Reliable before-fabrication forecasting of expected surface slope distributions for x-ray optics

Abstract

Numerical simulation of the performance of new beamlines and those under upgrade requires sophisticated and reliable information about the expected surface slope and height distributions of planned x-ray optics before they are fabricated. Obtaining such information should be based on the metrology data measured from existing mirrors that are made by the same vendor and technology; but, generally, with different sizes, slope and height rms variations. In this work, we demonstrate a method for highly reliable forecasting of the expected surface slope distributions of the prospective x-ray optics. The method is based on an autoregressive moving average (ARMA) modeling of the slope measurements with a limited number of parameters. With the found parameters of the ARMA model, the surface slope profile of an optic with the newly desired specification can reliably be forecast. We demonstrate the high accuracy of this type of forecasting by comparing the power spectral density distributions of the measured and forecast slope profiles.

Application of time-invariant linear filter approximation to parametrization of one- and two-dimensional surface metrology with high quality x-ray optics

“…9,10 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 Trustworthy ARMA modeling and forecasting based on a limited number of observations assumes statistical stability of the data used. The data are statistically stable if they are the result of a so called wide sense stationary (WSS) random process (see, for example, Ref.…”

“…1 reproduce the results of ARMA modeling performed in Refs. 9,10 The measured residual slope trace is shown with the short-dashed red line after subtracting the best-fit spherical surface shape with a radius of curvature of 1287.5 m. The trace consists of N=547 points measured with an increment of 0.2 x Δ = mm. The fitted slope trace, shown in Fig.…”

“…So far, following the early work, 9,10 we applied a standard ARMA modeling to the surface slope data. The ARMA modeling is inherently causal, assuming that the current value of the process depends only upon the past, as expressed with Eq.…”

“…The data in the table are the regression outputs generated by EViews 8 software. Note that the values of the ARMA parameters presented here are slightly different from that of the Ref., 9,10 where the software version 7 was used. However, the difference is well within the confidence interval for the parameters.…”

“…The surface description based on the AR model or the extended autoregressive moving average (ARMA) model 7,8 provided a way to replace the spectral estimation problem by that of parameter estimation. *vvyashchuk@lbl.gov; phone 1 510 495-2592; fax 1 510 486-7696; www-esg.lbl.gov/Production/OML/index.html In the recent work, 9,10 highly reliable forecasting of the expected surface slope distributions of the prospective x-ray optics based on an ARMA modeling has been demonstrated. In this case, ARMA modeling is applied to the surface slope metrology data obtained with existing optics fabricated by the same prospective vendor with the same technology.…”