Nonlinear coupled-mode theory for periodic waveguides and metamaterials with loss and gain

We develop a systematic procedure for deriving the coupled-mode equations describing the spatial evolution of the slowly-varying amplitudes of electromagnetic modes in nonlinear periodic structures with loss and gain. Our approach is rigorously based on the Lorentz reciprocity theorem, and therefore it can be applied to a broad range of non-magnetic structures with metal and dielectric components, including plasmonic waveguides and metamaterials. We verify our approach through a direct comparison with the previous analysis based on a direct perturbation theory for planar metal-dielectric structures, and furthermore demonstrate the application of our method to the fishnet metamaterials