Correlation analysis of surface slope metrology measurements of high quality x-ray optics

摘要

We discuss an application of correlation analysis to surface slope measurements of high quality x-ray optics with the aim of elicitation and, when possible, suppression of the instrumental systematic errors in the final metrology results. We describe and present the mathematical foundation for a novel method consisting of the randomization of the systematic error by the averaging of multiple measurements, specially arranged to mutually anti-correlate. We also discuss the possibility to apply correlation analysis to the entire residual surface slope distribution in order to find anti-correlation parameters of the distribution. In this case, repeated measurements with the corresponding change of the experimental arrangement (for example, position of the surface and/or its overall tilt) can be used to identify the origin of the observed anti-correlation features by analyzing the difference between the measurements. If the corresponding minimum of the auto-correlation function is due to a systematic error, averaging over the repeated measurements will provide an efficient suppression of the systematic error. If the observed anti-correlation properties are due to the polishing process, and therefore belong to the surface itself, we suggest that the possibility of re-polishing the surface based on the correlation analysis be considered. Throughout the present work we have discussed correlation analysis of surface slope metrology data. However, a similar consideration can be applied to surface topography in the height domain measured with other metrology instrumentation, for example: interferometers and interferometric microscopes.