The design and evaluation of the expected performance of optical systems requires sophisticated and\nreliable information about the surface topography of planned optical elements before they are fabricated. The\nproblem is especially severe in the case of x-ray optics for modern diffraction-limited-electron-ring and free-electron-\nlaser x-ray facilities, as well as x-ray astrophysics missions, such as the X-ray Surveyor under development.\nModern x-ray source facilities are reliant upon the availability of optics of unprecedented quality, with surface\nslope accuracy <0.1 �¼rad. The unprecedented high angular resolution and throughput of future x-ray space\nobservatories require high-quality optics of 100 m2 in total area. The uniqueness of the optics and limited number\nof proficient vendors make the fabrication extremely time-consuming and expensive, mostly due to the limitations\nin accuracy and measurement rate of metrology used in fabrication. We continue investigating the possibility of\nimproving metrology efficiency via comprehensive statistical treatment of a compact volume of metrology of\nsurface topography, which is considered the result of a stochastic polishing process. We suggest, verify,\nand discuss an analytical algorithm for identification of an optimal symmetric time-invariant linear filter\nmodel with a minimum number of parameters and smallest residual error. If successful, the modeling could\nprovide feedback to deterministic polishing processes, avoiding time-consuming, whole-scale metrology measurements\nover the entire optical surface with the resolution required to cover the entire desired spatial frequency\nrange. The modeling also allows forecasting of metrology data for optics made by the same vendor and technology.\nThe forecast data are vital for reliable specification for optical fabrication, evaluated from numerical simulation\nto be exactly adequate for the required system performance, avoiding both over- and underspecification.\n�© The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in\npart requires full attribution of the original publication, including its DOI. [DOI: 10.1117/1.OE.55.7.074106]
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