Impact of exposure time on optical-phase measurements in turbulence

In this paper, we explore the impact of exposure time on optical-phase measurements collected on light that has propagated through atmospheric-optical turbulence. We model the exposure time by phase averaging over a convective distance, and we quantify the associated impact of imposing an exposure time using the piston- and tilt-removed phase variance. We accomplish this analysis through the development of an analytic solution and wave-optics simulations. In turn, we show that the analytic solution and simulation results are in good agreement when U_cτ/D . 0.25, where U_c is the convective velocity, τ is the exposure time, and D is the aperture diameter. When U_cτ/D & 0.25, the analytic solution underestimates the piston- and tilt-removed phase variance relative to the simulation results, and we discuss these differences. This work, at large, informs wavefront sensing and adaptive-optics efforts, where either the wind speed is high, the system is on a high-speed platform, the beacon is on a high-speed platform, or the beacon signal is very faint thereby requiring long-exposure data collections.