Methodology for metrology and optimal figure tuning of bendable mirrors developed at the Advanced Light Source

We present a refined methodology developed at the Advanced Light Source for the optimal fine-tuning of mirror figures used in synchrotron beamlines. Our methodology involves meticulous metrology and the application of the method of characteristic functions and linear regression analysis to set the figure of a bendable x-ray mirror with exceptional precision, achieving a desired shape within a residual surface characterized by a radius of curvature of hundreds of kilometers. This high level of accuracy is crucial for the performance of synchrotron beamlines. We also address the suppression of systematic, drift, and random errors, which are critical for maintaining the achieved precision over time. By detailing the methods and mechanisms used to control these errors, we demonstrate the robustness and reliability of our approach, making it a significant advancement in the field of synchrotron optics.