Turbulent and AO corrected Point-Spread-Function as convolutive orders of the phase Power-Spectral-Density

Abstract: Ground-based astronomy is severely limited by the atmospheric turbulence, resulting in a large Point-Spread-Function (PSF) and poor imaging resolution. Even imaging with Adaptive Optics (AO) cannot completely correct the aberrated wavefront, and a residual turbulence still corrupts the observation. Thus the consequences of the turbulence on the PSF is of first interest when building any ground-based telescope. The Power Spectral Density (PSD) of a spatially stationary turbulent phase carries all the information needed for describing the long-exposure PSF. We then develop an analytical description of the long-exposure PSF as a series expansion of the aberrated phase PSD. Our description of the PSF given the PSD of the phase is a simple theoretical way to describe the impact of turbulence on the PSF. We also show accordance with previous papers when restricting our model to its first expansion order. Finally we derive applications of our formula to some particular cases, such as Kolmogorov or von-K\'arm\'an models, or the AO correction impact on the PSF.