Metrology of the Shape of Large-Size and Aspherical X-ray Mirrors with Subnanometer Accuracy

Abstract

This paper is devoted to the problems of metrology of high-precision X-ray mirrors. Approaches to the metrology of the shapes of flat and aspherical X-ray mirrors with the aim of achieving subnanometer accuracy are proposed. Algorithms for reconstructing X-ray mirrors with dimensions exceeding the interferometer aperture are given, and these are also applicable to aspherical mirrors where it is possible to decipher only part of the interferograms. Creating X-ray mirrors with subnanometer RMS error accuracy presents a number of difficulties, one of which is caused by the rings resulting from diffraction of the reference and working fronts during propagation through defects in the optical elements of the interferometer. The optimal approach for removing such rings is filtering of the interferometric data. In the course of the study, we considered both classical smoothing and, quite novel, filtering methods, applicable not only for processing interferometric data images, but also to microelectronic images, and for use with computed tomography. As well as these filters, we implemented our own filter based on Fourier filtering, using some specific parameters of frequency sampling, during which certain of the amplitudes decreased, while others remained unchanged. In addition to diffraction rings, when working to subnanometer accuracy, an error of the reference front occurs. As, the measurement methodology in our case is based on the presence of 1 interferometer with 1 reference surface, therefore, under these conditions, we based the problem on the condition of error asymmetry in the measurement zone, thus we proposed a mathematical model that allowed us to describe the behavior of the mirror in this zone. In addition, to avoid overloading the operating system, we used the methods of linear matrix inequalities to solve the optimization problem, since, in this case we would have to make an initial approximation, and then look for matrix derivatives for each value.