  The low efficiency of Raman spectroscopy can be overcome by placing the
active molecules in the vicinity of scatterers, typically rough surfaces or
nanostructures with various shapes. This surface-enhanced Raman scattering
(SERS) leads to substantial enhancement that depends on the scatterer that is
used. In this work, we find fundamental upper bounds on the Raman enhancement
for arbitrary-shaped scatterers, depending only on its material constants and
the separation distance from the molecule. According to our metric, silver is
optimal in visible wavelengths while aluminum is better in the near-UV region.
Our general analytical bound scales as the volume of the scatterer and the
inverse sixth power of the distance to the active molecule. Numerical
computations show that simple geometries fall short of the bounds, suggesting
further design opportunities for future improvement. For periodic scatterers,
we use two formulations to discover different bounds, and the tighter of the
two always must apply. Comparing these bounds suggests an optimal period
depending on the volume of the scatterer.
