Breakdown of light transport models in photonic scattering slabs with strong absorption and anisotropy

The radiative transfer equation (RTE) models the transport of light inside photonic scattering samples such as paint, foam and tissue. Analytic approximations to solve the RTE fail for samples with strong absorption and dominant anisotropic scattering and predict unphysical negative energy densities and the diffuse flux in the wrong direction. Here we fully characterize the unphysical regions of three popular approximations to the RTE for a slab, namely the $P_1$ approximation (or diffusion approximation), the $P_3$ approximation, and a popular modification to $P_3$ that corrects the forward scattering in the approximation. We find that the delta function correction to $P_3$ eliminates the unphysical range in the forward scattering. In addition, we compare the predictions of these analytical methods to exact Monte Carlo simulations for the physical and unphysical regions. We present maps of relative errors for the albedo and the anisotropy of the scatterers for a realistic index contrast typical of a polymer slab in air and optical thickness. The relative error maps provide a guideline for the accuracy of the analytical methods to interpret experiments on light transport in photonic scattering slabs. Our results show that the $P_1$ approximation is significantly inaccurate to extract transport parameters unless the sample scatters purely isotropic and elastic. The $P_3$ approximation exceeds $P_1$ in terms of accuracy in its physical range for moderate absorption, and the $P_3$ with the delta function correction is the most accurate approximation considered here for the forward direction.