Hydrodynamic model for second-harmonic generation at conductor surfaces with continuous profiles

We develop a hydrodynamic model for the calculation of second-harmonic generation (SHG) at the surface of conductors with arbitrary equilibrium electronic density profiles ${\mathit{n}}_0$(${\mathit{r}}_{\mathrm{\perp }}$). We apply our model to simple profiles and calculate the linear surface conductivity s(ω) and the nonlinear surface susceptibility tensor ${\mathrm{\chi }}_2^{\mathit{s}}$(2ω=ω+ω) for all ω, and we obtained the way they scale with the relevant bulk and surface parameters. The conductivity s(ω) displays a peak that corresponds to the dipolar surface plasmon at a frequency ${\mathrm{\omega }}_{\mathit{d}}$, which depends on the profile shape and width. The susceptibility (${\mathrm{\chi }}_2^{\mathit{s}}$ ${)}_{\mathrm{\perp }\mathrm{\perp }\mathrm{\perp }}$ has very large resonances at ${\mathrm{\omega }}_{\mathit{d}}$ and ${\mathrm{\omega }}_{\mathit{d}}$/2. The SHG efficiency is enhanced by several orders of magnitude at these resonances, suggesting that SHG spectroscopy might be a useful probe of surface collective modes.