Sum-of-squares bounds on surface-enhanced Raman scattering

Surface-enhanced Raman scattering (SERS) is a critical tool for chemical sensing and spectroscopy, and a key question is how to optimally design nanostructures for maximizing SERS. We present fundamental limits on spatially averaged SERS via periodic metasurfaces, derived using sum-of-squares (SOS) programming. This work represents the use of SOS techniques in optics, overcoming difficulties that prior bounding techniques have with regard to non-linear photonic processes with higher-order figures of merit. Our bounds on the ∫∥ E ∥ 4 d r SERS enhancement factor for 2D examples demonstrate remarkable tightness when compared with inverse-designed dielectric and metallic structures for both electrical field out-of-plane ( E z ) and in-plane ( H z ) polarizations. We show that delocalized high-Q guided modes can achieve significant, theoretically diverging SERS enhancement even in the presence of material loss. For metallic structures, we demonstrate a fundamental performance limitation for E z -polarized drive fields due to surface plasmon excitation restrictions. By varying the separation between Raman-active molecules and the metasurface design region, we also find material-dependent bounds on the maximum strength of field singularities. Our results offer insights into optimal metasurface design strategies for enhancing light–matter interactions, and our methodology may be adapted to the study of other nonlinear photonics design problems.