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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, F, is reconstructed using the imaginary part F, obtained from an experiment. The focus is on multiply (Q-)subtractive KK relations, which in contrast to the standard KK transformation, exploit information about F 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 F. We show that for the reconstruction of F 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 SiO.
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http://dx.doi.org/10.1016/j.saa.2022.122157 | DOI Listing |
Spectrochim Acta A Mol Biomol Spectrosc
March 2023
Institute for Theoretical Physics, Johannes Kepler University Linz, Altenbergerstraße 69, 4040 Linz, Austria.
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, F, is reconstructed using the imaginary part F, obtained from an experiment. The focus is on multiply (Q-)subtractive KK relations, which in contrast to the standard KK transformation, exploit information about F at a certain number Q of anchor frequencies.
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