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 n₀(r_⊥). We apply our model to simple profiles and calculate the linear surface conductivity s({ω}) and the nonlinear surface susceptibility tensor χ₂ˢ(2{ω}={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 ωd, which depends on the profile shape and width. The susceptibility (χ₂ˢ{)}_{{{⊥}}{{⊥}}{{⊥}}}$ has very large resonances at ${{{ω}}}d$ and ${{{ω}}}d$/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.