Polynomial-shaped electrodes for adaptive X-ray optics

For adaptive mirrors on X-ray beamlines and optical systems, we describe a strategy for arranging a small number of tangentially continuous electrodes with varying electrode widths tailored to improve mirror shape control. This approach, with patterned, compound electrodes, is well suited to the emerging class of lithium niobate mirrors that act as monolithic piezo-bimorphs, without the need for additional bonded or grown actuator elements. The proposed electrodes span the length of the mirror's clear aperture and have width profiles defined by polynomial shapes. Using a limited number of channels derived from Chebyshev polynomials, the actuators provide access to shape controls and correction modes of increasing polynomial order. We derive an analytic approach to optimize control voltages based on Euler-Bernoulli beam theory, and we show how the modeled characteristic response functions can be applied to fit arbitrary surfaces. Comparison with a conventional, linear array of 12 electrodes shows the relative benefit of each approach.