Interband Contributions to Optical Harmonic Generation at a Metal Surface

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 $\omega$ has been calculated. The calculation of the linear polarization or linear current density leads to the familiar expression for the dielectric constant $\epsilon \left(\omega \right)$. The part of the bilinear polarization varying as $e^{-2i\omega t}$ for free conduction electrons with a potential barrier at the surface is known to have the form ${\mathbf{P}}^{2\left(\omega \right)}\left(\mathrm{NL}\right)={\alpha }_{\mathrm{pl}}{\mathbf{E}}^{\left(\omega \right)}\times {\mathbf{H}}^{\left(\omega \right)}+{\beta }_{\mathrm{pl}}{\mathbf{E}}^{\left(\omega \right)}\mathbf{\nabla }·{\mathbf{E}}^{\left(\omega \right)},$ where ${\mathbf{E}}^{\left(\omega \right)}$ and ${\mathbf{H}}^{\left(\omega \right)}$ are, respectively, the electric and magnetic fields varying as $e^{-i\omega t}$. It is shown that the introduction of a periodic potential leads to the same form for ${\mathbf{P}}^{\left(2\mathrm{}\omega \right)}\left(\mathrm{NL}\right)$ for isotropic metals, with $\alpha$ and $\beta$ 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 ${\mathbf{P}}^{\left(2\mathrm{}\omega \right)}$ may be approximated by a form which may be completely specified in terms of $\epsilon \left(\omega \right)$ and $\epsilon \left(2\mathrm{}\omega \right)$. By solving Maxwell's equations for the fundamental as well as the second-harmonic fields for arbitrary $\epsilon$, $\alpha$, and $\beta$, 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 ${\mathbf{P}}^{\left(2\mathrm{}\omega \right)}\left(\mathrm{NL}\right)$ which cannot be expressed in terms of the linear dielectric constant alone.