New dispersion equations for insulators and semiconductors valid throughout radio-waves to extreme ultraviolet spectral range

Abstract

A new formulation of optical dispersion in terms of complex index of refraction, N(E) as a function of photon energy, E, applicable to insulators and semiconductors, is introduced. Previous formulations, developed over the past 200 years, are either valid only over a limited spectral range, or do not accurately fit experimental data, or do not conform to the Principle of Causality. The new formulation overcomes these shortcomings. Its validity is established based on theoretical and experimental considerations. The theoretical basis stems from consistency with Titchmarsh’s Theorem, a comprehensive mathematical expression of the Principle of Causality. In accordance with Titchmarsh’s Theorem, the expressions for the real and imaginary parts of N(E), the refractive index, n(E), and extinction coefficient, k(E), are shown to be skew reciprocal Hilbert Transforms. Also, the expression for the inverse Fourier Transform of N(E) is shown to be a complex-valued causal function of time. The causal function is identified as the polarization of the medium, P(t), if photons reach the medium at an initial time equal to zero. P(t) is associated with photon excitations of electrons and other quantum constituents of the media. The experimental basis of the formulation stems from demonstrating exceptionally close agreement with published experimental n(E) and k(E) data of different forms of insulators and semiconductors. Specifically, the formulation is applied to data of Water, Ice, SiO2-Glass, LiF, Polyethylene, a-Si, InSb, GaP, and As2Se3. Data for Water span thirteen orders of magnitude, and fourteen for Ice, over the spectral range of Radio-Waves to EUV. Data of the other materials involved smaller spectral ranges. Results indicate that fewer fitting parameters are needed over any segment of the electromagnetic spectrum, compared to other formulations. These theoretical and experimental findings suggest the formulation represents a universal description of optical dispersion of insulators and semiconductors.