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, SiO 2 -Glass, LiF, Polyethylene, a-Si, InSb, GaP, and As 2 Se 3 . 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.

Simulation of optical properties of semiconductor multilayers from extreme ultraviolet to far infrared

“…Both n and k of Si are consistent with the results of Forouhi et al 9 In Figure 2b, n of SiO 2 glass exhibits two peak values -one peak in the energy range of 0.06-0.16eV and another peak value of 2.45 at 10eV, which may be due to the inter-band absorption. 6 k of SiO 2 has almost a similar pattern as n; the behavior of both k and n, with photon energy, are consistent with those of Forouhi et al 6 In Figure 2c, the reflectance of Si has a high value of >60% in the energy range of 3.2-5.6eV (~ 220-400nm (UVA to UVC)). In Figure 2d, the reflectance of SiO 2 exhibits several peaks with values >20% from 0.06-5eV.…”

“…Extinction coefficient, k, characterizes the ability of the light beam to penetrate into the material. The Forouhi-Bloomer dispersion equation, 6 applicable to the entire wavelength from radio-wave to EUV spectral range, is proposed as in Eq. (I) and Eq.…”

“…Optical functions, n and k, are intrinsic material properties and may be thought of as the fingerprint of a material, and cannot be measured directly. 6 They are determined from extrinsic measurements that depend on their surface morphology, dopant type and dopant concentration/resistivity etc. (e.g., reflectance and transmittance, or ellipsometric parameters).…”

Effect of Ag2S nanoparticles on optical, photophysical, and electrical properties of P3HT thin films

“…The optical parameters were calculated by fitting the experimental data using ‘ new amorphous dispersion formula ’ given by [ 39,40 ] : with, where, n(ω ) is the refractive index at angular frequency ω, n ∞ is the refractive index when ω → ∞, and k(ω) is the extinction coefficient, The parameter f i is a dimensionless constant, which is proportional to the interband electron transition probabilities. [ 41 ] Spectroscopic ellipsometer data of fabricated films was fitted using the above model to yield a chi‐squared value χ 2 < 2 for all samples. The derived optical constants as a function of wavelength are shown in Figure 5(a, b).…”

THz-Photonics Transceivers By All-Dielectric Phonon-Polariton Nonlinear Nanoantennas

“…Apart from the linear solution of the fields at the pump optical frequencies ω 1 and ω 2 , to calculate the nonlinear DFG signal at ω 3 the knowledge of the linear material properties at the THz frequency is required. In order to consider a valid model for our simulations, we have implemented the so-called Forouhi-Bloomer (F-B) dispersion relations 27 . These are a set of equations for the complex refractive index N(E) = n(E) − ik(E) of insulators and semiconductors as a function of photon energy E, which are valid over wide ranges of the EM spectrum, spanning from radio waves up to extreme UV.…”