Anomalous Skin Effect for Specular Electron Scattering and Optical Experiments at Non-Normal Angles of Incidence

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The anomalous skin effect for specular electron scattering at the metal surface is studied, permitting the impinging plane wave to have an arbitrary angle of incidence. It is shown that the expressions for the surface impedance for a non-normal angle of incidence obtained by Reuter and Sondheimer as a generalization from their work at normal incidence are correct for $S$ polarization but incorrect for $P$ polarization. The correct surface impedance for $P$ polarization leads to an additional absorption peak in the frequency range ${10}^{\ensuremath{-}2}{\ensuremath{\omega}}_{p}\stackrel{\ensuremath{\sim}}{<}\ensuremath{\omega}\stackrel{\ensuremath{\sim}}{<}{\ensuremath{\omega}}_{p}$, where ${\ensuremath{\omega}}_{p}$ is the free-electron plasma frequency. This additional absorption, particularly pronounced for long electron lifetimes, is investigated in detail. One important conclusion drawn from this work is that, in general, optical experiments performed at non-normal angles of incidence cannot be analyzed in terms of a single complex frequency-dependent dielectric function. In the frequency range of the additional $P$ absorption, two such dielectric functions are needed, one function for describing $P$ polarization and a different function for describing $S$ polarization.