Stochastic analysis of surface metrology

Abstract

The design and evaluation of the expected performance of optical systems require sophisticated and reliable information about the surface topography for planned optical elements before they are fabricated. Modern x-ray source facilities are reliant upon the availability of optics with unprecedented quality (surface slope accuracy <0.1 μrad). The problem is especially complex in the case of x-ray optics, particularly for the X-ray Surveyor under development and other missions. The high angular resolution and throughput of future x-ray space observatories requires hundreds of square meters of high-quality optics. The uniqueness of the optics and limited number of proficient vendors makes the fabrication extremely time consuming and expensive, mostly due to the limitations in accuracy and measurement rate of metrology used in fabrication. We discuss improvements in metrology efficacy via comprehensive statistical analysis of a compact volume of metrology data. The data are considered stochastic, and a statistical model called invertible time-invariant linear filter (InTILF) is developed now for two-dimensional (2-D) surface profiles to provide compact description of the 2-D data in addition to one-dimensional data treated so far. The InTILF model captures stochastic patterns in the data and can be used as a quality metric and feedback to polishing processes, avoiding high-resolution metrology measurements over the entire optical surface. The modeling, implemented in our BeatMark™ software, allows simulating metrology data for optics made by the same vendor and technology. The data are vital for reliable specification for optical fabrication, to be exactly adequate for the required system performance.