@incollection{prahl95d,
  author = {S. A. Prahl},
  title = {The Diffusion Approximation in Three Dimensions},
  booktitle = {Optical-Thermal Response of Laser Irradiated Tissue},
  publisher = {Plenum Press},
  year = {1995},
  chapter = {7},
  pages = {207--231},
  editor = {A. J. Welch and M. J. C. van Gemert},
  abstract = {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 time-independent 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. \vskip2mm How do Green's functions help solve the problem of a finite beam incident on a turbid medium? As unscattered light propagates through the medium, it is scattered and becomes diffuse. This initial scattering event acts as a source of diffuse light. The Green's function describes the distribution resulting from a point source of diffuse light. Since the unscattered light decays exponentially with increasing depth in the slab, the Green's function for an irradiation point on the surface may be obtained by convolving the Green's function with the proper exponential function. Again using superposition, the response for an arbitrary source distribution is obtained by adding the contributions of all point irradiances. This description is not quite complete because it neglects the contribution from boundary conditions, however the analytic derivation in this chapter is complete. \vskip2mm The solutions for the searchlight problem are expressed as definite integrals or infinite series. There are a number of possible ways of obtaining solutions to the diffusion equation. Green's functions for a slab geometry [Reynolds 1976] have been known for some time. Somewhat surprisingly, the Green's function for a semi-infinite medium is not readily available in the literature and is included for completeness. The solutions for the semi-infinite and slab geometries are extended to include exponentially attenuating line sources. Finally, we present equations for calculating the internal fluence rates for finite beam irradiances (flat top and Gaussian) on slab and semi-infinite media with inhomogeneous Robin boundary conditions. \vskip2mm To avoid the usually complicated expressions that arise in solutions for a semi-infinite geometry, some authors use monopole and dipole methods. Both techniques generate solutions that satisfy the diffusion equation at the expense of satisfying the boundary conditions. The solutions and compromises inherent in using the dipole and monopole techniques are briefly discussed.},
}