  We present a general framework for inverse design of nanopatterned surfaces
that maximize spatially averaged surface-enhanced Raman (SERS) spectra from
molecules distributed randomly throughout a material or fluid, building upon a
recently proposed trace formulation for optimizing incoherent emission. This
leads to radically different designs than optimizing SERS emission at a single
known location, as we illustrate using several 2D design problems addressing
effects of hot-spot density, angular selectivity, and nonlinear damage. We
obtain optimized structures that perform about 4 times better than coating with
optimized spheres or bowtie structures and about 20 times better when the
nonlinear damage effects are included.
