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 root-mean-square variations, are vitally needed for numerical simulations of the performance of new x-ray beamlines and those under upgrade.
Highlights
Development of new beamlines for third-generation synchrotron radiation sources and free electron lasers is reliant upon the availability of x-ray optics of unprecedented quality, with a surface slope accuracy in the range of 0.1 to 0.2 μrad and a surface height error of less than 1 nm.[1,2,3,4,5] The uniqueness of the optics and the limited number of proficient vendors makes the fabrication of such optics extremely time consuming and expensive
We show that the time-invariant linear filter (TILF) approximation gains a better fitting accuracy and is free from the causality problem, compared to autoregressive moving average (ARMA) modeling of the surface metrology data
This paper is organized as follows: In Sec. 2, we briefly review the mathematical fundamentals of one-dimensional (1-D) ARMA modeling of topography of random rough surfaces
Summary
Development of new beamlines for third-generation synchrotron radiation sources and free electron lasers is reliant upon the availability of x-ray optics of unprecedented quality, with a surface slope accuracy in the range of 0.1 to 0.2 μrad and a surface height error of less than 1 nm.[1,2,3,4,5] The uniqueness of the optics and the limited number of proficient vendors makes the fabrication of such optics extremely time consuming and expensive. The PSD distributions appear as highly smoothed versions of the corresponding estimates via a direct digital Fourier transform (DFT).[9,10] The description of a rough surface, as the result of an ARMA stochastic process, provides a model-based mechanism for extrapolating the spectra outside the measured bandwidth.[9,10]. One-sided ARMA modeling, the “twosided symmetrical ARMA” model, depicted by Eq (9), is free of the limitations of the fixed direction (time flow) and causation This implies that the current value of the surface slope depends on the past and the future, in our case the neighboring points with the positive and negative lag values. In the case of causal TILFs (like AR and ARMA models), this can be intuitively understood as a result of averaging of the residual noises of the fits with the corresponding causal filters of the direct and reversed processes. Assuming that the residual noises are not mutually correlated, one should expect a suppression of the variance of the averaged residual noise by a factor of 2 with respect to the corresponding causal filter [compared with the variance of the second sum in Eq (9)]
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