and random atomic positions enables one to characterize the optical response of the system in terms of a hierarchy of strongly interacting, nearby atomic pairs. The shifts of the resonance frequencies arising from the near-field interactions then effectively yield an inhomogeneously broadened optical medium, where the amount of broadening linearly scales with density. This implies that light of any given wavelength only interacts with at most $\sim 1$ near-resonant atom per reduced cubic wavelength $\lambda_0^3/(2\pi)^3$, regardless of the physical atomic density, thus limiting the optical response ([Fig. \[fig:scatt\_cross\_sec\]]{}-d).

Our results are potentially significant on a number of fronts. First, they provide a convincing picture of why typical theories for optical response, based upon a smooth density approximation, fail for dense, near-resonant atomic media, due to the important role of granularity and nearest-neighbor interactions. Furthermore, our results show the promise of a bottom-up approach to understanding the physical limits of refractive index, starting from objects (isolated atoms) whose optical responses are both huge and exquisitely understood. Separately, the existence of a fundamental mechanism that results in inhomogeneous broadening (i.e. dephasing) and saturation of optical properties at high densities, which occurs even for perfect, stationary atoms, should impose fundamental bounds on the 