Description: Numerical simulation of the performance of new x-ray mirror performed by NASA and those under upgrade requires sophisticated and reliable information about the expected surface slope and height distributions of prospective x-ray optics before the optics are fabricated. Ideally, such information has to be based on the metrology data obtained from existing optics fabricated by the same vendor and technology, but, generally, with different sizes and slope and height rms variations. It has been demonstrated that an optical surface can be thought of as a stationary uniform random process. It was further shown that an autoregressive moving average (ARMA) modeling of one-dimensional (1D) slope measurements allows highly confident fitting of the metrology x-ray mirrors data with a limited number of parameters. With the parameters of the ARMA model, the surface slope profile of an optic with the newly desired specification can be forecast reliably. However, ARMA models are causal and do not allow for generalization from one dimension to two. We propose to generalize the method from processing of one dimensional profile data to two dimensional surface data with invertible time-invariant linear filter (InTILF). This approach will also allow to parameterize surface metrology of high quality x-ray optics optimally. Our preliminary studies indicate that the InTILF approximation has all advantages of one-sided AR and ARMA modeling, but it additionally gains in terms of fewer filter parameters and better spectral accuracy. The envisioned software can also be used to analyze a polishing process and as a feedback for polishing tools.

Benefits: The envisioned analytic software will facilitate development of new x-ray optics of unprecedented quality with surface slope precision of less than 0.1-0.2 μrad and surface height error of less than 1 nm.1-5 generation. This optics is essential for new generation of beamlines and free electron lasers (FELs). The uniqueness of this expensive optics fabrication creates a requirement to provide the compact characterization for it based on existing metrology data. We will develop theoretical base and software tools to characterize the surface morphology. The Software will capture statistical properties of the surface and enable the user to generate thousands of random statistically equivalent surfaces to use for system performance prediction. Statistical equivalence will be defined in terms of fidelity to spectral properties of initial surface under characterization.

The envisioned analytic software will facilitate development of surface analysis and modeling for a variety of applications. Any application involving an interpolation of spectal analysis of a sparsely sampled surface would benefit. The software can be used for quality control, both front and back end in fabrication of smooth components, for example optics and lithography. The method can help the ongoing effort of seafloor interpolation for cartography. Any application that creates a requirement to provide the compact characterization of surface metrology data and assessment and comparizon of surface properties. We will develop theoretical base and software tools to characterize the surface morphology. The Software will capture statistical properties of the surface and enable the user to generate thousands of random statistically equivalent surfaces to use for assessment and prediction. Statistical equivalence will be defined in terms of fidelity to spectral properties of initial surface under testing and characterization.