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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 aberratedwavefront, and a residual turbulence still corrupts the observation. Thus theconsequences of the turbulence on the PSF is of first interest when buildingany ground-based telescope. The Power Spectral Density (PSD) of a spatiallystationary turbulent phase carries all the information needed for describingthe long-exposure PSF. We then develop an analytical description of thelong-exposure PSF as a series expansion of the aberrated phase PSD. Ourdescription of the PSF given the PSD of the phase is a simple theoretical wayto describe the impact of turbulence on the PSF. We also show accordance withprevious papers when restricting our model to its first expansion order.Finally we derive applications of our formula to some particular cases, such asKolmogorov or von-K\'arm\'an models, or the AO correction impact on the PSF.

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