  We apply a rigorous eigenmode analysis to study the electromagnetic
properties of linear and weakly nonlinear metamaterials. The nonlinear response
can be totally described by the linear eigenmodes when weak nonlinearities are
attributed to metamaterials. We use this theory to interpret intrinsic
second-harmonic spectroscopy on metallic metamaterials. Our study indicates
that metamaterial eigenmodes play a critical role in optimizing a nonlinear
metamaterial response to the extent that a poorly optimized modal pattern
overwhelms the widely recognized benefits of plasmonic resonant field
enhancements.
