Anisotropic extension of the Parratt formalism

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.

Figures (9)

• Figure 1: Multilayer sample and coordinate-system
• Figure 2: Comparison of X-ray reflectometry results for [Ni(7 nm)/Ti(8 nm)] $_{N_\text{rep}}$/Glass multilayer systems for different bilayer repetition numbers $N_\text{rep}.$ The reflectivity was computed using both the transfer matrix method $\left(R_\text{Transfer}\right)$ and the generalized Parratt algorithm $\left(R_\text{Parratt}\right)$ described here. The ratio of these results $\left(R_\text{T}/R_\text{P} = R_\text{Transfer}/R_\text{Parratt}\right)$ is also plotted. The results of numerical instabilities are apparent in the total reflection region and near the first Bragg-peak for $N_\text{rep} = 900.$
• Figure 3: Comparison of neutron reflectometry results for [Cr(4 nm)/Fe(6 nm)]$_{N_\text{rep}}$/MgO multilayer systems for different bilayer repetition numbers $N_\text{rep}.$ The reflectivity was computed using both the transfer matrix method $\left(R_\text{Transfer}\right)$ and the generalized Parratt algorithm $\left(R_\text{Parratt}\right)$ described here. The ratio of these results $\left(R_\text{T}/R_\text{P} = R_\text{Transfer}/R_\text{Parratt}\right)$ is also plotted. The results of numerical instabilities are apparent in and near the total reflection region.
• Figure 4: Comparison of polarised neutron reflectometry results for [Cr(4 nm)/Fe(6 nm)/Cr(4 nm)/Fe(6 nm)/Cr(4 nm)/Fe(6 nm)]$_{N_\text{rep}}$/MgO multilayer systems for different hexalayer repetition numbers $N_\text{rep}.$ In the hexalayers, the iron layers are magnetised in different directions (30°, 120°, 200° measured from the plane of incidence) in the sample plane. The reflectivities for four combined polarizer-analyzer states $(-\,-, -+, +-, ++)$ were computed using both the transfer matrix method $\left(R_\text{Transfer}\right)$ and the generalized Parratt algorithm $\left(R_\text{Parratt}\right)$ described here. The ratio of these results $\left(R_\text{T}/R_\text{P} = R_\text{Transfer}/R_\text{Parratt}\right)$ is also plotted. The results of numerical instabilities are apparent in and near the total reflection region.
• Figure 5: It is the same as Fig. \ref{['fig:FeCrStructAniz']}, but calculated for different repetition numbers $N_\text{rep}.$