X-ray astronomy offers the opportunity to observe important phenomena, including the early accretion of massive black holes and detecting diffuse ionized intergalactic gas that is heated to X-ray temperatures (>106). One of the technical challenges facing X-ray astronomy is fabricating optics that are properly shaped and smooth enough to produce quality images. Surface defects on the order of the wavelength of the observed spectrum and up to the size of the optical surface must be polished out of the mirrors without leaving a detectable pattern because the detectable signal is on the order of magnitude of the noise. This leads to a cycle of polishing and metrology that adds time and expense to optics fabrication.
Innovative, computationally efficient numerical methods that detect patterns in metrology data have the potential to reduce the number of polishing cycles by directing the Computer Numerical Control (CNC) polishing machines to remove specific defects without creating new polishing artifacts. This leads to reduced cost for X-ray astronomy projects, including NASA's proposed X-ray Surveyor Mission.
The Invertible Time Invariant Linear Filter (InTILF) method for X-ray metrology data analysis allows for the detection of a faint pattern left on a surface of an X-ray mirror by polishing tools. The InTILF filters are sets of small numbers of coefficients, and they provide an optimal, statistically valid description of the surface data. The information contained in the description can be used for surface quality analysis for repolishing, surface quality comparison, surface data simulations, and as feedback to the polishing tools. InTILF analysis can be used for polishing parameter optimization. This will lead to optimization of the entire polishing and metrology cycle, open the exclusive X-ray market to smaller optics producers, and bring significant cost savings to NASA. Moreover, the parameterization and optimization of the mirror production cycle will position it for eventual automation and further significant reduction in manufacturing cost.
The InTILF algorithm helps detect patterns left on a surface by a polishing tool; the ideal surface won't have any. The information on the pattern is indicative of the surface quality and informs repolishing if needed. The coefficients of the InTILF are found analytically by solving a system of equations that states the equality of Autocovariance Functions (ACF) of the data and the model. In some cases the equations are solved approximately, and then the ACF of the model and the data are numerically very close, which also ensures that Power Spectral Density (PSD) of the model is very close to that of the data. Generalization of the method to 2D stochastic fields also was developed to be used with 2D metrology data.
The approach is new because traditionally, the frequency analysis (Fourier Transform) technique was used to study patterns in metrology data. However, with the advances of deterministic polishing methods, the ultrafine surfaces they produce are better processed and the traces of polishing are faint. An ideal mirror surface is devoid of pattern entirely. As the polishing methods advance, the pattern becomes very faint, and its signal is buried in the noise of small and random polishing artifacts. The character of metrology data becomes stochastic. Statistical methods are best suited to study stochastic signals, but they were not previously used to study metrology data.