Modeling of surface metrology of state-of-the-art x-ray mirrors as a result of stochastic polishing process

The design and evaluation of the expected performance of optical systems requires sophisticated and reliable information about the surface topography of planned optical elements before they are fabricated. The problem is especially severe in the case of x-ray optics for modern diffraction-limited-electron-ring and free-electron-laser x-ray facilities, as well as x-ray astrophysics missions, such as the X-ray Surveyor under development. Modern x-ray source facilities are reliant upon the availability of optics of unprecedented quality, with surface slope accuracy $<0.1μrad$. The unprecedented high angular resolution and throughput of future x-ray space observatories require high-quality optics of $100m2$ in total area. The uniqueness of the optics and limited number of proficient vendors make the fabrication extremely time-consuming and expensive, mostly due to the limitations in accuracy and measurement rate of metrology used in fabrication. We continue investigating the possibility of improving metrology efficiency via comprehensive statistical treatment of a compact volume of metrology of surface topography, which is considered the result of a stochastic polishing process. We suggest, verify, and discuss an analytical algorithm for identification of an optimal symmetric time-invariant linear filter model with a minimum number of parameters and smallest residual error. If successful, the modeling could provide feedback to deterministic polishing processes, avoiding time-consuming, whole-scale metrology measurements over the entire optical surface with the resolution required to cover the entire desired spatial frequency range. The modeling also allows forecasting of metrology data for optics made by the same vendor and technology. The forecast data are vital for reliable specification for optical fabrication, evaluated from numerical simulation to be exactly adequate for the required system performance, avoiding both over- and underspecification.