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

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.