Abstract

We present a Monte Carlo method for propagating partially coherent fields through complex deterministic optical systems. A Gaussian copula is used to synthesize a random source with an arbitrary spatial coherence function. Physical optics and Monte Carlo predictions of the first- and second-order statistics of the field are shown for coherent and partially coherent sources for free-space propagation, imaging using a binary Fresnel zone plate, and propagation through a limiting aperture. Excellent agreement between the physical optics and Monte Carlo predictions is demonstrated in all cases. Convergence criteria are presented for judging the quality of the Monte Carlo predictions.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  6. B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824-830 (1983).
    [CrossRef] [PubMed]
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  8. L. Tsang, J. A. Kong, K.H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
    [CrossRef]
  9. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).
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    [CrossRef]
  20. J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves (Hilger, 1986).
  21. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1970).
  22. N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. 44, 335-341 (1949).
    [CrossRef] [PubMed]
  23. M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154 (1989).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  25. T. P. Moffitt and S. A. Prahl, “Sized-fiber reflectometry for measuring local optical properties,” IEEE J. Sel. Top. Quantum Electron. 7, 952-958 (2001).
    [CrossRef]
  26. P. R. Bargo, S. A. Prahl, and S. L. Jacques, “Optical properties effects upon the collection efficiency of optical fibers in different probe configurations,” IEEE J. Sel. Top. Quantum Electron. 9, 314-321 (2003).
    [CrossRef]
  27. S. A. Carp, S. A. Prahl, and V. Venugopalan, “Radiative transport in the delta-P1 approximation: Accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media,” J. Biomed. Opt. 9, 632-647 (2004).
    [CrossRef] [PubMed]
  28. B. D. Cameron, M. J. Rakovic, M. Mehrubeoglu, G. W. Kattawar, S. Rastegar, L. V. Wang, and G. Coté, “Measurement and calculation of the two-dimensional backscattering Mueller matrix of a turbid medium,” Opt. Lett. 23, 485-487 (1998).
    [CrossRef]
  29. S. Bartel and A. H. Hielscher, “Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. 39, 1580-1588 (2000).
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  30. M. Xu, “Electric field Monte Carlo simulation of polarized light propagation in turbid media,” Opt. Express 26, 6530-6539 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
  33. J. C. Ramella-Roman, S. A. Prahl, and S. L. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part II,” Opt. Express 13, 10392-10405 (2005).
    [CrossRef] [PubMed]
  34. D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, and T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025-3044 (1998).
    [CrossRef] [PubMed]
  35. G. Yao and 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]
  36. Q. Lu, X. Gan, M. Gu, and Q. Luo, “Monte Carlo modeling of optical coherence tomography imaging through turbid media,” Appl. Opt. 43, 1628-1637 (2004).
    [CrossRef] [PubMed]
  37. M. Y. Kirillin, I. V. Meglinskii, and A. V. Priezzhev, “Effect of photons of different scattering orders on the formation of a signal in optical low-coherence tomography of highly scattering media,” Quantum Electron. 36, 247-252 (2006).
    [CrossRef]
  38. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).
  39. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [CrossRef]
  40. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611-1618 (2005).
    [CrossRef]

2008 (1)

2006 (1)

M. Y. Kirillin, I. V. Meglinskii, and A. V. Priezzhev, “Effect of photons of different scattering orders on the formation of a signal in optical low-coherence tomography of highly scattering media,” Quantum Electron. 36, 247-252 (2006).
[CrossRef]

2005 (5)

2004 (5)

D. G. Fischer and T. D. Visser, “Spatial correlation properties of focused partially coherent light,” J. Opt. Soc. Am. A 21, 2097-2102 (2004).
[CrossRef]

C. Mujat and A. Dogariu, “Statistics of partially coherent beams: a numerical analysis,” J. Opt. Soc. Am. A 21, 1000-1003 (2004).
[CrossRef]

S. A. Carp, S. A. Prahl, and V. Venugopalan, “Radiative transport in the delta-P1 approximation: Accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media,” J. Biomed. Opt. 9, 632-647 (2004).
[CrossRef] [PubMed]

M. Xu, “Electric field Monte Carlo simulation of polarized light propagation in turbid media,” Opt. Express 26, 6530-6539 (2004).
[CrossRef]

Q. Lu, X. Gan, M. Gu, and Q. Luo, “Monte Carlo modeling of optical coherence tomography imaging through turbid media,” Appl. Opt. 43, 1628-1637 (2004).
[CrossRef] [PubMed]

2003 (1)

P. R. Bargo, S. A. Prahl, and S. L. Jacques, “Optical properties effects upon the collection efficiency of optical fibers in different probe configurations,” IEEE J. Sel. Top. Quantum Electron. 9, 314-321 (2003).
[CrossRef]

2002 (2)

2001 (1)

T. P. Moffitt and S. A. Prahl, “Sized-fiber reflectometry for measuring local optical properties,” IEEE J. Sel. Top. Quantum Electron. 7, 952-958 (2001).
[CrossRef]

2000 (1)

1999 (1)

G. Yao and 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]

1998 (2)

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

B. D. Cameron, M. J. Rakovic, M. Mehrubeoglu, G. W. Kattawar, S. Rastegar, L. V. Wang, and G. Coté, “Measurement and calculation of the two-dimensional backscattering Mueller matrix of a turbid medium,” Opt. Lett. 23, 485-487 (1998).
[CrossRef]

1995 (1)

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

1994 (1)

1989 (2)

A. Oulamara, G. Tribillon, and J. Duvernoy, “Biological activity measurement on botanical specimen surfaces using a temporal decorrelation effect of laser speckle,” J. Mod. Opt. 36, 165-179 (1989).
[CrossRef]

M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154 (1989).
[CrossRef] [PubMed]

1983 (1)

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

1949 (1)

N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. 44, 335-341 (1949).
[CrossRef] [PubMed]

Adam, G.

B. C. Wilson and 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.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

Ao, C. O.

L. Tsang, J. A. Kong, K.H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Bargo, P. R.

P. R. Bargo, S. A. Prahl, and S. L. Jacques, “Optical properties effects upon the collection efficiency of optical fibers in different probe configurations,” IEEE J. Sel. Top. Quantum Electron. 9, 314-321 (2003).
[CrossRef]

Bartel, S.

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1970).

Cairns, B.

B. Cairns, “An investigation of radiative transfer and multiple scattering,” Ph.D. thesis (University of Rochester, 1992). (Available from UMI Dissertation Information Service, 300 N. Zeeb Road, Ann Arbor, Michigan 48106).

Cameron, B. D.

Carp, S. A.

S. A. Carp, S. A. Prahl, and V. Venugopalan, “Radiative transport in the delta-P1 approximation: Accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media,” J. Biomed. Opt. 9, 632-647 (2004).
[CrossRef] [PubMed]

Chen, Z.

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

Coté, G.

Côté, D.

Ding, K.H.

L. Tsang, J. A. Kong, K.H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Ding, K.-H.

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

Dogariu, A.

Duncan, D. D.

Duvernoy, J.

A. Oulamara, G. Tribillon, and J. Duvernoy, “Biological activity measurement on botanical specimen surfaces using a temporal decorrelation effect of laser speckle,” J. Mod. Opt. 36, 165-179 (1989).
[CrossRef]

Fischer, D. G.

Gan, X.

Gbur, G.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

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

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975), pp. 9-75.
[CrossRef]

Gu, M.

Hielscher, A. H.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

Jacques, S. L.

J. C. Ramella-Roman, S. A. Prahl, and S. L. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part I,” Opt. Express 13, 4420-4438 (2005).
[CrossRef] [PubMed]

J. C. Ramella-Roman, S. A. Prahl, and S. L. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part II,” Opt. Express 13, 10392-10405 (2005).
[CrossRef] [PubMed]

P. R. Bargo, S. A. Prahl, and S. L. Jacques, “Optical properties effects upon the collection efficiency of optical fibers in different probe configurations,” IEEE J. Sel. Top. Quantum Electron. 9, 314-321 (2003).
[CrossRef]

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

M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154 (1989).
[CrossRef] [PubMed]

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in SPIE Proceedings of Dosimetry of Laser Radiation in Medicine and Biology, G.J.Müller and D.H.Sliney, eds., Institute Series 5 (SPIE Press, 1989), pp. 102-111.

Ji, L.

Jorgensen, T. M.

Kattawar, G. W.

Keijzer, M.

M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154 (1989).
[CrossRef] [PubMed]

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in SPIE Proceedings of Dosimetry of Laser Radiation in Medicine and Biology, G.J.Müller and D.H.Sliney, eds., Institute Series 5 (SPIE Press, 1989), pp. 102-111.

Kirillin, M. Y.

M. Y. Kirillin, I. V. Meglinskii, and A. V. Priezzhev, “Effect of photons of different scattering orders on the formation of a signal in optical low-coherence tomography of highly scattering media,” Quantum Electron. 36, 247-252 (2006).
[CrossRef]

Kirkpatrick, S. J.

Kong, J. A.

L. Tsang, J. A. Kong, K.H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

Lindmo, T.

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

Lu, Q.

Luo, Q.

Meglinskii, I. V.

M. Y. Kirillin, I. V. Meglinskii, and A. V. Priezzhev, “Effect of photons of different scattering orders on the formation of a signal in optical low-coherence tomography of highly scattering media,” Quantum Electron. 36, 247-252 (2006).
[CrossRef]

Mehrubeoglu, M.

Metropolis, N.

N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. 44, 335-341 (1949).
[CrossRef] [PubMed]

Miao, Q.

Milner, T. E.

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

Moffitt, T. P.

T. P. Moffitt and S. A. Prahl, “Sized-fiber reflectometry for measuring local optical properties,” IEEE J. Sel. Top. Quantum Electron. 7, 952-958 (2001).
[CrossRef]

Mujat, C.

Nelson, J. S.

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

Nelson, R. B.

R. B. Nelson, An Introduction to Copulas (Springer-Verlag, 1999).

Oulamara, A.

A. Oulamara, G. Tribillon, and J. Duvernoy, “Biological activity measurement on botanical specimen surfaces using a temporal decorrelation effect of laser speckle,” J. Mod. Opt. 36, 165-179 (1989).
[CrossRef]

Papoulis, A.

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

Pillai, S. U.

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

Ponomarenko, S. A.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

Prahl, S. A.

J. C. Ramella-Roman, S. A. Prahl, and S. L. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part II,” Opt. Express 13, 10392-10405 (2005).
[CrossRef] [PubMed]

J. C. Ramella-Roman, S. A. Prahl, and S. L. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part I,” Opt. Express 13, 4420-4438 (2005).
[CrossRef] [PubMed]

S. A. Carp, S. A. Prahl, and V. Venugopalan, “Radiative transport in the delta-P1 approximation: Accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media,” J. Biomed. Opt. 9, 632-647 (2004).
[CrossRef] [PubMed]

P. R. Bargo, S. A. Prahl, and S. L. Jacques, “Optical properties effects upon the collection efficiency of optical fibers in different probe configurations,” IEEE J. Sel. Top. Quantum Electron. 9, 314-321 (2003).
[CrossRef]

T. P. Moffitt and S. A. Prahl, “Sized-fiber reflectometry for measuring local optical properties,” IEEE J. Sel. Top. Quantum Electron. 7, 952-958 (2001).
[CrossRef]

M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154 (1989).
[CrossRef] [PubMed]

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in SPIE Proceedings of Dosimetry of Laser Radiation in Medicine and Biology, G.J.Müller and D.H.Sliney, eds., Institute Series 5 (SPIE Press, 1989), pp. 102-111.

Priezzhev, A. V.

M. Y. Kirillin, I. V. Meglinskii, and A. V. Priezzhev, “Effect of photons of different scattering orders on the formation of a signal in optical low-coherence tomography of highly scattering media,” Quantum Electron. 36, 247-252 (2006).
[CrossRef]

Rakovic, M. J.

Ramella-Roman, J. C.

Rastegar, S.

Roychowdhury, H.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

Smithies, D. J.

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

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves (Hilger, 1986).

Tribillon, G.

A. Oulamara, G. Tribillon, and J. Duvernoy, “Biological activity measurement on botanical specimen surfaces using a temporal decorrelation effect of laser speckle,” J. Mod. Opt. 36, 165-179 (1989).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, K.H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

Tycho, A.

Ulam, S.

N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. 44, 335-341 (1949).
[CrossRef] [PubMed]

Venugopalan, V.

S. A. Carp, S. A. Prahl, and V. Venugopalan, “Radiative transport in the delta-P1 approximation: Accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media,” J. Biomed. Opt. 9, 632-647 (2004).
[CrossRef] [PubMed]

Visser, T. D.

Vitkin, I. A.

Wang, L.

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

Wang, L. V.

Wang, R.

Welch, A. J.

M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154 (1989).
[CrossRef] [PubMed]

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in SPIE Proceedings of Dosimetry of Laser Radiation in Medicine and Biology, G.J.Müller and D.H.Sliney, eds., Institute Series 5 (SPIE Press, 1989), pp. 102-111.

Wilson, B. C.

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

Wolf, E.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (2002).
[CrossRef]

D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11, 1128-1135 (1994).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1970).

Wu, J.

Xiong, G.

Xu, M.

M. Xu, “Electric field Monte Carlo simulation of polarized light propagation in turbid media,” Opt. Express 26, 6530-6539 (2004).
[CrossRef]

Xue, P.

Yao, G.

G. Yao and 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.

Zheng, L.

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

Appl. Opt. (3)

Comput. Methods Programs Biomed. (1)

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

IEEE J. Sel. Top. Quantum Electron. (2)

T. P. Moffitt and S. A. Prahl, “Sized-fiber reflectometry for measuring local optical properties,” IEEE J. Sel. Top. Quantum Electron. 7, 952-958 (2001).
[CrossRef]

P. R. Bargo, S. A. Prahl, and S. L. Jacques, “Optical properties effects upon the collection efficiency of optical fibers in different probe configurations,” IEEE J. Sel. Top. Quantum Electron. 9, 314-321 (2003).
[CrossRef]

J. Am. Stat. Assoc. (1)

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Figures (12)

Fig. 1
Fig. 1

(a) Illustration of the field realization cube in terms of two spatial and one temporal dimension. (b) Reinterpretation of the field realization cube in terms of an ensemble of spatial line sources.

Fig. 2
Fig. 2

Ensemble members of the stochastic source field process.

Fig. 3
Fig. 3

Illustration of Monte Carlo approximation of the Huygens wavefront.

Fig. 4
Fig. 4

Illustration of Fresnel zone plate imaging architecture.

Fig. 5
Fig. 5

Free-space architecture.

Fig. 6
Fig. 6

Alternate simulation configuration with interposed limiting aperture.

Fig. 7
Fig. 7

Monte Carlo and physical optics calculation of imaging of the point source with the Fresnel zone plate. The zone plate is 50 λ in diameter with a central (clear) zone of 14 λ diameter; object and image distances are each 100 λ .

Fig. 8
Fig. 8

Source plane field characterization calculated over all ensemble members. (a) Intensity, with effective source length l s = 69.8 λ (residual with respect to Gaussian source weighting shown at bottom); (b) real part of complex coherence factor, with coherence length d c = 50.4 λ .

Fig. 9
Fig. 9

Observation plane field characterization calculated over all ensemble members and as calculated using the generalized van Cittert–Zernike theorem. (a) Intensity with effective length l o = 983 λ ; (b) real part of the complex coherence factor, with coherence length d c = 684 λ .

Fig. 10
Fig. 10

Comparison of Monte Carlo and physical optics results for free-space propagation. Monto Carlo calculations used 50,000 rays/source pixel ( 2.55 M realization ) and an 8 λ observation plane pixel size. In each figure the physical optics calculation is shown at the top with the absolute residual ( physical optics Monte Carlo ) shown at the bottom. (a) Intensity, (b) real part of complex coherence factor.

Fig. 11
Fig. 11

Observation plane field characterization, by physical optics and Monte Carlo calculations, calculated over all ensemble members for configuration shown in Fig. 6. Monte Carlo calculations used 2.58 M rays/realization and a 4 λ observation plane pixel size. In each figure the physical optics calculation is shown at the top with the absolute residual ( physical optics Monte Carlo ) shown at the bottom. (a) Intensity with effective image length of 221 λ , (b) real part of the complex coherence factor.

Fig. 12
Fig. 12

Intensity contrast for the intermediate-aperture configuration. (a) Contrast (as calculated over all ensemble members) for physical optics and Monte Carlo results shown in comparison with theoretical value of unity. (b) Observation plane intensity, top, and excess contrast, bottom (difference between that for Monte Carlo and physical optics).

Equations (29)

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H ( x , y ) = C [ F ( x ) , G ( y ) ] .
Y 1 = 2 ln X 1 cos ( 2 π X 2 ) ,
Y 2 = 2 ln X 1 sin ( 2 π X 2 ) ,
f Y 1 , Y 2 = 1 2 π exp { 1 2 ( y 1 2 + y 2 2 ) } = 1 2 π exp { 1 2 y 1 2 } 1 2 π exp { 1 2 y 2 2 } ,
Z 1 Z 2 = 1 2 1 1 1 1 1 + r 0 0 1 r Y 1 Y 2 ,
f Z 1 , Z 2 = 1 2 π 1 r 2 exp { 1 2 ( 1 r 2 ) ( z 1 2 2 r z 1 z 2 + z 2 2 ) } .
T 1 = F Z ( Z 1 ) ; T 2 = F Z ( Z 2 ) ,
μ = exp ( 1 2 σ Δ ϕ 2 ) ,
r 1 k E { ( T 11 μ 11 ) ( T 1 k μ 1 k ) } σ 11 σ 1 k = 1 + r 2 ,
var ( ϕ 1 ϕ 2 ) = ( 2 π m ) 2 ( 1 r 1 k ) 6 ,
μ 1 k = exp [ ( 2 π m ) 2 12 ( 1 r 1 k ) ] ,
r = cos ( π k 1 k max 1 ) ,
μ 1 k = exp { ( 2 π m ) 2 6 sin 2 [ π 4 ( k 1 k max 1 ) ] } ,
μ k c k , k c + k U k c k U k c + k * U k c k 2 1 2 U k c + k 2 1 2 = exp { ( 2 π m ) 2 6 sin 2 [ π 4 ( 2 k k max 1 ) ] } ,
U o ( x o ) = i π λ U s ( x s ) z r o s H 1 ( 1 ) ( k r o s ) d x s ,
r o s = [ ( x 0 x s ) 2 + z 2 ] 1 2 ,
U ̃ o ( f x ) = U ̃ s ( f x ) H ( f x , z ) ,
H ( f x , z ) = exp { i z [ k 2 ( 2 π f x ) 2 ] 1 2 }
H ( f x , z ) = exp ( i k z i π λ z f x 2 ) .
I ( x ) = U ( x ) 2 ,
μ ( Δ x ) = U ( x Δ x 2 ) U * ( x + Δ x 2 ) U ( x Δ x 2 ) 2 1 2 U ( x + Δ x 2 ) 2 1 2 .
d c = μ ( Δ x ) d Δ x .
I o ( x o ) μ s ( Δ x s ) exp { i 2 π x o Δ x s λ z } d Δ x s
μ o ( Δ x o ) I s ( x s ) exp { i 2 π x s Δ x o λ z } d x s .
B j e i β j = A j e μ a l j e i ( ϕ j + k l j ) ,
U ( x n ) = j B j e i β j = j A j e μ a l j e i ( ϕ j + k l j ) .
U ( x n ) = j A j e i ( ϕ j + k l j ) , l j = d cos θ j ,
RMS I ( N r , N ) = 1 n o i = 1 n o [ I MC ( x i , N r , N ) I PO ( x i ) ] 2 ,
RMS μ ( N r , N ) = 1 n o i = 1 n o [ μ MC ( x i , N r , N ) μ PO ( x i ) ] 2 ,

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