Fig. 1: Theory of nonlinear vibrational coupling in plasmonic nanocavities.

a Schematic of 80 nm nanoparticle-on-mirror (NPoM) containing 1.3 nm-thick self-assembled monolayer (SAM) of biphenyl-4-thiol (BPT) molecules, showing benzene ring stretch at 1586 cm⁻¹. b Nanogap supports localized plasmon modes (red). BPT molecular Raman dipoles (s, s') interact via their image molecules (pink arrow) and through plasmon modes (blue arrow). c Complex self-interaction Green’s function (\({{{\rm{Re}}}}\{G\}\) = red, \({{{\rm{Im}}}}\{G\}\) = blue) for response produced by a vertical dipole in the gap centre. Scattering cross section (grey) shows dominant localized plasmonic (10) (lowest order) and (20) (second order) modes (vertical dashed) at λ₍₁₀₎ ≈ 830 nm, λ₍₂₀₎ ≈ 670 nm, and a peak in ImG at \({\lambda }_{{PPM}}\,\approx \, 520\) nm identified as the plasmon pseudo-mode (PPM), originating from overlapping higher order modes. Dashed curves show G when only a single-optical-mode is considered in the model. d Two-point Green’s function between spatially separated locations in the gap (separation ρ), at \({\lambda }_{\left(20\right)}\) = 670 nm as obtained numerically (solid line) and with an analytic model based on image dipoles (dashed lines, see Supplementary Note S3.1). Inset shows midgap electric near-field at \({\lambda }_{(20)}\), with a scale bar 10 nm.