The transmission coefficient distribution of highly scattering sparse random media

We consider the distribution of the transmission coefficients, i.e. the singular values of the modal transmission matrix, for 2D random media with periodic boundary conditions composed of a large number of point-like non-absorbing scatterers. The scatterers are placed at random locations in the medium and have random refractive indices that are drawn from an arbitrary, known distribution. We construct a randomized model for the scattering matrix that retains scatterer dependent properties essential to reproduce the transmission coefficient distribution and analytically characterize the distribution of this matrix as a function of the refractive index distribution, the number of modes, and the number of scatterers. We show that the derived distribution agrees remarkably well with results obtained using a numerically rigorous spectrally accurate simulation. Analysis of the derived distribution provides the strongest principled justification yet of why we should expect perfect transmission in such random media regardless of the refractive index distribution of the constituent scatterers. The analysis suggests a sparsity condition under which random media will exhibit a perfect transmission-supporting universal transmission coefficient distribution in the deep medium limit.