Sum-of-Squares Bounds on Surface-Enhanced Raman Scattering

TL;DR

This paper establishes fundamental limits on the enhancement of surface-enhanced Raman scattering (SERS) using sum-of-squares programming, providing insights into optimal nanostructure design for maximizing light-matter interactions.

Contribution

It introduces the first application of SOS techniques to optics, deriving tight bounds on SERS enhancement for periodic metasurfaces and revealing fundamental performance limitations.

Findings

Bounds on SERS enhancement are remarkably tight compared to inverse-designed structures.

High-Q guided modes can theoretically achieve diverging SERS enhancement despite material loss.

Metallic structures face fundamental limitations for E_z polarized drive fields due to surface plasmon restrictions.

Abstract

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 first use of SOS techniques to optics, overcoming difficulties that prior bounding techniques have with regards to non-linear photonic processes with higher order figures of merit. Our bounds on the $\int \lVert \mathbf{E} \rVert^4 \text{d} \mathbf{r}$ 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.