A general theoretical analysis is made of the performance of optical systems that are designed to give diffraction-limited images of astronomical objects by compensating effects of atmospheric seeing in real time. The heart of any such system is a feedback algorithm, which expresses the controlled displacements of mirror surfaces as functions of the output of optical sensors. The statistical behavior of the system is calculated, assuming the feedback algorithm to be linear but otherwise unrestricted. The effect of feedback in diminishing atmospheric noise while amplifying photon noise is worked out in detail, taking into account photon–photon correlations. A figure of merit for system performance, namely a statistical average of a positive quadratic function of phase errors over mirror surfaces, is arbitrarily chosen. By use of this figure of merit, the optimum feedback algorithm for any given optical system is explicitly determined. The optimum algorithm is independent of the quadratic function of phase errors that is chosen as the figure of merit. Applications of the general theory to particular optical systems are briefly discussed. In principle, systems optimized in this way should be able to give images of arbitrarily high resolution of astronomical objects brighter than about magnitude 14.
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