On the Kramers–Kronig transform with logarithmic kernel for the reflection phase in the Drude model

Abstract

We consider the Kramers-Kronig transform (KKT) with logarithmic kernel to obtain the reflection phase and subsequently the complex refractive index of a bulk mirror from reflectance. However, it remains some confusion on the formulation for this analysis.Assuming the damped Drude model for the dielectric constant and the oblique incidence case, we calculate the additional terms: phase at zero frequency and Blashke factor and we propose a reformulated KKT within this model. Absolute reflectance in the s-polarization case of a gold film is measured between 40 and 350 eV for various glancing angles using synchrotron radiation and its complex refractive index is deduced using the reformulated KKT that we propose. The results are discussed with respect to the data available in the literature.2

Phase retrieval of reflection and transmission coefficients from Kramers–Kronig relations

“…Several authors have worked on this topic in various situations of multilayers [10,11]. However, as pointed out in the literature, serious difficulties remain and result from the existence of zeros in reflection spectra since they produce branch points and cuts when the complex logarithm is used [12][13][14] to separate the phase from the modulus. In addition, to our knowledge, a rigorous proof of the possibility to use Kramers-Kronig relation at oblique incidence has not been established.…”

Effect of reflection and refraction on NEXAFS spectra measured in TEY mode

“…As an example, the Blashke factors correspond to the occurrence of singularities in the complex coefficient of reflection r along the imaginary frequency axis. As follows from the literature (Toll, 1956;Stern, 1963;Young, 1977;Plaskett & Schatz, 1963;Vinogradov et al, 1988;Sheik-Bahae, 2005;Lucarini et al, 2005;Nash et al, 1995;Smith, 1977;Lee & Sindoni, 1997;André et al, 2010), the value of additional terms depends strongly on the behavior of the dielectric constant and on the conditions of the measurements such as oblique or normal incidence, type of the polarization of the incident beam, material of the sample and nature of the mirror (massive or layered system). Nevertheless, in the case of an isotropic massive film and s-polarized radiation, the phase shift , appearing after reflection of the wave at the surface of the mirror, changes in the limits À 0.…”

“…It is known (Toll, 1956;Stern, 1963;Young, 1977;Plaskett & Schatz, 1963;Vinogradov et al, 1988;Sheik-Bahae, 2005;Lucarini et al, 2005;Nash et al, 1995;Smith, 1977;Lee & Sindoni, 1997;André et al, 2010) that generally an integral in the form (8) ambiguously defines the dependence between the research papers imaginary and real parts of a function, i.e. additional arbitrary constants (zero-frequency phase term, Blashke factors) along with their own dispersive integral are possible.…”