

Blanca, C. M.

Bloembergen, N.

N. Bloembergen, “Laser-induced electric breakdown in solids,” IEEE J. Quantum Electron. QE-10, 375–386 (1974).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).

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]

Chen, Z.

D. J. Smithies, T. Lindmo, Z. Chen, S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulations,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Cheong, W.

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

Denk, W.

W. Denk, J. Strickler, W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Dong, K.

Dror, I.

Dunn, A. K.

Engelhardt, R.

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.

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]

Glatstain, E.

J. Manyak, A. Russo, P. Smith, E. Glatstain, “Photodynamic therapy,” J. Clin. Oncol. 6, 380–391 (1988).
[PubMed]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

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.

Harris, T. E.

H. Kahn, T. E. Harris, “Estimation of particle transmission by random sampling,” in Monte Carlo Method, Vol. 12 of National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, D.C., 1951), pp. 27–30.

Hee, M. R.

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]

Hu, X. H.

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, 1998).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, in IEEE/OUP Series on Electromagnetic Wave Theory (IEEE Press, New York, 1997).

Jacques, S. L.

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

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]

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model for light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. 155 of SPIE Institute Series (SPIE, Bellingham, Wash., 1998) pp. 102–111.

Jørgensen, T. M.

Kahn, H.

H. Kahn, T. E. Harris, “Estimation of particle transmission by random sampling,” in Monte Carlo Method, Vol. 12 of National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, D.C., 1951), pp. 27–30.

Keijzer, M.

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model for light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. 155 of SPIE Institute Series (SPIE, Bellingham, Wash., 1998) pp. 102–111.

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]

J. Schmitt, A. Knüttel, M. Yadlowski, “Confocal microscopy in turbid media,” J. Opt. Soc. A 11, 2226–2235 (1994).
[CrossRef]

Kopeika, N. S.

Liang, G.

Lin, C. P.

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]

Lindmo, T.

D. J. Smithies, T. Lindmo, Z. Chen, S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulations,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Lu, J. Q.

Manyak, J.

J. Manyak, A. Russo, P. Smith, E. Glatstain, “Photodynamic therapy,” J. Clin. Oncol. 6, 380–391 (1988).
[PubMed]

Milner, T. E.

D. J. Smithies, T. Lindmo, Z. Chen, S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulations,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Nelson, S.

D. J. Smithies, T. Lindmo, Z. Chen, S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulations,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Pan, Y.

Prahl, S.

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

Prahl, S. A.

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model for light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. 155 of SPIE Institute Series (SPIE, Bellingham, Wash., 1998) pp. 102–111.

Puliafito, C. A.

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]

Richards-Kortum, R.

Rosperich, J.

Russo, A.

J. Manyak, A. Russo, P. Smith, E. Glatstain, “Photodynamic therapy,” J. Clin. Oncol. 6, 380–391 (1988).
[PubMed]

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Saloma, C.

Sandrov, A.

Schmitt, J.

J. Schmitt, A. Knüttel, M. Yadlowski, “Confocal microscopy in turbid media,” J. Opt. Soc. A 11, 2226–2235 (1994).
[CrossRef]

Schmitt, J. M.

Schuman, J. S.

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]

Siegman, A. E.

Smith, P.

J. Manyak, A. Russo, P. Smith, E. Glatstain, “Photodynamic therapy,” J. Clin. Oncol. 6, 380–391 (1988).
[PubMed]

Smithies, D. J.

D. J. Smithies, T. Lindmo, Z. Chen, S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulations,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Smithpeter, C.

Song, Z.

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]

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]

Strickler, J.

W. Denk, J. Strickler, W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Sturesson, C.

C. Sturesson, S. Andersson-Engels, “Mathematical modeling of dynamic cooling and pre-heating, used to increase the depth of selective damage to blood vessels in laser treatment of port wine stains,” Phys. Med. Biol. 41, 413–428 (1996).
[CrossRef] [PubMed]

Swanson, E. A.

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]

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Thrane, L.

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

H. T. Yura, L. Thrane, P. E. Andersen, “Closed-form solution for the Wigner phase-space distribution function for diffuse reflection and small-angle scattering in a random medium,” J. Opt. Soc. Am. A 17, 2464–2474 (2000).
[CrossRef]

L. Thrane, Optical Coherence Tomography: Modeling and Applications, Ph.D. dissertation (Risø National Laboratory, Roskilde, Denmark, 2001).

H. T. Yura, L. Thrane, “The effects of multiple scattering on axial resolution in optical coherence tomography,” in Conference on Lasers and Electro-Optics, Vol. 73 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2002), paper CThI5.

Tycho, A.

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]

Wang, L. V.

L. V. Wang, G. Liang, “Absorption distribution of an optical beam focused into a turbid medium,” Appl. Opt. 38, 4951–4958 (1999).
[CrossRef]

G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef] [PubMed]

Wang, L.-H.

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Webb, W.

W. Denk, J. Strickler, W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Welch, A. J.

A. K. Dunn, C. Smithpeter, A. J. Welch, R. Richards-Kortum, “Sources of contrast in confocal reflectance imaging,” Appl. Opt. 35, 3441–3446 (1996).
[CrossRef] [PubMed]

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

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model for light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. 155 of SPIE Institute Series (SPIE, Bellingham, Wash., 1998) pp. 102–111.

Wilson, B. C.

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Yadlowski, M.

J. Schmitt, A. Knüttel, M. Yadlowski, “Confocal microscopy in turbid media,” J. Opt. Soc. A 11, 2226–2235 (1994).
[CrossRef]

Yao, G.

G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef] [PubMed]

Yura, H. T.

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

H. T. Yura, L. Thrane, P. E. Andersen, “Closed-form solution for the Wigner phase-space distribution function for diffuse reflection and small-angle scattering in a random medium,” J. Opt. Soc. Am. A 17, 2464–2474 (2000).
[CrossRef]

H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
[CrossRef]

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

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Figures (9)

Fig. 1
Fig. 1

Sample arm setup of the OCT system. Lenses L1 and L2 are considered to be identical and perfect and to have infinite radius. The setup is essentially a 4F system.

Fig. 2
Fig. 2

Transverse intensity distributions in the focal plane for case 1 (z f = 0.5 mm). (a), (b) distributions for g = 0.92 and g = 0.99, respectively. Symbols indicate results of the EHF model; curves represent results of the MC simulation. The distributions are normalized according to the value obtained with (q, z, g, μ s ) = (0, 0, 0.99, 5 mm-1): filled squares and dotted curves, μ s = 5 mm-1; filled downward triangles and solid curves, μ s = 15 mm-1; filled upward triangles and dashed curves, μ s = 30 mm-1.

Fig. 3
Fig. 3

Transverse intensity distributions in the focal plane for case 2 (z f = 0.5 mm). (a), (b) Distributions for g = 0.92 and g = 0.99, respectively. Symbols indicate results of the EHF model; curves represent results of the MC simulation. The distributions are normalized according to the value obtained with (q, z, g, μ s ) = (0, 0, 0.99, 5 mm-1): filled squares and dotted curves, μ s = 5 mm-1; filled downward triangles and solid curves, μ s = 15 mm-1; filled upward triangles and dashed curves, μ s = 30 mm-1.

Fig. 4
Fig. 4

Heterodyne efficiency factors estimated from the EHF model and the MC method presented here for two cases of the anisotropy parameter g. (a), (b), (c), (d) estimated values for geometries 1, 2, 3, and 4, respectively. In all cases z f = 0.5 mm, except for case 4, for which z f = 1.0 mm. Solid and dotted curves, results of the EHF model for g = 0.99 and g = 0.92, respectively. Dashed-double-dotted and dashed curves, results of the MC simulations for g = 0.99 and g = 0.92, respectively. Filled diamonds and filled squares, actual data points obtained by the MC simulation method. As a reference, the exponential reduction in signal that is due to scattering obtained by a single-scatter model is shown as dashed-dotted curves.

Fig. 5
Fig. 5

Relative numerical difference between the results of the EHF model and the MC approach that are depicted in Fig. 7 below for a representative selection of the geometries considered here. The ratio ΨEHFMC is plotted for case 1 and g = 0.99 by filled diamonds and a solid curve; for case 2 and g = 0.92 by filled squares and a dashed-double-dotted curve; for case 2 and g = 0.92 by filled upward triangles and a dashed curve; and for case 5 and g = 0.92 by filled downward triangles and a dotted curve.

Fig. 6
Fig. 6

Heterodyne efficiency factors estimated by the EHF model and the new MC method when the sample consists of two layers and the beam is given according to case 2 (z f = 0.5 mm). The top layer has a constant scattering coefficient, and Ψ is plotted as a function of the scattering coefficient in the second layer. Dashed curve, the result of applying the EHF model; filled squares and the solid curve, the result of applying the MC method.

Fig. 7
Fig. 7

Transverse intensity distributions in the focal plane for beams described by (a) case 1 and (b) case 3 propagating in a random medium according to the text. Symbols, results from the EHF-intensity model; curves, results of the MC simulation. The distributions are normalized with the value obtained for (q, μ s ) = (0, 5 mm-1). Filled squares and dotted curve, μ s = 5 mm-1; filled downward triangles and solid curves, μ s = 15 mm-1; filled upward triangles and dashed curves, μ s = 30 mm-1.

Fig. 8
Fig. 8

Axial focus of a beam described by case 5 (z f = 1.0 mm) in Table 1 propagating in a random medium described in the text. All distributions have been normalized to unity for (q, z) = (0, 0). (a) Axial intensity estimated by the geometrical-focus method. Dashed curve, larger grid; dotted curve, smaller grid (see text). (b) Similar curves obtained with the hyperboloid method. Solid curve, the intensity distribution obtained from the intensity-EHF model. (c) Transverse intensity distribution (small grid) in the focal plane: dotted curve, geometrical-focus method; dashed curve, hyperboloid method; solid curve, intensity-EHF model.

Fig. 9
Fig. 9

Heterodyne efficiency factors estimated by the EHF model and a MC method including an angular criterion for case 2 (z f = 0.5 mm), respectively. Dotted curve, result of applying the EHF model; filled squares and dashed-dotted curve, result of applying the MC method.

Tables (1)

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Table 1 Beam Geometries Chosen for Numerical Investigations

Equations (35)

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i2=2α2|gτ|2 Re ΓRr1, r2ΓSr1, r2dr1dr2Ψri02,
ρ03μszfλπθrms1+nd/zf,
Ψp=i2i02= IRpISpd2p IRpIS0pd2p=Ψr,
w2z=wf21+ z-zf22z02,
xz=±wf2z0z-zf.
wphoton2z=C2wf21+z-zf22z02,
pr=2/w02exp-r2/w02rdr,
r=w0 ln1/ξ=w-d ln1/ξ,
Ψp= IRpISpd2p IRpIS0pd2p.
Irθr=IT cos θr.
θr=arcsinξ,
φr=2πζ,
IS,m=wmΔp2,
ΨMC=mM IRpmIS,mΔp2i02=mM IRpmwmi02,
str=μszf1-g.
Iq= k2πB2KρexpikBρ·qΓPTρd2ρ.
K ρ=exp-ikABρ·RUSiR+ρ/2×USi*R-ρ/2d2R,
R=½ r+r, ρ=r-r,
ΓPTr-r=expiφr, q-iφr, q=exp-μsz1-bφ|r-r|,
bφx=π/2n1f1-gkxzerf1-gkxz2n1f,
Kρ=PS exp-ρ24w02,
Ψr= |Kρ|2|ΓPTρ|2d2ρ |Kρ|2d2ρ.
Ψp=i2i02= IRpISpd2p IRpIS0pd2p,
Ψr=Re ΓRr1, r2ΓSr1, r2dr1dr2Re ΓRr1, r2ΓS0r1, r2dr1dr2=Ψp.
USr, p= UrrGr-pr, pd2r,
G0r, p=-k2πBexp-ik2BAr2-2r·p+Dp2,
Urr= ηqUqqGfq, rexpiφq, rd2q,
ΓSp1, p2= Gfq, rGf*q, r×Gr-pr, p1Gr-p*r, p2×UqqUq*qηqηq×expiφq, r-iφq, r×d2rd2rd2qd2q,
ΓSp1, p2= Gfq, rGf*q, rGr-pr, p1×Gr-p*r, p2Iqq×ΓPTr-rd2rd2rd2q,
ΓSp2, p2=exp-ik2fp22-p12-ρ·p1+p2+2R·p-p1|ΓPTρ|2K-ρd2ρd2R,
expim·u+vd2m=2π2δu+v.
ΓSp2, p2=δp2-p1exp-ik2fp22-p12-ρ·p1+p2|ΓPTρ|2K-ρd2ρ,
ISp=Alensexp-ikfρ·p|ΓPTρ|2K-ρd2ρ.
i2=exp-ik2f-2ρ·p|ΓPTρ|2K-ρ×IRpd2ρd2p = |ΓPTρ|2|Kρ|2d2ρ,
Ψp= |ΓPTρ|2|Kρ|2d2ρ |Kρ|2d2ρ.

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