Unimorph piezoelectric actuators would be used to control shapes of membrane mirrors.

Astudy has been performed to establish a foundation for the analysis, design, and further development of inplane- actuated deformable membrane mirrors for lightweight spaceborne telescopes. It is envisioned that the telescopes, having typical mirror diameters of 20 m or larger, would be stowed compactly for launch and transport, then deployed in orbit around the Earth for use in surveillance of the Earth and in exploration of deep space.

Of Three Types of In-Plane Actuators that could be used on deformable mirrors, the unimorph type was considered for membrane optics in the study reported here. In each case, the arrows depict expansion or contraction in the piezoelectric layer.

In the study, the underlying differential equations for a unimorph-actuated membrane were developed, using a plate-membrane model to represent the elastic behavior of the membrane, along with applicable assumptions of quasi-static piezoelectric theory and a piezoelectric-thermal analogy. A finite-element model corresponding to the differential equations was developed and used to generate theoretical predictions for a 0.127-m diameter deformable mirror testbed. A boundary tension field needed for the finite-element model was determined by use of laservibrometer data. A nonlinear solution technique was used to incorporate the stiffening of the membrane by applied tension. Deformation data calculated by use of influence functions derived from the finite-element model for the static case were compared with experimental deformation data, then a least-squares approach was followed in creating an influencefunction matrix, which, in turn, was incorporated into a quasi-static control algorithm. In a subsequent test of the algorithm, simultaneous tracking of the Zernike tip, tilt, and defocus modes was demonstrated. (Zernike modes are so named because they are characterized by orthonormal polynomials, developed by Frits Zernike, that arise in the mathematical treatment of wavefronts in optical systems having circular pupils.)