An error function minimization approach for the inverse problem of adaptive mirrors tuning

Adaptive x-ray optics are more and more used in synchrotron beamlines, and it is probable that they will be considered for the future high-power free-electron laser sources, as the European XFEL now under construction in Hamburg, or similar projects now in discussion. These facilities will deliver a high power x-ray beam, with an expected high heat load delivered on the optics. For this reason, bendable mirrors are required to actively compensate the resulting wavefront distortion. On top of that, the mirror could have also intrinsic surface defects, as polishing errors or mounting stresses. In order to be able to correct the mirror surface with a high precision to maintain its challenging requirements, the mirror surface is usually characterized with a high accuracy metrology to calculate the actuators pulse functions and to assess its initial shape. After that, singular value decomposition (SVD) is used to find the signals to be applied into the actuators, to reach the desired surface deformation or correction. But in some cases this approach could be not robust enough for the needed performance. We present here a comparison between the classical SVD method and an error function minimization based on root-mean-square calculation. Some examples are provided, using a simulation of the European XFEL mirrors design as a case of study, and performances of the algorithms are evaluated in order to reach the ultimate quality in different scenarios. The approach could be easily generalized to other situations as well.