  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.
