Analysis of reflectance data using the Kramers-Kronig Relations

Abstract

Kramers-Kronig Relations (KKR) are a well-known tool to interpret reflectance spectra by reconstructing the reflected wave's phase from its modulus with the help of a dispersion relation. However, a unscrupulous application is only possible in the case of semiinfinite media at perpendicular incidence.

Here the method is generalized to oblique incidence and layered structures. We discuss the KKR's theoretical foundations and their significance for the phase retrieval problem. Firstly, convergence problems occur, in particular in the case of Fabry-Perot-interferences of layered samples. They are removed by a special normalization procedure which is discussed in detail for IR spectra. Since the phase-retrieval procedure is based on a transformation of the “un-physical” quantity ln R, KKR cannot be applied straightforward. Additional terms have to be considered due to zeroes of the sample's reflection coefficient in the physical half plane. These terms are explained. A method is introduced to evaluate them with the help of few additional data. Finally we discuss an alternative procedure allowing the phase retrieval without further data using physical confinements. As examples the methods are applied to reflectance spectra of BaF₂ and NaCl measured to investigate the materials' multiphonon processes.