Closed-form expressions for the centroid-tilt error due to scintillation

In adaptive optics, the tracker and wavefront sensor commonly measure irradiance centroids (in their respective focal planes) to estimate the turbulence-degraded wavefront in the pupil plane. Several factors affect the accuracy of these centroid measurements, including noise, speckle, and scintillation. The centroid-tilt or ``C-tilt'' errors due to these factors have been studied by numerous researchers; however, to our knowledge, closed-form expressions for the C-tilt error due to scintillation have not been found. In this paper, we derive such expressions, assuming spherical-wave illumination of the pupil, a path-invariant index of refraction structure constant $C_n^2$, and Kolmogorov turbulence. We compare the analytically predicted C-tilt values to those obtained by wave-optics simulations. The agreement, at large, is quite good over a wide range of conditions. As a result, researchers and engineers will find this analysis useful when quantifying the performance of centroid-based trackers and wavefront sensors, both of which are critical adaptive-optics components.