Thickness dependence of piezo-bimorph adaptive mirror bending.

A new generation of adaptive x-ray optics (AXO) is being installed on high-coherent-flux x-ray beamlines worldwide to correct and control the optical wavefront with sub-nm precision. These ultra-smooth mirrors achieve high reflectivities at glancing angles of incidence and can be hundreds of mm long. One type of adaptive x-ray mirror relies on piezoelectric ceramic strips which are segmented into channels and actuated to induce local, longitudinal bending, generating one-dimensional shape changes in the mirror substrate. A recently described mirror model uses a three-layer geometry with parallel actuators on the front and back surfaces of a thicker mirror substrate. By analogy to a solved problem in the thermal actuation of a tri-metal strip, we show that the achievable bending radius varies approximately as the square of the substrate thickness. We provide an analytic solution and simulate bending using a finite-element model.