  We study strong light-mediated resonant dipole-dipole interactions in
two-dimensional planar lattices of cold atoms. We provide a detailed analysis
for the description of the dipolar point emitter lattice plane as a
"super-atom", whose response is similar to electromagnetically-induced
transparency, but which exhibits an ultra-narrow collective size-dependent
subradiant resonance linewidth. The super-atom model provides intuitively
simple descriptions for the spectral response of the array, including the
complete reflection, full transmission, narrow Fano resonances, and asymptotic
expressions for the resonance linewidths of the collective eigenmodes. We
propose a protocol to transfer almost the entire radiative excitation to a
single correlated subradiant eigenmode in a lattice and show that the medium
obtained by stacked lattice arrays can form a cooperative magnetometer. Such a
magnetometer utilizes similar principles as magnetometers based on the
electromagnetically-induced transparency. The accuracy of the cooperative
magnetometer, however, is not limited by the single-atom resonance linewidth,
but the much narrower collective linewidth that results from the strong
dipole-dipole interactions.
