A unified framework for the numerical evaluation of the Q-subtractive Kramers-Kronig relations and application to the reconstruction of optical constants of quartz

Abstract

The direct usage of the Kramers-Kronig (KK) relations is complicated by two factors: limited frequency range of the available spectra and experimental errors. Here, we reconsider the application of the KK relations to experimental data for the construction of a self-consistent set of optical constants over a wide spectral range: the real part of the complex optical constant, F₁, is reconstructed using the imaginary part F₂, obtained from an experiment. The focus is on multiply (Q-)subtractive KK relations, which in contrast to the standard KK transformation, exploit information about F₁ at a certain number Q of anchor frequencies. We develop a general mathematical framework of the Q-subtractive KK relations and analyze all sources of errors contributing to the inaccuracy of the reconstructed F₁. We show that for the reconstruction of F₁ only a single evaluation of the standard KK relation is needed together with a correction term given by an approximate evaluation of the error in the standard KK. It is demonstrated that in the classical form of the Q-subtractive KK relations, this correction term coincides with the Lagrange interpolation polynomial of the error with nodes at the anchor frequencies. Another correction term can also be constructed as a lower degree polynomial through a least squares fit, a particular realization of which is taking the average of Q singly subtractive KK relations. As a result, recommendations for the application of Q-subtractive KK relations are given. The accuracy of the considered approaches is illustrated on synthetic examples and experimental data of fused SiO₂.

Keywords: Extinction coefficient; Kramers–Kronig relations; Multiply subtractive; Numerical evaluation; Refractive index.