Application of the time-invariant linear filter approximation to parametrization of surface metrology with high-quality x-ray optics

Abstract

Abstract. We investigate the time-invariant linear filter (TILF) approach to optimally parameterize the surface metrology of high-quality x-ray optics considered as a result of a stationary uniform random process. The approach is a generalization of autoregressive moving average (ARMA) modeling of one-dimensional slope measurements with x-ray mirrors considered. We show that the suggested TILF approximation has all the advantages of one-sided autoregressive and ARMA modeling, allowing a high degree of confidence when fitting the metrology data with a limited number of parameters. Compared to ARMA modeling, the TILF approximation gains in terms of better fitting accuracy and the absence of the causality limitation. Moreover, the TILF approach can be directly generalized to two-dimensional random fields. With the determined model parameters, the surface topography of prospective beamline optics can be reliably forecast before they are fabricated. These forecast metrology data, containing essential and reliable statistical information about the existing optics which are fabricated by the same vendor and technology, but generally, have different sizes, and slope and height rootmean-square variations, are vitally needed for numerical simulations of the performance of new x-ray beamlines and those under upgrade. © The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

DABAM: an open-source database of X-ray mirrors metrology

“…Church (1984) has suggested the so-called ABC model that solves some of the problems seen. A better modern approach is a method based on an autoregressive moving average (ARMA) modeling of the surface metrology (Yashchuk & Yashchuk, 2012;Yashchuk et al, 2014Yashchuk et al, , 2015a resulting from a stochastic polishing process. It provides a reliable way to describe, model and parametrize the measured residual slopes profiles of the X-ray mirrors.…”

Modeling of surface metrology of state-of-the-art x-ray mirrors as a result of stochastic polishing process

“…In our previous work, 10,11 we have suggested a simple way of fixing the causality problem in ARMA modeling.…”

“…2.1 and in our previous publications, 10,11 we model surface slope measurements with a TILF, which is built based on symmetrization of the ARMA process determined with EViews 8 software. 19 Here, we present an original algorithm for direct optimization of the TILF model without involving results of the ARMA modeling.…”

“…Second, the residual slope variation is treated as a result of a stochastic polishing by fitting with an autoregressive moving average (ARMA) and an extension of ARMA to time-invariant linear filter (TILF) modeling. 10,11 The modeling allows a high degree of confidence in description of the surface topography data (and the polishing process) with a limited number of parameters.…”

Specification of x-ray mirrors in terms of system performance: a new twist to an old plot

“…38,39 Description of a rough surface as the result of an ARMA stochastic process provides a modelbased mechanism for extrapolating the spectra outside the measured bandwidth. 38,39 Recent publications 38,39,55,56 describe a successful application of ARMA modeling to the experimental surface slope data for a 1280 m spherical reference mirror. 57,58 In the next section, we apply ARMA modeling to the results of slope metrology with the mirror, shown in Fig.…”