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.
With respect to a given mirror membrane, "in-plane-actuated" as used here signifies that actuators would be bonded to the membrane, where they would, variously, expand or contract along the membrane surface upon application of suitable voltages and would thereby locally impart bending moments to deform the mirror surface (see figure). The actuator voltages would be chosen so that the deformations would form the mirror into an acceptably close approximation of a precise shape needed for optical control. For the purpose of this study, it was assumed that actuators for outerspace membrane optics would be, more specifically, unimorph piezoelectric actuators, each bonded to one surface of the membrane (usually, opposite the mirror surface).

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.)

Analytical solutions to plate-membrane and beam-string ordinary differential equations representing the deformable-mirror equations were developed. A simplified approach to modeling in axisymmetric cases also was developed. Significantly, it was shown both analytically and through numerical analysis that the result of static actuation for a mirror with a discrete electrode pattern and a high tension- to-stiffness ratio is simply a localized piston displacement (a displacement perpendicular to the surface of the membrane) in the region of the actuator.

Next, a static control strategy denoted the modal transformation method was developed. The method was implemented in a finite-element simulation to demonstrate the capability to form Zernike surfaces within a clear aperture region by use of a number of statically actuated structural modes.

The problem of scaling for membrane optics was addressed. Linear mathematical models were shown to correctly represent the behaviors of small-scale laboratory models, but fully nonlinear models were found to be necessary for describing the behaviors of larger-aperture membrane mirrors. The results of this part of the study have been interpreted to suggest that nonlinear effects must be considered in feasibility studies for future largeaperture membrane mirrors for telescopes.

This work was done by Michael J. Shepard of the Air Force Institute of Technology for the Air Force Research Laboratory. For more information, download the Technical Support Package (free white paper) at under the Photonics category. AFRL-0005

This Brief includes a Technical Support Package (TSP).
Study of Membrane Optics for Lightweight Space

(reference AFRL-0005) is currently available for download from the TSP library.

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This article first appeared in the August, 2007 issue of Defense Tech Briefs Magazine.

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