We theoretically investigate the optical response of many cold atoms within a pair of identical planar lattices mostly in a dense regime using microscopic electrodynamics numerical simulations. We find cooperative optical properties of the double atomic lattices to a normal incidence of low-intensity radiation at particular ranges of sample parameters. The main findings are that resonance shifts, linewidths, and transmittance could be engineered by varying layer-to-layer distance, lattice constant, and parameters for incident light. We also propose a bilayer atomic lattice could be modeled as two effective superatoms. Simulation results qualitatively agree with the ones from an isotropic infinite lattice model in the ranges of external parameters for which a mean-field approximation is not valid. Hopefully, our bilayer lattice samples might be exploited for implementing an optical switch and a photonic device in quantum computing.