RISE: robust iterative surface extension for sub-nanometer X-ray mirror fabrication

Abstract

Precision optics have been widely required in many advanced technological applications. X-ray mirrors, as an example, serve as the key optical components at synchrotron radiation and free electron laser facilities. They are rectangular silicon or glass substrates where a rectangular Clear Aperture (CA) needs to be polished to sub-nanometer Root Mean Squared (RMS) to keep the imaging capability of the incoming X-ray wavefront at the diffraction limit. The convolutional polishing model requires a CA to be extended with extra data, from which the dwell time is calculated via deconvolution. However, since deconvolution is very sensitive to boundary errors and noise, the existing surface extension methods can hardly fulfill the sub-nanometer requirement. On one hand, the figure errors in a CA were improperly modeled during the extension, leading to continuity issues along the boundary. On the other hand, uncorrectable high-frequency errors and noise were also extended. In this study, we propose a novel Robust Iterative Surface Extension (RISE) method that resolves these problems with a data fitting strategy. RISE models the figure errors in a CA with orthogonal polynomials and ensures that only correctable errors are fit and extended. Combined with boundary conditions, an iterative refinement of dwell time is then proposed to compensate the errors brought by the extension and deconvolution, which drastically reduces the estimated figure error residuals in a CA while the increase of total dwell time is negligible. To our best knowledge, RISE is the first data fitting-based surface extension method and is the first to optimize dwell time based on iterative extension. An experimental verification of RISE is given by fabricating two elliptic cylinders (10 mm × 80 mm CAs) starting from a sphere with a radius of curvature around 173 m using ion beam figuring. The figure errors in the two CAs greatly improved from 204.96 nm RMS and 190.28 nm RMS to 0.62 nm RMS and 0.71 nm RMS, respectively, which proves that RISE is an effective method for sub-nanometer level X-ray mirror fabrication.

A comprehensive review of dwell time optimization methods in computer-controlled optical surfacing

“…To avoid this issue, it is necessary to choose a 𝑧 𝑒 (x) that is at least larger than the CA by the diameter (rather than the radius) of the TIF. If the extra area is unavailable during measurement, surface extrapolation [32,45,70,71] is necessary to generate the missing data before applying the FIM. It is worth noting that, in this study, to ensure fair comparison without edge effects, we analytically extend the nominal removal maps shown in Fig.…”

“…The function-form methods include the function-form iterative method (FIM) [10,37,38,39,40,41], Bayesian method (BAM) [28,42], Fourier transform methods (FTM) [20,38,43], robust iterative Fourier transform-based dwell time optimization algorithm (RIFTA) [44], robust iterative surface extension-based method (RISE) [45], and others [46,47,48]. The matrix-form methods comprise the matrixform iterative method (MIM) [49], Tikhonov-regularized methods [11,16,17,29,30,31,50,51,52,53,54,55,56,57,58], constrained linear least-squares methods (CLLS) [30,59,60], and universal dwell time optimization method (UDO) [32].…”

Statistical Tool Size Study for Computer-Controlled Optical Surfacing

“…Computer-controlled optical surfacing (CCOS) [1,2] systems have been successfully used to fabricate high-precision optics in various cutting-edge applications, such as telescopes for space exploration [3][4][5], X-ray mirrors for synchrotron radiation and free-electron laser facilities [6][7][8][9][10], and optics in EUV lithography [11,12]. Different CCOS systems use different tools, which can be adopted based on the requirements for the precision and shape of the desired optical surface.…”

High-Precision Ion Beam Figuring of X-Ray Plane Mirrors for the Bendable KB Focusing System

“…They first proposed an effective 1D-IBF method with improved calibration of coordinate correspondence and dwell time calculation [13]. Several new dwell time calculations and optimization algorithms were further developed to reduce the estimated figure error residuals and enhance the computation efficiency [14,15]. These methods were demonstrated in their own-built IBF system.…”

“…These methods were demonstrated in their own-built IBF system. Two elliptical mirrors of 80 mm in length were fabricated from cylinders which showed 0.62 and 0.71 nm RMS figure error, respectively [14]. The figure error of a flat mirror over an aperture of 92.3 mm × 15.7 mm was reduced from 6.32 to 0.2 nm RMS [15].…”