The Diffusion Approximation in Three Dimensions

Abstract

The diffusion approximation of the radiative transport equation isused extensively because closed-form analytical solutions can beobtained. The previous chapter gave closed-form solutions to theone-dimensional diffusion equation. In this chapter, the classicsearchlight problem of a finite beam of light normally incident ona slab or semi-infinite medium will be solved in the time-independentdiffusion approximation. The solution follows naturally once theGreen's function for the problem is known, and so the Green's functionsubject to homogeneous Robin boundary conditions will be given forsemi-infinite and slab geometries. The diffuse radiant fluence ratesare then found for impulse, flat (constant), and Gaussian shapedfinite beam irradiances. \vskip2mm How do Green's functions helpsolve the problem of a finite beam incident on a turbid medium? Asunscattered light propagates through the medium, it is scatteredand becomes diffuse. This initial scattering event acts as a sourceof diffuse light. The Green's function describes the distributionresulting from a point source of diffuse light. Since the unscatteredlight decays exponentially with increasing depth in the slab, theGreen's function for an irradiation point on the surface may be obtainedby convolving the Green's function with the proper exponential function.Again using superposition, the response for an arbitrary source distributionis obtained by adding the contributions of all point irradiances.This description is not quite complete because it neglects the contributionfrom boundary conditions, however the analytic derivation in thischapter is complete. \vskip2mm The solutions for the searchlightproblem are expressed as definite integrals or infinite series. Thereare a number of possible ways of obtaining solutions to the diffusionequation. Green's functions for a slab geometry [Reynolds 1976] havebeen known for some time. Somewhat surprisingly, the Green's functionfor a semi-infinite medium is not readily available in the literatureand is included for completeness. The solutions for the semi-infiniteand slab geometries are extended to include exponentially attenuatingline sources. Finally, we present equations for calculating the internalfluence rates for finite beam irradiances (flat top and Gaussian)on slab and semi-infinite media with inhomogeneous Robin boundaryconditions. \vskip2mm To avoid the usually complicated expressionsthat arise in solutions for a semi-infinite geometry, some authorsuse monopole and dipole methods. Both techniques generate solutionsthat satisfy the diffusion equation at the expense of satisfyingthe boundary conditions. The solutions and compromises inherent inusing the dipole and monopole techniques are briefly discussed.