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 structures with metal and dielectric non-magnetic components, including plasmonic waveguides and metamaterials. We verify our approach through a direct comparison with the earlier analysis based on a direct perturbation theory for planar metal-dielectric structures, and furthermore demonstrate the application of our method to the three-dimentional multilayer fishnet metamaterials.
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