Probing the position-dependent optical energy fluence rate in three-dimensional scattering samples

The accurate determination of the position-dependent energy fluence rate of scattered light (which is proportional to the energy density) is crucial to the understanding of transport in anisotropically scattering and absorbing samples, such as biological tissue, seawater, atmospheric turbulent layers, and light-emitting diodes. While Monte Carlo simulations are precise, their long computation time is not desirable. Common analytical approximations to the radiative transfer equation (RTE) fail to predict light transport and could even give unphysical results. Therefore, we experimentally probe the position-dependent energy fluence rate of light inside scattering samples where the widely used $P_1$ and $P_3$ approximations to the RTE fail. The samples are three-dimensional (3D) aqueous suspensions of anisotropically scattering and both absorbing and nonabsorbing spherical scatterers, namely, microspheres ($r=0.5\mu \mathrm{m}$) with and without absorbing dye. To probe the energy fluence rate, we detect the emission of quantum-dot reporter particles that are excited by the incident light and that are contained in a thin capillary. By scanning the capillary through the sample, we access the position dependence. We present a comprehensive discussion of experimental limitations and of both random and systematic errors. Our observations agree well with the Monte Carlo simulations and the $P_3$ approximation of the RTE with a correction for forward scattering. In contrast, the $P_1$ and the $P_3$ approximations deviate increasingly from our observations, ultimately even predicting unphysical negative energies.