Multiple light scattering: mean number of scatterings and related problems

Abstract

This is a discussion of V. A. Ambartsumyan’s studies of the mean number of scatterings for photons in scattering media and of further work and development in this area, especially at Ambartsumyan’s St. Petersburg school. The following questions are discussed briefly: (a) the traditional method for calculating the number of scatterings from the source function and critiques of this method. (b) The equation for the number N(τ; τ0 ) of scatterings for a photon born at optical depth τ in a plane layer of optical thickness τ0 and its use for calculating the number $\bar{N}$ of scatterings, averaged over the entire ensemble of photons for a medium with arbitrary internal sources. These questions are first considered for the case of monochromatic scattering, and then for scattering in a spectral line with complete frequency redistribution (CFR). (c) The mean path length $\bar{T}$ for a resonance line photon in a scattering medium with CFR and continuum absorption: the basic equations and asymptotic behavior of an optically thick layer. (d) A review of calculations of $\bar{N}$ and $\bar{T}$ in media that are so thick that the CFR approximation breaks down and the effects of partial frequency redistribution (PFR) become dominant. The presentation is at a semiquantitative level in many parts of this paper, with stress on physical significance rather than the mathematics, through the use of approximate and asymptotic solutions.