Shaped electrodes for adaptive X-ray optics

Lithium niobate has emerged as a promising substrate material for adaptive X-ray mirrors. Heat treatment induces ferroelectric domain inversion, yielding two oppositely poled, layered regions that enable lithium niobate to behave as a monolithic bimorph in the presence of an applied electric field. In addition to fabrication simplicity, this well known material has favorable actuation and shape-control properties without hysteresis, creep, and drift. Following Timoshenko's approach to the bi-metallic strip thermostat, we derive an analytic description of shaping continuous electrode widths to produce a spatially varying surface curvature when a voltage is applied. Through comparison with finite-element analysis, the method is demonstrated for uniform electrode widths that produce cylindrical surfaces, and for varying electrode widths that produce plane-parabolic and plane-elliptical surfaces from initially flat substrates. A proposed longitudinally continuous, compound electrode pattern shows how a single flat mirror could be shaped to achieve multiple profiles with a single applied voltage.