  Context. Adaptive optics (AO) is now a tool commonly deployed in astronomy.
The real time correction of the atmospheric turbulence that AO enables allows
telescopes to perform close to the diffraction limit at the core of their point
spread function (PSF). Among other factors, AO-corrected PSFs depend on the
ability of the wavefront corrector (WFC), generally a deformable mirror, to fit
the incident wavefront corrugations. Aims. In this work, we focus on this error
introduced by the WFC, the so-called fitting error. To date, analytical models
only depend on the WFC cut-off frequency, and Monte Carlo simulations are the
only solution for studying the impact of the WFC influence function shape on
the AO-corrected PSF. We aim to develop an analytical model accounting for the
influence function shape. Methods. We first obtain a general analytical model
of the fitting error structure function. With additional hypotheses, we then
derive an analytical model of the AO-corrected power spectral density. These
two analytical solutions are compared with Monte Carlo simulations on different
ideal profiles (piston, pyramid, Gaussian) as well as with real hardware (DM192
from ALPAO). Results. Our analytical predictions show a very good agreement
with the Monte Carlo simulations. We show that in the image plane, the depth of
the correction as well as the transition profile between the AO-corrected area
and the remaining turbulent halo depend on the influence functions of the WFC.
We also show that the generally assumed hypothesis of stationarity of the AO
correction is actually not met. Conclusions. As the fitting error is the
intrinsic optimal limit of an AO system, our analytical model allows for the
assessment of the theoretical limits of extreme AO systems limited by the WFC
in high-contrast imaging through a context where other errors become
comparable.
