Parameter detection of thin films from their X-ray reflectivity by support vector machines

Abstract

Reflectivity measurements are used in thin film investigations for determining the density and the thickness of layered structures and the roughness of external and internal surfaces. From the mathematical point of view the deduction of these parameters from a measured reflectivity curve represents an inverse problem. At present, curve fitting procedures, based to a large extent on expert knowledge are commonly used in practice. These techniques are very time consuming and suffer from a low degree of automation. In this paper, we present a new method for the evaluation of reflectivity curves by the sparse approximation of multivariate vector-valued function mapping the reflectivity curves directly onto the thin film parameter set. This is the first method which solves the problem in a reasonable amount of time. Our approach utilizes an extended version of the optical matrix method as well as support vector machines for regression working in parallel. The solution of the corresponding quadratic programming problem makes use of the SVMTorch algorithm. We present numerical investigations to assess the performance of our method using models of practical relevance. It is concluded that the approximation by support vector machines represents a very promising tool in X-ray reflectivity investigations and seems also to be applicable for a much broader range of parameter detection problems in X-ray analysis.