Focusing and energy deposition inside random media

The degree of control over waves transmitted through random media is determined by characteristics of the singular values of the transmission matrix. This Letter explores focusing and energy deposition in the interior of disordered samples and shows that these are determined by the singular values of the matrix relating the field channels inside a medium to the incident channels. Through calculations and simulations, we discovered that the variation with depth of the maximal energy density and the contrast in optimal focusing are determined by the participation number M(z) of the energy density eigenvalues, while its inverse gives the variance of the energy density at z in a single configuration.