  We show that the spatial distribution of the energy density of optimally
shaped waves inside a scattering medium can be described by considering only a
few of the lowest eigenfunctions of the diffusion equation. Taking into account
only the fundamental eigenfunction, the total internal energy inside the sample
is underestimated by only 2%. The spatial distribution of the shaped energy
density is very similar to the fundamental eigenfunction, up to a cosine
distance of about 0.01. We obtained the energy density inside a quasi-1D
disordered waveguide by numerical calculation of the joined scattering matrix.
Computing the transmission-averaged energy density over all transmission
channels yields the ensemble averaged energy density of shaped waves. From the
averaged energy density obtained, we reconstruct its spatial distribution using
the eigenfunctions of the diffusion equation. The results from our study have
exciting applications in controlled biomedical imaging, efficient light
harvesting in solar cells, enhanced energy conversion in solid-state lighting,
and low threshold random lasers.
