The adaptive optics point-spread function at moderate and high strehl ratios

We describe the symmetries present in the point-spread function (PSF) corrected by an adaptive optics (AO) system to Strehl ratios of about 60% or greater. We expand the PSF in powers of the the residual phase disturbance over an arbitrarily shaped and apodized entrance aperture. We make no assumptions about aperture symmetry or apodization in our analysis. We show that for traditional unapodized aperture geometries, bright speckles pinned to the bright Airy rings are part of an antisymmetric first order perturbation of the perfect PSF which arise from the term that is first order in the residual phase error. This first order term redistributes power within each bright ring, but contributes no power in regions where the perfect image would have no light. because it is modulated by the square root of the perfectly corrected PSF. It also vanishes at the center of the image. There are two symmetric second degree terms, one is negative at the center, and is also modulated by the perfect image's field strength - it reduces to the Marechal approximation at the center of the PSF. The other second degree term is non-negative everywhere, zero at the image center, and can be responsible for an extended halo - it limits the dynamic range in the dark portions of the image. These features can be exploited by appropriate telescope and instrument design, observing strategies, and data reduction methods to improve the dynamic range of AO observations.