Applications of Power Series Solutions of Membrane Equilibrium Equations to the Optical Evaluation of Membrane Mirrors with Curvature
Final rept. Dec 97-Dec 98
AIR FORCE RESEARCH LAB KIRTLAND AFB NMDIRECTED ENERGY DIRECTORATE
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Power series solutions of the Hencky-Campbell equilibrium equations for an initially planar circular membrane in a state of uniform pre-strain are used to analyze the performance of an in-house laboratory membrane mirror curved by a uniform vacuum pressure applied to one side. The results provide theoretical verification of the improvement in surface figure achievable by controlling the amount of pre-strain. In particular, a pre-strain of only 0.25, corresponding to a uniform radial displacement of the edge of an 11-inch diameter membrane mirror by about 0.014 inches, is predicted to achieve nearly 95 reduction in the maximum deviation of an f8 membrane from a reference sphere. These results are in good agreement with laboratory measurements of spherical aberration using a Hartmann wavefront sensor. The model was also used in an extensive investigation of the effects of control parameters pressure and pre-strain on the improvement of surface figure for lower f-number systems. The results indicate that comparable improvement in systems of f-number less than f6 would require membrane materials capable of withstanding pre-strains on the order of 2.0 or greater.