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 ω has been calculated. The calculation of the linear polarization or linear current density leads to the familiar expression for the dielectric constant ε(ω). The part of the bilinear polarization varying as e^-2iωt for free conduction electrons with a potential barrier at the surface is known to have the form P^2(ω)(NL)=α_plE^(ω) ×H^(ω)+β_plE^(ω)∇·E^(ω), where E^(ω) and H^(ω) are, respectively, the electric and magnetic fields varying as e^-iωt. It is shown that the introduction of a periodic potential leads to the same form for P^(2ω)(NL) for isotropic metals, with α and β 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 P^(2ω) may be approximated by a form which may be completely specified in terms of ε(ω) and ε(2ω). By solving Maxwell's equations for the fundamental as well as the second-harmonic fields for arbitrary ε, α, and β, 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 P^(2ω)(NL) which cannot be expressed in terms of the linear dielectric constant alone.