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Monoatomically thin polarizable sheets

We consider a flat lattice of dipoles modeled by harmonic oscillators interacting with the electromagnetic field in dipole approximation. Eliminating the variables from the coupled equations of motion, we come to effective Maxwell equations. These allow for taking the lattice spacing $a$ to zero. As a result, we obtain reflection coefficients for the scattering of electromagnetic waves off the sheet. These are a generalization of that known from the hydrodynamic model. For instance, we get a non trivial scattering for polarizability perpendicular to the sheet. Also we show that the case of a sheet polarizable parallel to the sheet can be obtained in a natural way from a plasma layer of finite thickness. As an alternative approach we discuss the elimination of the electromagnetic fields resulting in effective equations for the oscillators. These show, for $a\to 0$, divergent behavior, resulting from the electrostatic interaction of the dipoles.