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We study the optical response of a 2D square lattice of atoms using classicalelectrodynamics. Due to dipole-dipole interactions, the lattice atoms polarizeas if the lattice were an atom with up to three resonance frequencies, withcooperatively shifted resonances and altered transition linewidths. We showthat when the distance between two 2D lattices is large enough and Braggreflections are absent, the lattices interact among themselves as if theyradiated a plane wave whose amplitude is in accordance with the radiation froma dipole moment continuously distributed in the lattice plane. We employ theseresults to study light propagation in stacks of 2D lattices, drawing on simplequalitative pictures of the response of a 2D lattice and light propagation in1D waveguides. We show that a stack of 2D lattices may emulate regularly spacedatoms in a lossless 1D waveguide, and argue that in a suitable geometry theresonance shifts characteristic of 1D and 2D lattice structures may completelycancel to eliminate density dependent resonance shifts of atoms bound to a 3Dlattice. A generalization to the case of anisotropic polarizability, such as inthe presence of a magnetic field, reveals light frequencies induced by themagnetic field for which the lattice is either completely transparent, orcompletely opaque.

Light propagation in systems involving two-dimensional atomic lattices