Abstract

An innovative method of phase compensation employs a segmented primary mirror in place of a conventional deformable mirror. Segmentation of the primary mirror offers an inexpensive method to produce large, active telescopes or beam directors. The success of this approach for adaptive optics hinges on the ability to control the segments. The segmented mirror will adopt the requisite conjugate beacon phase front to the level of precision of the wave-front-sensor measurements if it is made to behave like a continuous deformable membrane. We show that this is equivalent to applying the measured wave-front slopes directly to the segments and then matching adjacent edge midpoints. This discrete linear system of equations for the segment pistons is singular, and no general solution exists. We have succeeded in analytically solving for the segmented-mirror configuration that minimizes the sum of the squares of the differences in height of adjacent segment midpoints. This solution results from the identification of constraints on the surface. The constraints have a natural geometric interpretation as continuity loops. The knowledge of this constructive solution eliminates the need to do iterations or the need to develop iterative control algorithms. This solution functional can be found in advance for any particular mirror design and relates segment-midpoint height differences to the measured input tilt field in a fully deterministic and unique way.

© 1994 Optical Society of America

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