Theoretical model of differential piston variance under atmospheric turbulence and its application on piston detection of segmented mirrors

The phase of a plane wave is severely distorted by atmospheric turbulence. Thus, co-phasing detection of segmented mirrors not only includes intrinsic error between mirrors, but also error induced by turbulence. Long-exposure or an average of multiple short-exposure frames has been adopted to suppress the influence of turbulence. However, the influence of turbulence on the differential piston between two sub-apertures is challenging to quantify owing to the complexity of the problem of light propagation through turbulence. In this paper, theoretical models of differential piston variance that incorporate the influence of exposure time, wind, outer scale, and multi-layered turbulence have been proposed. Specifically, a model that only considers the differential piston caused by turbulence and a model that considers the differential piston induced by turbulence when the global tilt is removed have been derived. Removing the global tilt has been proven to be effective in mitigating the influence of turbulence and reducing the exposure time of piston detection methods. The influence of turbulence on the differential piston has been quantified and illustrated by using a simplified turbulence profile and wind vertical distribution data from the Fuxian Lake Solar Observatory as an example. The minimum exposure time for piston detection can hence be determined. A numerical experiment has been conducted to cross-validate our theory and PSF-based piston detection method. Our models can serve as a theoretical guidance on strategies to reduce the influence of turbulence on the piston detection of segmented mirrors.