Variance of focal-plane centroids

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Abstract
The variance of the unbiased focal-plane estimates of image centroids from a point source is derived for arbitrary strength of scattering by using path-integral methods. The Markov approximation and narrow angular scattering are assumed. Results are presented for an ideal focusing telescope with an aperture transmission that produces no phase errors. Calculations of the leading term are performed for a circular telescope aperture and the case of homogeneous atmospheric turbulence described by the Hill spectrum [J. Fluid Mech. 88, 541–562 (1978)]. When there are irradiance fluctuations, the variance of the unbiased estimate of image centroids depends on the choice of the origin of the weighting vector. The variance of the centroid of a single image depends on three physical parameters related to the Fresnel scale, the inner scale, and the strength of scattering. The variance of the separation between two images from two separated telescope apertures depends on the same parameters in addition to the normalized separation of the apertures. The important contributions from propagation distance and spectral wave number are identified for the different parameter regimes. The magnitude of centroid motion compared to the average image size is also discussed.

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