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

TL;DR

This paper introduces a reformulated Kramers-Kronig transform with a logarithmic kernel for accurately deriving the complex refractive index from reflectance data in the Drude model, addressing existing formulation ambiguities.

Contribution

The authors propose a new formulation of the Kramers-Kronig transform tailored for the Drude model, including additional terms for improved accuracy in reflection phase analysis.

Findings

Reformulated KKT improves phase and index calculations.

Experimental data supports the new formulation's validity.

Results align with existing literature data.

Abstract

We use 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, there remains some confusion regarding 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.