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 xray 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.

Binary pseudorandom array test standard optimized for characterization of large field-of-view optical interferometers

“…A natural drawback of the BPRA generation procedure based on 'folding' of the long 1D random sequence is the possible asymmetry in randomness in two directions, along the rows and along the columns. Therefore, we specially check if the distribution of the PSD variations is random (white noise like) by using a statistical software EViews8 [17], used at the XROL also for stochastical modeling of the results of surface slope metrology with x-ray mirrors (see Refs [18] and references therein). Figure 2 illustrates the application of a standard histogram and statistics test of the EViews8 software to one of twelve BPRA generated with the Mathematica pseudo-RBG [16].…”