Bistable optical transmission through arrays of atoms in free space

We determine the transmission of light through a planar atomic array beyond limit low intensity that displays optical bistability in mean-field regime. develop theory describing intrinsic bistability, which is supported purely by resonant dipole-dipole interactions free space, showing how bistable amplitudes exhibit both strong cooperative and weak single-atom responses they depend on underlying collective excitation eigenmodes. Similarities with cavities are highlighted, while recurrent scattering between atoms takes role cavity mirrors. Our numerics analytic estimates show sharp variation extinction, reflectivity, group delays array, incident completely extinguished up to critical well limit. analysis paves way for nonlinear optics cooperatively responding dense ensembles.