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

Abstract

Abstract. In the early 1990s, Church and Takacs pointed out that the specification of surface figure and finish of x-ray mirrors must be based on their performance in the beamline optical system. We demonstrate the limitations of specification, characterization, and performance evaluation based on conventional statistical approaches, including root-mean-square roughness and residual slope variation, evaluated over spatial frequency bandwidths that are system specific, and a more refined description of the surface morphology based on the power spectral density distribution. We show that these limitations are fatal, especially in the case of highly collimated coherent x-ray beams, like beams from x-ray free electron lasers (XFELs). The limitations arise due to the deterministic character of the surface profile data for a definite mirror, while the specific correlation properties of the surface are essential for the performance of the entire x-ray optical system. As a possible way to overcome the problem, we treat a method, suggested by Yashchuk and Yashchuk in 2012, based on an autoregressive moving average modeling of the slope measurements with a limited number of parameters. The effectiveness of the approach is demonstrated with an example specific to the x-ray optical systems under design at the European XFEL. © 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.

Stochastic analysis of surface metrology

“…The forecast data are vital for reliable specification for optical fabrication, with evaluation from numerical simulation being necessary and sufficient for the required system performance, avoiding both over-and underspecification. 12,13 In this work, we continue investigations, started in Refs. 8-14; we consider surface slope metrology data sto-chastic and stationary and use a compact volume of metrol-ogy data to develop the TILF model 15 and determine its parameters.…”

“…Models of AR type turned out to be most suited for description of the surface slope data measured with high quality x-ray mirrors as surfaces under testing. [8][9][10][11][12][13][14] In our previous work, 10,11 we have described the construc-tion of InTILF models of AR, MA, and ARMA types (sym-metric ARMA models) and determination of its coefficients. We have shown that the optimal InTILF model of a given type (AR, MA, or ARMA) and given number of coefficients can be derived analytically using the ACF of the data.…”

“…Based on the determined parameters (coefficients of the optimal InTILF), one can generate syn-thetic, statistically equivalent data needed, for example, for numerical simulation of optical system (beamline) perfor-Yðt1; t2Þ ¼ ðk ;k Þ∈M aðk1; k2ÞXðt1 − k1; t2 − k2Þ: (6) mance and fabrication specification of x-ray optics before purchasing. 12,13 The software is designed to process 1-D and 2-D data in formats of the main commercially offered surface The goal of the modeling is to find an optimal filter, deter-mined by a set of coefficients Aopt a l1; l2 , such that the model Y best fit the data X with minimum possible difference between X and its model Y:…”

Ex situ metrology and data analysis for optimization of beamline performance of aspherical pre-shaped x-ray mirrors at the advanced light source

“…The mirror substrates made of single-crystal silicon have overall dimensions of 160 (length) × 50 (width) × 50 (thickness) mm 3 with clear apertures (CA) of 150 mm × 20 mm for SM1 and SM2 mirrors, and 300 × 50 × 75 mm 3 1. The Wolter Type I optical system consisting of a pair of mirrors, one hyperbolic and one elliptical.…”

“…The development of fully coherent free electron lasers and diffraction limited storage ring x-ray sources has brought to a focus the need for higher performing x-ray optics with unprecedented tolerances for surface slope and height errors and roughness. [1][2][3][4][5] For example, the proposed beamlines for the future upgraded Advanced Light Source, ALS-U, 6,7 require mirrors characterized by a residual slope error of <100 nrad (root-mean-square, RMS) and a height error of <1-2 nm (peakto-valley, PV). Such high quality aspherical x-ray mirrors are now available from a number of the most advanced vendors utilizing deterministic polishing techniques.…”