Temporally averaged speckle noise in wavefront sensors for beam projection in weak turbulence

Adaptive optics (AO) compensation for imaging or coherent illumination of a remote object relies on accurate sensing of atmospheric aberrations. When a coherent beacon is projected onto the object to enable wavefront sensing, the reflected reference wave will exhibit random variation in phase and amplitude characteristics of laser speckle. In a Shack-Hartmann wavefront sensor (SHWFS) measurement, speckle effects cause fluctuations in the intensity of focal spots and errors in the position of their centroids relative to those expected from purely atmospheric phase aberrations. The resulting error in wavefront measurements negatively impacts the quality of atmospheric phase conjugation. This paper characterizes the effect of reflected laser speckle on the accuracy of SHWFS measurements for ground-to-space beam projection systems in weak turbulence conditions. We show via simulation that the speckle-induced error in centroiding depends on the ratio between beacon diameter and the diffraction-limited resolution of the lenslet and confirm these results with experimental data. We provide experimental validation that averaging of SHWFS lenslet spot intensities over speckle realizations converges to the incoherent intensity as expected. We further show that the effects of shot noise and speckle noise add in quadrature, simplifying noise analysis. Finally, we characterize the effect of temporal averaging under typical conditions of target motion and integration time. This work provides a straightforward set of relations that can help investigators more accurately estimate the required integration time for wavefront sensing in the presence of laser speckle.