Time-domain perturbation analysis of a scattering slab

Propagation of light in a homogeneous scattering slab is conveniently modelled with a diffusion equation. This approach can be extended to a heterogeneous slab through a perturbation analysis. Within BornOs approximation, the effect of an inclusion on the transmitted light is described by space-time integrals. Closed-form time integration is possible, which reduces the perturbation expressions to volume integrals over the inclusion. These can be useful to model small inclusions over which the integrand can be considered as constant. In the case of cubic inclusions with sides parallel and perpendicular to the boundaries of the surrounding slab, closed-form volume integration over the inclusion can be performed instead. Only time integrals are left, which reduces the numerical work. Numerical examples are presented. It is shown that inclusions with different volume and contrast with regards to the surrounding medium can produce the same effect on the transmitted light and are thus indistinguishable. The perturbation analysis has been used to assess the possibility of obtaining some longitudinal localization of an inclusion by using source beams and detectors of different sizes. Calculation results are also compared to experimental measurements to illustrate the validity of this analysis in the presence of small perturbations.