Path-length distributions, scattering, and absorption in refractive spheres and slabs

We derive the path-length distribution (PLD) of light rays diffusely incident on a refractive sphere and slab in the geometric optics case. Refraction affects the width of the distribution and internal reflection introduces longer path lengths. Polarization is also taken into account and has a minor effect on the shape. For the slab, we also consider the effect of a small amount of scattering in the medium which adds to the tail of the PLD and significantly increases the mean path length. The absorption is calculated from the PLD, from which we approximate the absorption in the slab and sphere to the second order in the absorption coefficient, and also to the second order in the scattering coefficient for the slab.