The mutual coherence function (MCF) of a wave propagated through atmospheric turbulence and corrected by zonal adaptive optics is derived for a segmented corrector with either piston only or piston with a local tilt fit. We begin with an intuitive approach which is based on replacing the outer scale of turbulence in the uncorrected MCF with a quantity proportional to the segment size. This approach is especially useful in deriving asymptotes. A rigorous approach provides the full curves, and reduces to a single curve for piston only and a single curve for piston with tilt, after proper parametrization. The effect of amplitude fluctuations is also considered, since scintillation limits the ultimate degree of correction as the segment size goes to zero. Although the analysis is for a segmented corrector, the results apply fairly well to continuous corrector with actuator spacing equal to d/√3, where d is the segment diameter.
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