Diffusion of collimated, narrow beam waves in discrete random media

The three-dimensional diffusion of a narrow beam wave in discrete random media is discussed. Mismatched boundary conditions are taken into account for surfaces on which the reflection of diffuse light occurs. An analytical expression is derived for the average diffuse intensity in terms of the sum of the residual values under practical situations of interest. The spatial spreading of beam waves for nonabsorption particles only slightly increases with an increase in the mean cosine of the scattering angle. A comparison with previously reported Monte Carlo and experimental results of the beam width versus the optical depth shows the validity of the analytical solutions obtained here. The effects of a mismatched boundary are shown to increase the transmitted diffuse flux rather than the spatial spreading.