X-ray beam-shaping via deformable mirrors: Analytical computation of the required mirror profile

X-ray mirrors with high focusing performances are in use in both mirror modules for X-ray telescopes and in synchrotron and FEL (Free Electron Laser) beamlines. A degradation of the focus sharpness arises in general from geometrical deformations and surface roughness, the former usually described by geometrical optics and the latter by physical optics. In general, technological developments are aimed at a very tight focusing, which requires the mirror profile to comply with the nominal shape as much as possible and to keep the roughness at a negligible level. However, a deliberate deformation of the mirror can be made to endow the focus with a desired size and distribution, via piezo actuators as done at the EIS-TIMEX beamline of FERMI@Elettra. The resulting profile can be characterized with a Long Trace Profilometer and correlated with the expected optical quality via a wavefront propagation code. However, if the roughness contribution can be neglected, the computation can be performed via a ray-tracing routine, and, under opportune assumptions, the focal spot profile (the Point Spread Function, PSF) can even be predicted analytically. The advantage of this approach is that the analytical relation can be reversed; i.e., from the desired PSF the required mirror profile can be computed easily, thereby avoiding the use of complex and time-consuming numerical codes. The method can also be suited in the case of spatially inhomogeneous beam intensities, as commonly experienced at synchrotrons and FELs. In this work we expose the analytical method and the application to the beam shaping problem.