Angular reflectance of a highly forward scattering medium at grazing incidence of light

We study the angular distribution of light diffusely reflected from a turbid medium with large (compared to the light wavelength) inhomogeneities. Using Monte Carlo radiative transfer simulations, we calculate the azimuthally averaged bidirectional reflectance for an optically thick plane-parallel medium and analyze its dependence on the parameters of the scattering phase function. To model single scattering in the medium, we take advantage of the Reynolds-McCormick phase function. For grazing angles of incidence, we find that the angular distribution of reflected light becomes very sensitive to the angular profile of the scattering phase function. The more elongated the phase function, the more pronounced the peak that arises around the specular reflection angle. Comparison of our numerical results with an analytic solution of the radiative transfer equation is performed, and it is shown that the bidirectional reflectance can be decomposed into two contributions, namely, the diffusion contribution and the contribution from light experiencing multiple scattering through small angles. The latter relates directly to the angular profile of the scattering phase function and is responsible for the peak in the angular distribution of reflected light. An explicit analytic formula for the azimuthally averaged bidirectional reflectance is obtained.