Light propagation in atomic stratified media: breakdown of the transfer-matrix method at high density

The transfer-matrix method is a standard approach to wave propagation in stratified media. With the advent of cold-atom-based quantum and photonic technologies, several experiments and many proposals consider light propagation in one-dimensional optical lattices, using the transfer matrices as the main tool for the simulation. Here, we study the validity of this method by comparing its results to the microscopic coupled-dipole model, which is exact in the linear-optics regime. We show that the transfer-matrix method works very well at low density, even for thin disordered slices, and breaks down at high density because the dipole-dipole interaction induces a collective response from the atoms such that the properties of one layer are influenced by the others. We determine the boundary values of atomic densities for which this method is still applicable for describing experiments. Our findings are relevant for experimental realizations using ultra-cold atoms.