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 www.defensetechbriefs.com/tsp under the Photonics category. AFRL-0005

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