Nonlinear optical response of a gold surface in the visible range: A study by two-color sum-frequency generation spectroscopy. II. Model for metal nonlinear susceptibility

Abstract

We present a modeling of the nonlinear optical response of a metal surface in order to account for recent experimental results from two-color Sum-Frequency Generation (SFG) experiments on gold. The model allows calculating the surface and bulk contributions and explicitly separates free and bound electron terms. Contrary to the other contributions, the perpendicular surface component is strongly model-dependent through the surface electron density profiles. We consider three electron density schemes at the surface, with free and bound electrons overlapping or spilling out of the bulk, for its calculation. The calculated SFG signals from the metal rely only on bulk quantities and do not need an explicit definition of the density profiles. In the particular case of gold, when the free electrons overlap with the bound ones or spill out of the bulk, the free electron response completely dominates through the perpendicular surface terms. When the bound electrons spill out, the situation is more balanced, still in favor of the free electrons, with lower amplitudes and different dispersion line shapes. As for silver, the free electron contributions dominate and the calculated slow amplitude growth from blue to red follows the experimental trends.

Determining nonlinear optical coefficients of metals by multiple angle of incidence heterodyne-detected sum-frequency generation spectroscopy

“…The second-order nonlinear optical response of a metal surface has been studied soon after the first reported evidence of second harmonic generation (SHG), [29][30][31] and continues to be of interest in the SHG and SFG community. [32][33][34][35][36][37][38][39][40] The description of the nonlinear gold response presented here follows from the general phenomenological model of surface and bulk SFG response, 27,29,[41][42][43] analogous to SHG models, such as that developed by Mizrahi and Sipe. 44 For a polycrystalline cubic material, the bulk and surface behave essentially like an isotropic system.…”

“…This formulation leads to a single, compact but intricate, bulk contribution to the effective nonlinear susceptibility that is added to the surface terms. 27,42,43 In the isotropic case, the bulk source is parametrized by the non-vanishing coefficients of the quadrupolar nonlinear susceptibility defined by…”

“…3 from the electronic properties of gold, 43 while a more rigorous description requires separating the contributions from free and bound electrons. 27,62 Surface and bulk effective susceptibilities, calculated separately, reconstruct the total SFG response from gold. 28 The second formulation for the bulk terms describes the bulk as an infinite stack of thin plates, just like the surface terms.…”

“…Among the bulk separable terms of the equivalent nonlinear surface susceptibility, those that do not vary with θ behave exactly like a surface term in our experiment. Following our previous work, 27 we have calculated them for both the free and bound electron populations of gold, and display their magnitudes in Fig. 2.…”

“…thiolated) self-assembled molecular layers, its chemical stability in air and its intense SFG nonresonant response (due to contributions from its free and bound electrons. 27,28 ) We begin by introducing some formalism that relates the surface fields to the incidence and reflected SFG fields. We also describe the quadrupolar response from the bulk of the material.…”