Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source

The images method is widely used to solve the diffusion equation for the propagation of light within a semi-infinite medium (z>0) in which both scattering and absorption occur. The results are widely used both to analyze data and to illustrate the failings of the diffusion approximation. We emphasize that the images method does not properly obey the extrapolation boundary conditions of diffusion theory, namely, (1-ze d/dz)φ(z)|z=0=0; instead, it approximates these by φ(-ze)=0. The images method also does not describe the source of diffusing photons produced by the exponential attenuation of a collimated incident beam. By correcting these defects and comparing with a simulation, we show that most of the error in the images solution is due to the source and boundary treatment rather than to the diffusion approximation. Our prediction for backscattered intensity versus distance from the source point should be useful for improved analysis of experimental data, especially in cases of nonzero wall reflectivity.

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