An analytic model for natural guide star wide field adaptive optics

Wide-Field Adaptive Optics (WFAO) is an AO mode in which one deformable mirror is used to achieve modest adaptive optics correction of the atmospheric turbulence, but in a much wider field of view than classical AO. At the heart of the concept is the desire to trade image quality at the center of the field of view for better image quality at the edge of a wide field (typically similar to10') and is also called Improved Seeing AO (ISAO) or Ground Layer AO (GLAO) in the literature. An analytical (Fourier domain) model allows us to rapidly derive requirements on the number, brightness and distribution of guide stars for a WFAO system running on an 8-m or 30-m telescope, as well as basic AO system requirements such as loop rate and DM actuator density. In this paper we derive the Fourier domain filter that describes WFAO and present a method for evaluating WFAO performance and sky coverage. We test our performance evaluation on a pathological case, computing the scientifically relevant metric, radius of 50% encircled energy for a typical C-n(2) profile.