Interband Contributions to Optical Harmonic Generation at a Metal Surface

Abstract

The effect of interband transitions of electrons on the linear as well as the bilinear polarization induced in a metal by a light wave of (circular) frequency $\ensuremath{\omega}$ has been calculated. The calculation of the linear polarization or linear current density leads to the familiar expression for the dielectric constant $\ensuremath{\epsilon}(\ensuremath{\omega})$. The part of the bilinear polarization varying as ${e}^{\ensuremath{-}2i\ensuremath{\omega}t}$ for free conduction electrons with a potential barrier at the surface is known to have the form ${\mathbf{P}}^{2(\ensuremath{\omega})}(\mathrm{NL})={\ensuremath{\alpha}}_{\mathrm{pl}}{\mathbf{E}}^{(\ensuremath{\omega})}\ifmmode\times\else\texttimes\fi{}{\mathbf{H}}^{(\ensuremath{\omega})}+{\ensuremath{\beta}}_{\mathrm{pl}}{\mathbf{E}}^{(\ensuremath{\omega})}\mathbf{\ensuremath{\nabla}}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{E}}^{(\ensuremath{\omega})},$ where ${\mathbf{E}}^{(\ensuremath{\omega})}$ and ${\mathbf{H}}^{(\ensuremath{\omega})}$ are, respectively, the electric and magnetic fields varying as ${e}^{\ensuremath{-}i\ensuremath{\omega}t}$. It is shown that the introduction of a periodic potential leads to the same form for ${\mathbf{P}}^{(2\ensuremath{\omega})}(\mathrm{NL})$ for isotropic metals, with $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ now containing both the intraband and interband contributions. Except near a resonance for interband transitions involving at least 3 bands, it is found that in the long-wavelength limit for the light wave the general expression for ${\mathbf{P}}^{(2\ensuremath{\omega})}$ may be approximated by a form which may be completely specified in terms of $\ensuremath{\epsilon}(\ensuremath{\omega})$ and $\ensuremath{\epsilon}(2\ensuremath{\omega})$. By solving Maxwell's equations for the fundamental as well as the second-harmonic fields for arbitrary $\ensuremath{\epsilon}$, $\ensuremath{\alpha}$, and $\ensuremath{\beta}$, general expressions for both linear and bilinear reflection coefficients have been derived. These phenomenological solutions can be used to determine experimentally the residual 3-band contributions to ${\mathbf{P}}^{(2\ensuremath{\omega})}(\mathrm{NL})$ which cannot be expressed in terms of the linear dielectric constant alone.