Time-dependent radiative transfer and pulse evolution

Abstract

The time-dependent radiative transfer equation in an absorbing and scattering medium is recast as an evolution equation that is similar to the global formulation of Preisendorfer. All the properties of radiative transfer are embodied in the evolution operator, which can be studied and manipulated in the Dirac operator notation. The reciprocity theorem is obtained in a simple way from the fundamental components of the operator, and in a homogeneous and globally isotropic medium this leads to the statement of the theorem in terms of the radiometric distributions of a laser and a spherical point source. A path integral expression is derived for the evolution operator by using the Dirac notation. By collapsing the Hilbert space of Euclidean and angular degrees of freedom into a finite-sized subspace, an explicit finite-difference numerical scheme is obtained. A feature of the scheme is causal interpolation, which adjusts the spatial interpolation on a step-by-step basis in order to guarantee that the speed of propagation measured from the calculation is the physical (causal) speed to within the spatial and temporal resolution of the calculation. This finite-difference scheme is unconditionally stable and with only weak conditions is consistent. Lax’s theorem guarantees that the scheme converges to the continuous solution as the spatial, temporal, and angular grids become arbitrarily fine. Example results from a code written to execute this algorithm are presented. In this example grid and ray affects are illustrated. Finally, the scheme is modified to account for sources of radiant energy located within the medium, and an algorithm is presented to include reflections from surfaces within the medium.