  We present a rigorous method for deriving limits on Raman scattering in
structured media. We exploit this framework to constrain the maximum Raman
signal resulting from a planewave incident on two experimentally relevant
systems, consisting of either a single Raman molecule in the vicinity of a
structured medium or a designable Raman medium (a distribution of Raman
molecules). Results pertaining to metallic and dielectric structures illustrate
the efficacy of structural optimization and the importance of accounting and
co-designing for the nonlinear interplay between pump and signal fields. In
particular, we show that treating the pump-focusing and signal-extraction
processes separately, as has been done in prior works, leads to highly
unrealistic predictions of achievable enhancements. The formulation could
readily find applications in guiding further improvements on surface-enhanced
Raman scattering (SERS) sensitivity and Raman-assisted lasing.
