Dynamic Newton–gradient-direction-type algorithm for multilayer structure determination using grazing X-ray specular scattering: numerical simulation and analysis

Abstract

A new dynamic iterative algorithm code for retrieving macroscopic multilayer structure parameters (the layer thickness and complex refraction index for each layer, the surface roughness and the interface roughness between the layers) from specular scattering angular scan data is proposed. The use of conventional direct methods, particularly the well known Newton algorithm and gradient-direction-type algorithm operating dynamically to minimize the error functional in a least-squares fashion, is explored. Such an approach works well and seems to be effective in solving the inverse problem in the high-resolution X-ray reflectometry (HRXR) method. In order to demonstrate some features of the proposed iterative algorithm, numerical calculations for retrieving three-layer structure parameters are carried out using simulated HRXR angular scan data. The calculations indicate clearly that the dynamic iterative algorithm is convergent and capable of yielding the true solution. It is important that the performance coefficient for successful iterative cycles for the absolute minimization of the HRXR error functional is quite high even if the initial values of the search parameters are chosen rather far from the true values. It is particularly noteworthy that the relative number of successful iterative cycles is of the order of 90-40% when only moderately accurate initial parameter values, varying by +/-10-40% from the true values, are presumed.