Anisotropic extension of the Parratt formalism

TL;DR

This paper extends the Parratt formalism to anisotropic multilayer systems, providing a stable and accurate method for neutron and X-ray reflectometry analysis, overcoming numerical issues of previous matrix-based approaches.

Contribution

It introduces a generalized Parratt method for anisotropic systems that avoids numerical instabilities inherent in transfer matrix methods.

Findings

The new method is numerically stable for thick samples at grazing angles.

Derived formulas for reflectivity and transmissivity applicable to anisotropic layers.

Comparison shows improved stability and accuracy over existing methods.

Abstract

Neutron and X-ray reflectometry are important methods for studying thin multilayer systems. The Parratt method and the method of characteristic matrices, also referred to as transfer matrices, are used for simulation, evaluation of experimental results, and designing optical systems, like mirrors. The Parratt method had been derived for isotropic systems. The method of characteristic matrices can also handle anisotropic problems, but it is burdened with numerical instabilities, which may arise in the case of thick samples at grazing angle incidence. In this paper, we derive a generalized Parratt method applicable to anisotropic systems. Furthermore, as we show, this is devoid of the numerical instabilities arising in the method of characteristic matrices. We derive formulae for both reflectivity and transmissivity. The stability of the new approach is demonstrated by comparing calculated results obtained via different methods. The problem of rough interfaces is also addressed, and the results gained by different approximations for some systems are compared.