Coherent light propagation through cold atomic clouds beyond the independent scattering approximation

We calculate the relative permittivity of a cold atomic gas under weak probe illumination, up to second order in the density. Within the framework of a diagrammatic representation method, we identify all the second order diagrams that enter into the description of the relative permittivity for coherent light transmission. These diagrams originate from pairwise position correlation and recurrent scattering. Using coupled dipole equations, we numerically simulate the coherent transmission with scalar and vector waves, and find good agreement with the perturbative calculations. We applied this perturbative expansion approach to a classical gas at rest, but the method is extendable to thermal gas with finite atomic motion and to quantum gases where non-trivial pair correlations can be naturally included.