Advertisement

Abstract

This chapter describes the adding-doubling method for solving the radiative transport equation. The advantages and disadvantages of the method are presented, followed by sections describing its theory and computer implementation. A detailed example is given with intermediate numerical results. Accurate tables with values of reflection and transmission for slabs of varying thicknesses with mismatched boundaries are given.

Keywords

Phase Function Optical Thickness Quadrature Point Reflection Function Light Transport 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Prahl SA, Gemert MJC van, Welch AJ. “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. 32: 559–568 (1993).ADSCrossRefGoogle Scholar
  2. 2.
    Pickering JW, Prahl SA, Wieringen N van, Beek JB, Sterenborg HJCM, Gemert MJC van. “A double integrating sphere system for measuring the optical properties of tissue,” Appl. Opt. 32: 399–410 (1993).ADSCrossRefGoogle Scholar
  3. 3.
    Parrish JA. “New concepts in therapeutic photomedicine: Photochemistry, optical targeting and the therapeutic window,” J. Invest. Dermatol. 77: 44–50 (1981).Google Scholar
  4. 4.
    Cheong WF, Prahl SA, Welch AJ. “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26: 2166–2185 (1990).ADSCrossRefGoogle Scholar
  5. 5.
    Jacques SL, Alter CA, Prahl SA. “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1: 309–333 (1987).Google Scholar
  6. 6.
    Ishimaru A. Wave Propagation and Scattering in Random Media, vol. 1. Academic Press, New York, 1978.Google Scholar
  7. 7.
    Case KM, Zweifel PF. Linear Transport Theory, Addison-Wesley, Reading, MA, 1967.zbMATHGoogle Scholar
  8. 8.
    Wilson BC, Adam G. “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10: 824–830 (1983).CrossRefGoogle Scholar
  9. 9.
    Prahl SA, Keijzer M, Jacques SL, Welch AJ. “A Monte Carlo model of light propagation in tissue,” in Müller GJ, Sliney DH (eds.), Dosimetry of Laser Radiation in Medicine and Biology, Vol. IS 5 (Bellingham, WA) SPIE Optical Engineering Press, 1989, pp. 102–111.Google Scholar
  10. 10.
    Keijzer M, Jacques SL, Prahl SA, Welch AJ. “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9: 148–154 (1989).CrossRefGoogle Scholar
  11. 11.
    Reynolds L, Johnson CC, Ishimaru A. “Diffuse reflectance from a finite blood medium: Applications to the modeling of fiber optic catheters,” Appl. Opt. 15: 2059–2067 (1976).ADSCrossRefGoogle Scholar
  12. 12.
    Bonner RF, Nossal R, Havlin S, Weiss G.H. “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4: 423–432 (1987).ADSCrossRefGoogle Scholar
  13. 13.
    Kubelka P. “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38: 448–457 (1948).MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Welch AJ, Yoon G, Gemert van MJC, “Practical models of light distribution in laser irradiated tissue,” Lasers Surg. Med. 6: 488–493 (1987).CrossRefGoogle Scholar
  15. 15.
    Chandrasekhar S. Radiative Transfer, Dover, New York, 1960.Google Scholar
  16. 16.
    Liou KN. “A numerical experiment on Chandrasekhar’s discrete ordinate method for radiative transfer: Applications to cloudy and hazy atmospheres,” J. Atmos. Sci. 30: 1303–1326 (1973).ADSCrossRefGoogle Scholar
  17. 17.
    Hulst HC van de. Multiple Light Scattering, Vol. 1, Academic Press, New York, 1980.Google Scholar
  18. 18.
    Plass GN, Kattawar GW, Catchings FE. “Matrix operator theory of radiative transfer. 1: Rayleigh scattering,” Appl. Opt. 12: 314–329 (1973).ADSCrossRefGoogle Scholar
  19. 19.
    Hulst HC van de. “A new look at multiple scattering,” Tech. Rep., NASA Institute for Space Studies, New York, 1962.Google Scholar
  20. 20.
    Irvine WM. “Multiple scattering in planetary atmospheres,” Icarus 25: 175–204 (1975).ADSCrossRefGoogle Scholar
  21. 21.
    Grant IP, Hunt GE. “Discrete space theory of radiative transfer I. Fundamentals,” Proc. R. Soc. London A A313: 183–197 (1969).MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Grant IP Hung GE. “Discrete space theory of radiative transfer II. Stability and non-negativity,” Proc. R. Soc. London A 313: 199–216 (1969).ADSCrossRefGoogle Scholar
  23. 23.
    Hansen JE, Travis LD. “Light scattering in planetary atmospheres,” Space Sci. Rev. 16: 525–610 (1974).ADSCrossRefGoogle Scholar
  24. 24.
    Wiscombe WJ. “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transfer 16: 637–658 (1976).ADSCrossRefGoogle Scholar
  25. 25.
    Prahl SA. “Light transport in tissue,” PhD thesis, University of Texas at Austin, 1988.Google Scholar
  26. 26.
    Michels HH. “Abscissas and weight coefficients for Lobatto quadrature,” Math. Comput. 17: 237–244 (1963).Google Scholar
  27. 27.
    Hildebrand FB. Introduction to Numerical Analysis, Dover, New York, 1974.zbMATHGoogle Scholar
  28. 28.
    Press WH, Flannery BP, Teukolsky SA, Vetterling WT. Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, New York, 1986.Google Scholar
  29. 29.
    Hulst HC van de, Davis MM. Proc. Koninkl. Nederl. Akad. Wet. B64: 220 (1961).Google Scholar
  30. 30.
    Wiscombe WJ. “The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34: 362–377 (1977).CrossRefGoogle Scholar
  31. 31.
    Joseph JH, Wiscombe WJ, Weinman JA. “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33: 2452–2459 (1976).ADSCrossRefGoogle Scholar
  32. 32.
    Grant IP, Hunt GE. “Solution of radiative transfer problems using the invariant Sn method,” Mon. Not. R. Astron. Soc. 141: 27–41 (1968).ADSGoogle Scholar
  33. 33.
    Irvine WM. “Multiple scattering by large particles,” Astrophys. J. 142: 1463–1475 (1965).ADSGoogle Scholar
  34. 34.
    Wiscombe WJ. “Doubling initialization revisited,” J. Quant. Spectrosc. Radiat. Transfer 18: 245–248 (1977).ADSCrossRefGoogle Scholar
  35. 35.
    Priesendorfer R. Hydrologic Optics, U.S. Dept. of Commerce, Washington, D.C., 1976.Google Scholar
  36. 36.
    Hulst HC van de. Multiple Light Scattering, Vol. 2, Academic Press, New York, 1980.Google Scholar
  37. 37.
    Giovanelli RG. “Reflection by semi-infinite diffusers,” Opt. Acta 2: 153–162 (1955).MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Scott A. Prahl
    • 1
  1. 1.Oregon Medical Laser CenterSt. Vincent HospitalPortlandUSA

Personalised recommendations