Scintillation-induced centroid jitter: analytical solutions

Atmospheric induced amplitude fluctuations, known as scintillation, impose limitations on active tracking and wavefront-sensing performance over near-horizontal propagation paths. These sensors typically use centroid tracking to estimate the aperture-averaged phase gradient (G-tilt). G-tilt, in practice, is a phase-only measurement, whereas centroid tracking includes both the phase and amplitude components. For a nonuniform beam, centroid tracking will measure the irradiance-weighted average phase gradient (C-tilt). In a closed-loop system, the angular position of the centroid is used to conjugate tilt and reduce system jitter. Of particular interest are the effects of scintillation on the estimation of G-tilt from the centroid angular position. Scintillation will cause an error in the estimation of the G-tilt, and this error can be quantified by the noise-equivalent angle (NEA). The two main objectives of this work are to formulate a closed-form expression for (1) the NEA due to scintillation, and (2) the difference between G-tilt and C-tilt in the weak-to-moderate scintillation regime. The derived solutions are based on the first-order Rytov approximation. As such, the difference will be quantified by deriving a mean-squared error between the desired measurement (G-tilt) and the estimator (C-tilt).