Abstract

This paper develops a method for calculating the angular distribution (AD) of multiply scattered photons through isotropic turbid slabs. Extension to anisotropic scattering is also discussed. Previous studies have recognized that the AD of multiply scattered photons is critical for many applications, such as the design of imaging optics and estimation of image quality. This paper therefore develops a closed-from method that can accurately calculate the AD over a wide range of conditions. Other virtues of the method include its simplicity in implementation and its prospective for extension to anisotropic scattering.

©2011 Optical Society of America

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References

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  1. J. C. Hebden, D. J. Hall, M. Firbank, and D. T. Delpy, “Time-Resolved Optical Imaging of a Solid Tissue-Equivalent Phantom,” Appl. Opt. 34(34), 8038–8047 (1995).
    [Crossref] [PubMed]
  2. R. M. Measures, Laser Remote Sensing: Fundamentals and applications (Krieger Publishing Company, 1992).
  3. M. A. Linne, M. Paciaroni, J. R. Gord, and T. R. Meyer, “Ballistic Imaging of The Liquid Core for a Steady Jet in Crossflow,” Appl. Opt. 44(31), 6627–6634 (2005).
    [Crossref] [PubMed]
  4. J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, and J. Reintjes, “Achievable Spatial Resolution of Time-Resolved Transillumination Imaging Systems Which Utilize Multiply Scattered Light,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(1), 1142–1155 (1996).
    [Crossref] [PubMed]
  5. J. A. Moon and J. Reintjes, “Image Resolution by Use of Multiply Scattered Light,” Opt. Lett. 19(8), 521–523 (1994).
    [Crossref] [PubMed]
  6. J. A. Moon, R. Mahon, M. D. Duncan, and J. Reintjes, “Resolution Limits for Imaging through Turbid Media with Diffuse Light,” Opt. Lett. 18(19), 1591–1593 (1993).
    [Crossref] [PubMed]
  7. M. Paciaroni and M. Linne, “Single-Shot, Two-Dimensional Ballistic Imaging Through Scattering Media,” Appl. Opt. 43(26), 5100–5109 (2004).
    [Crossref] [PubMed]
  8. A. A. Kokhanovsky, “Analytical Solutions of Multiple Light Scattering Problems: A Review,” Meas. Sci. Technol. 13(3), 233–240 (2002).
    [Crossref]
  9. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, Englewood Cliffs, New Jersey, 1991).
  10. D. Contini, F. Martelli, and G. Zaccanti, “Photon Migration Through a Turbid Slab Described by a Model Based on Diffusion Approximation. 2. Theory,” Appl. Opt. 36(19), 4587–4599 (1997).
    [Crossref] [PubMed]
  11. A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon Path-Length Distributions for Transmission through Optically Turbid Slabs,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(2), 810–818 (1993).
    [Crossref] [PubMed]
  12. E. Berrocal, D. L. Sedarsky, M. E. Paciaroni, I. V. Meglinski, and M. A. Linne, “Laser Light Scattering in Turbid Media Part I: Experimental and Simulated Results for The Spatial Intensity Distribution,” Opt. Express 15(17), 10649–10665 (2007).
    [Crossref] [PubMed]
  13. M. D. King and Harshvardhan, “Comparative Accuracy of Selected Multiple Scattering Approximations,” J. Atmos. Sci. 43(8), 784–801 (1986).
    [Crossref]
  14. R. F. Bonner, R. Nossal, S. Havlin, and G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A 4(3), 423–432 (1987).
    [Crossref] [PubMed]
  15. K.-N. Liou, An Introduction to Atmospheric Radiation (Academic Press Ltd., 2002).
  16. I. M. Sobol, The Monte Carlo Method (The University of Chicago Press, 1967).
  17. 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(12), 4420–4438 (2005).
    [Crossref] [PubMed]
  18. E. Berrocal, D. L. Sedarsky, M. E. Paciaroni, I. V. Meglinski, and M. A. Linne, “Laser Light Scattering in Turbid Media Part II: Spatial and Temporal Analysis of Individual Scattering Orders via Monte Carlo Simulation,” Opt. Express 17(16), 13792–13809 (2009).
    [Crossref] [PubMed]
  19. Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables(Dover, 1965).

2009 (1)

2007 (1)

2005 (2)

2004 (1)

2002 (1)

A. A. Kokhanovsky, “Analytical Solutions of Multiple Light Scattering Problems: A Review,” Meas. Sci. Technol. 13(3), 233–240 (2002).
[Crossref]

1997 (1)

1996 (1)

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, and J. Reintjes, “Achievable Spatial Resolution of Time-Resolved Transillumination Imaging Systems Which Utilize Multiply Scattered Light,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(1), 1142–1155 (1996).
[Crossref] [PubMed]

1995 (1)

1994 (1)

1993 (2)

J. A. Moon, R. Mahon, M. D. Duncan, and J. Reintjes, “Resolution Limits for Imaging through Turbid Media with Diffuse Light,” Opt. Lett. 18(19), 1591–1593 (1993).
[Crossref] [PubMed]

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon Path-Length Distributions for Transmission through Optically Turbid Slabs,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(2), 810–818 (1993).
[Crossref] [PubMed]

1987 (1)

1986 (1)

M. D. King and Harshvardhan, “Comparative Accuracy of Selected Multiple Scattering Approximations,” J. Atmos. Sci. 43(8), 784–801 (1986).
[Crossref]

Bashkansky, M.

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, and J. Reintjes, “Achievable Spatial Resolution of Time-Resolved Transillumination Imaging Systems Which Utilize Multiply Scattered Light,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(1), 1142–1155 (1996).
[Crossref] [PubMed]

Battle, P. R.

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, and J. Reintjes, “Achievable Spatial Resolution of Time-Resolved Transillumination Imaging Systems Which Utilize Multiply Scattered Light,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(1), 1142–1155 (1996).
[Crossref] [PubMed]

Berrocal, E.

Bonner, R. F.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon Path-Length Distributions for Transmission through Optically Turbid Slabs,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(2), 810–818 (1993).
[Crossref] [PubMed]

R. F. Bonner, R. Nossal, S. Havlin, and G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A 4(3), 423–432 (1987).
[Crossref] [PubMed]

Contini, D.

Delpy, D. T.

Duncan, M. D.

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, and J. Reintjes, “Achievable Spatial Resolution of Time-Resolved Transillumination Imaging Systems Which Utilize Multiply Scattered Light,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(1), 1142–1155 (1996).
[Crossref] [PubMed]

J. A. Moon, R. Mahon, M. D. Duncan, and J. Reintjes, “Resolution Limits for Imaging through Turbid Media with Diffuse Light,” Opt. Lett. 18(19), 1591–1593 (1993).
[Crossref] [PubMed]

Firbank, M.

Gandjbakhche, A. H.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon Path-Length Distributions for Transmission through Optically Turbid Slabs,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(2), 810–818 (1993).
[Crossref] [PubMed]

Gord, J. R.

Hall, D. J.

Harshvardhan,

M. D. King and Harshvardhan, “Comparative Accuracy of Selected Multiple Scattering Approximations,” J. Atmos. Sci. 43(8), 784–801 (1986).
[Crossref]

Havlin, S.

Hebden, J. C.

Jacques, S. L.

King, M. D.

M. D. King and Harshvardhan, “Comparative Accuracy of Selected Multiple Scattering Approximations,” J. Atmos. Sci. 43(8), 784–801 (1986).
[Crossref]

Kokhanovsky, A. A.

A. A. Kokhanovsky, “Analytical Solutions of Multiple Light Scattering Problems: A Review,” Meas. Sci. Technol. 13(3), 233–240 (2002).
[Crossref]

Linne, M.

Linne, M. A.

Mahon, R.

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, and J. Reintjes, “Achievable Spatial Resolution of Time-Resolved Transillumination Imaging Systems Which Utilize Multiply Scattered Light,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(1), 1142–1155 (1996).
[Crossref] [PubMed]

J. A. Moon, R. Mahon, M. D. Duncan, and J. Reintjes, “Resolution Limits for Imaging through Turbid Media with Diffuse Light,” Opt. Lett. 18(19), 1591–1593 (1993).
[Crossref] [PubMed]

Martelli, F.

Meglinski, I. V.

Meyer, T. R.

Moon, J. A.

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, and J. Reintjes, “Achievable Spatial Resolution of Time-Resolved Transillumination Imaging Systems Which Utilize Multiply Scattered Light,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(1), 1142–1155 (1996).
[Crossref] [PubMed]

J. A. Moon and J. Reintjes, “Image Resolution by Use of Multiply Scattered Light,” Opt. Lett. 19(8), 521–523 (1994).
[Crossref] [PubMed]

J. A. Moon, R. Mahon, M. D. Duncan, and J. Reintjes, “Resolution Limits for Imaging through Turbid Media with Diffuse Light,” Opt. Lett. 18(19), 1591–1593 (1993).
[Crossref] [PubMed]

Nossal, R.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon Path-Length Distributions for Transmission through Optically Turbid Slabs,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(2), 810–818 (1993).
[Crossref] [PubMed]

R. F. Bonner, R. Nossal, S. Havlin, and G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A 4(3), 423–432 (1987).
[Crossref] [PubMed]

Paciaroni, M.

Paciaroni, M. E.

Prahl, S. A.

Ramella-Roman, J. C.

Reintjes, J.

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, and J. Reintjes, “Achievable Spatial Resolution of Time-Resolved Transillumination Imaging Systems Which Utilize Multiply Scattered Light,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(1), 1142–1155 (1996).
[Crossref] [PubMed]

J. A. Moon and J. Reintjes, “Image Resolution by Use of Multiply Scattered Light,” Opt. Lett. 19(8), 521–523 (1994).
[Crossref] [PubMed]

J. A. Moon, R. Mahon, M. D. Duncan, and J. Reintjes, “Resolution Limits for Imaging through Turbid Media with Diffuse Light,” Opt. Lett. 18(19), 1591–1593 (1993).
[Crossref] [PubMed]

Sedarsky, D. L.

Weiss, G. H.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon Path-Length Distributions for Transmission through Optically Turbid Slabs,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(2), 810–818 (1993).
[Crossref] [PubMed]

R. F. Bonner, R. Nossal, S. Havlin, and G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A 4(3), 423–432 (1987).
[Crossref] [PubMed]

Zaccanti, G.

Appl. Opt. (4)

J. Atmos. Sci. (1)

M. D. King and Harshvardhan, “Comparative Accuracy of Selected Multiple Scattering Approximations,” J. Atmos. Sci. 43(8), 784–801 (1986).
[Crossref]

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

Meas. Sci. Technol. (1)

A. A. Kokhanovsky, “Analytical Solutions of Multiple Light Scattering Problems: A Review,” Meas. Sci. Technol. 13(3), 233–240 (2002).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (2)

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, and J. Reintjes, “Achievable Spatial Resolution of Time-Resolved Transillumination Imaging Systems Which Utilize Multiply Scattered Light,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(1), 1142–1155 (1996).
[Crossref] [PubMed]

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon Path-Length Distributions for Transmission through Optically Turbid Slabs,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(2), 810–818 (1993).
[Crossref] [PubMed]

Other (5)

R. M. Measures, Laser Remote Sensing: Fundamentals and applications (Krieger Publishing Company, 1992).

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, Englewood Cliffs, New Jersey, 1991).

K.-N. Liou, An Introduction to Atmospheric Radiation (Academic Press Ltd., 2002).

I. M. Sobol, The Monte Carlo Method (The University of Chicago Press, 1967).

Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables(Dover, 1965).

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Figures (4)

Fig. 1
Fig. 1 Schematic of the problem and the configuration of MC simulation.

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Fig. 2
Fig. 2 Panel (a). Comparison of P against MC data. Panels (b) and (c). Comparison of Qz against MC data under various OD.

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Fig. 3
Fig. 3 AD of transmitted photons calculated using Eq. (5) compared to MC data. The symbols represent the MC data (with ballistic photons excluded), the solid lines the calculation using Eq. (5).

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Fig. 4
Fig. 4 Panel (a). Comparison of the average cosine at various OD. Panel (b). Fraction of transmitted photons within given acceptance angles. Panel (c). Error relative to the MC data.

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Equations (6)

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

Q T (θ)=C ∫ 0 Z P(z,θ)× Q z (z) dz
P(θ,z)=A(z) e −z/cosθ
Q z (z)=B⋅ e −z
C= 2 /(Z+ 2 )= 2 /(OD+ 2 )
Q T (θ)= 2 OD+ 2 ∫ 0 Z e −z(1/cosθ+1) (1− e −z )( e −z −z× ∫ 1 ∞ t −1 e −zt dt) dz
cos(θ) ¯ = ∫ 0 π/2 Q T (θ)sinθcosθdθ / ∫ 0 π/2 Q T (θ)sinθdθ

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