Optical dispersion equations for metals applicable to the Far-IR through EUV spectral range

Abstract

Abstract The long-standing problem of finding a general formulation of optical dispersion for metals, valid over a wide spectral range of photon energy E is tackled. To this end, equations for refractive index n and extinction coefficient k as functions of E are developed. Functions n E and k E respectively represent real and imaginary parts of complex index of refraction N E . Previous formulations, most of which are based on various combinations of Drude and Lorentz models, are either useable only over a limited spectral range or do not accurately fit experimental data. The formulation overcomes these shortcomings by exploiting concepts set forth by Forouhi and Bloomer in 1986, 1988 and 2019 publications pertaining to optical dispersion of semiconductors and insulators. These concepts are centered on time-dependent perturbation theory and consistency with principle of causality. The new expression for k E is based on three types of events initiated by photon-electron interactions in metals: intraband electron dipole transitions; interband electron dipole transitions; inelastic collisions of electrons. Expression for n E is obtained as Hilbert transform of k E . It is demonstrated that the new dispersion equations satisfy Titchmarsh’s Theorem, a mathematical theorem that conveys the principle of causality, which in turn establishes the theoretical validity of the formulation. Equations for n E and k E are fitted to published experimental data of metals encompassing all metal groups of the periodic table. Reported data span various ranges of energy, from far-infrared to extreme-ultraviolet. Excellent fits between calculated and experimental spectra are achieved. Having established consistency with Titchmarsh’s Theorem and by extension, causality, plus agreement with experimental findings suggests this formulation represents a valid description of optical dispersion of metals.