Ex situ tuning of bendable x-ray mirrors for optimal beamline performance

We extend analytical and numerical methods recently developed at the Advanced Light Source (ALS) optical metrology laboratory (OML) for optimal tuning and calibration of bendable x-ray optics based on ex situ measurements with surface slope profilers [Opt. Eng. 48(8), 083601 (2009); Proc. SPIE 8141, 8141-19 (2011)]. We minimize the rms variation of residual slope deviations from ideal surface figure. Previously, our adjustment assumed the deviations were weighted equally across the optic. In this work, we analyze the case when the mirror length is significant with respect to the imaging conjugate. This corresponds, for example, to high de-magnification by bendable Kirkpatrick Baez mirror pairs, used near the ends of synchrotron and free electron laser beamlines for micro- and nano-focusing that often results in a very short mirror to image distance, of the same order of magnitude as the mirror’s length. In this case, contributions to focal distortion of residual errors of mirror surface figure (appearing due to mechanical alignment tolerances, sagittal shaping errors, and the limited number of adjustable parameters inherent in a two-couple bender) strongly depend on position across the optic. Specifically, the downstream deviations from exact shape should be weighted less because the rays have a shorter path to travel to the image. Here, we derive an analytical expression for the weighting function and present a mathematical background for the bending adjustment procedure for optimization of the mirror’s beamline performance. The efficacy of the optimization is demonstrated for a short-focus mirror used for diffraction limited focusing at ALS beamline 12.3.2.