Photon transport in thin disordered slabs

Recent experiments on the transport of an ultrashort pulse of light through multiply scattering slabs whose thickness L is only a few times the transport mean free path $l^{*}$, have shown a reduction in the effective diffusion constant with decreasing optical thickness $\tau = L/l^{*}$ [ R.H.J. Kop {\it et.al.,} Phys.Rev. Lett. {\bf 79}, 4369 (1997)]. While the diffusion approximation is accurate for optically thick media, it is not valid in the regime where all the intensity is not yet converted to a diffuse flux, as is the case for slabs where $L < 8l^{*}$. We examine using Monte Carlo simulations, photon transport in optically `thin' slabs whose thickness L is only a few times $l^{*}$, with particles of different scattering anisotropies. Our simulations bring out clearly a physical picture for the observed reduction in the photon diffusion coefficient in terms of looping paths for the photons surviving in the slab, and also examine the role played by the scattering anisotropy of the particles. We also describe the use of the Ornstein-Uhlenbeck process, which interpolates between the short time ballistic and long time diffusive regimes.