Photon path distributions in optically thin slabs

Abstract

The probability distribution function of photon path length in a scattering medium contains valuable information on that medium. While strongly scattering optically thick media have been extensively studied, in particular, with resort to the diffusion approximation, optically thin media have received much less attention. Here, we derive the probability distribution functions for the lengths of singly- and twice-scattered photon paths in an isotropically scattering slab of optical thickness τ, for both reflected and transmitted photons. We show that, in the case of an optically thin slab, these photons dominate the overall response of the medium. We confirm that the second moment of the distribution deviates from the ballistic limit in the case of collimated illumination. Interestingly, we show that under diffuse illumination, the second moment of the distribution is dominated by unscattered transmitted photons, hence is proportional to lnτ, and independent of the phase function. Higher moments of order n (≥3) scale as H n τn−2. When only reflected or transmitted photons are considered, the second moment scales as H2τ−1, whatever the illumination and viewing conditions. This provides direct access to τ. These theoretical results are extensively supported by Monte Carlo ray-tracing simulations. Extension to anisotropic scattering using these same simulations shows that the results hold, given a scaling factor for collimated illumination, and without any dependence on the phase function for diffuse illumination. These results overall demonstrate that the optical thickness of an optically thin slab can be estimated from the second moment of the distribution. Along with the fact that under diffuse illumination the geometrical thickness can be derived from the first moment of the distribution, this proves that the extinction coefficient of the medium can be estimated from the combination of both moments. This study thus opens new perspectives for non-invasive characterization of optically thin media either in the laboratory or by remote sensing.