The Diffusion Approximation in Three Dimensions

  • Scott A. Prahl
Part of the Lasers, Photonics, and Electro-Optics book series (LPEO)


The diffusion approximation of the radiative transport equation is used extensively because closed-form analytical solutions can be obtained. The previous chapter gave closed-form solutions to the one-dimensional diffusion equation. In this chapter, the classic searchlight problem of a finite beam of light normally incident on a slab or semi-infinite medium will be solved in the timeindependent diffusion approximation. The solution follows naturally once the Green’s function for the problem is known, and so the Green’s function subject to homogeneous Robin boundary conditions will be given for semi-infinite and slab geometries. The diffuse radiant fluence rates are then found for impulse, flat (constant), and Gaussian shaped finite beam irradiances.


Diffusion Equation Gaussian Beam Diffusion Approximation Robin Boundary Condition Slab Geometry 
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  1. 1.
    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
  2. 2.
    Roach GF. Green’s Functions, Cambridge University Press, Cambridge, 1982.zbMATHGoogle Scholar
  3. 3.
    Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products, Academic Press, New York, 1980.zbMATHGoogle Scholar
  4. 4.
    Grossweiner LI, Karagiannes JL, Johnson PW, Zhang Z. “Gaussian beam spread in biological tissues,” Appl. Opt. 29: 379–383 (1990).ADSCrossRefGoogle Scholar
  5. 5.
    Fretterd RJ, Longini RL. “Diffusion dipole source,” J. Opt. Soc. Am. 63: 336–337 (1973).ADSCrossRefGoogle Scholar
  6. 6.
    Hirko RJ, Fretterd RJ, Longini RL. “Application of the diffusion dipole to modelling the optical characteristics of blood,” Med. Biol. Eng. 13: 192–195 (1975).CrossRefGoogle Scholar
  7. 7.
    Eason G, Veitch AR, Nisbet RM, Turnbull FW. “The theory of the back-scattering of light by blood,” J. Phys. D: Appl. Phys.11: 1463–1479 (1978).ADSCrossRefGoogle Scholar
  8. 8.
    Patterson MS, Schwartz E, Wilson BC. “Quantitative reflectance spectrophotometry for the noninvasive measurement of photosensitizer concentration in tissue during photodynamic therapy,” in Photodynamic Therapy Mechanisms, Vol. 1065, SPIE Optical Engineering Press, 1989, pp. 115–122.Google Scholar
  9. 9.
    Allen V, McKenzie AL. “The modifiied diffusion dipole model,” Phys. Med. Biol. 36: 1621— 1638 (1991).Google Scholar
  10. 10.
    Farrell TJ, Patterson MS, Wilson B. “A diffusion theory model of spatially resolved, steadystate reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19: 879–888 (1992).CrossRefGoogle Scholar
  11. 11.
    Ishimaru A. Wave Propagation and Scattering in Random Media, Vo1. 1, Academic Press, New York, 1978.Google Scholar
  12. 12.
    Case KM, Zweifel PF. Linear Transport Theory, Addison-Wesley, Reading, MA, 1967.zbMATHGoogle Scholar
  13. 13.
    Moulton JD. “Diffusion modeling of picosecond laser pulse propagation in turbid media,” Master’s thesis, McMaster University, Hamilton, Ontario, Canada, 1990.Google Scholar
  14. 14.
    Joseph JH, Wiscombe WJ, Weinman JA. “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33: 2452–2459 (1976).ADSCrossRefGoogle Scholar
  15. 15.
    Yoon G, Prahl SA, Welch AJ. “Accuracies of the diffusion approximation and its similarity relations for laser irradiated biological media,” Appl. Opt. 28: 2250–2255 (1989).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Scott A. Prahl
    • 1
  1. 1.Oregon Medical Laser CenterPortlandUSA

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