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