Curvefitting Imaginary Components of Optical Properties: Restrictions on the Lineshape Due to Causality

Abstract

The Kramers–Kronig transformation has been extensively applied in optical spectroscopy to calculate the real component of an optical quantity from the imaginary component, such as the real refractive index from the imaginary component, or vice versa. In this paper, the traditional proof of the Kramers–Kronig transformation, and its application to the complex refractive index, complex dielectric constant, and complex molar polarizability, are reviewed. Often the imaginary components of these quantities are fitted with standard lineshapes such as the Gaussian, Lorentzian, or Classical Damped Harmonic Oscillator (CDHO) lineshapes. It is shown that the usual Gaussian and Lorentzian lineshapes do not meet the physical criteria of these imaginary components nor the conditions of the Kramers–Kronig transformation since they are not odd functions of wavenumber. However, the CDHO lineshape meets the physical criteria of the imaginary components of these optical quantities and the Kramers–Kronig transformation. Modifications are presented that make the Gaussian and Lorentzian odd. The Gaussian decays so fast that the modification is not needed in practice; however, the Lorentzian is much slower to decay and thus modification is necessary whenever fitting peaks below ∼250 cm−1. Since the computational difference between the usual Lorentzian and modified Lorentzian is negligible, the author recommends that only the modified Lorentzian be used when fitting bands with a Lorentzian lineshape.