  The accurate determination of the position-dependent energy fluence rate of
light is crucial for 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. Here, 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. We study samples that contain anisotropically scattering and
absorbing spherical scatterers, namely microspheres ($r = 0.5$ $\rm{\mu}$m)
with and without absorbing dye. To probe the position-dependent energy fluence
rate, we detect the emission of quantum dots that are excited by the incident
light and that are contained in a thin capillary. By scanning the sample using
the capillary, we access the position dependence. We present a comprehensive
analysis of experimental limitations and (systematic) errors. Our measured
observations are compared to the results of analytical approximations of the
solution of the radiative transfer equation and to Monte Carlo simulations. Our
observations are found to agree well with the Monte Carlo simulations. The
$P_3$ approximation with a correction for forward scattering also agrees with
our observations, whereas the $P_1$ and the $P_3$ approximations deviate
increasingly from our observations, ultimately even predicting unphysical
negative energies.
