The turbulence in the atmosphere perturbs the incoming light wavefront from an astronomical source and blurs the images for ground-based telescopes whose resolution is therefore limited by the seeing angle (typically one arc-second). The Hubble space telescope (HST) or any space projects are an efficient but still expensive way to get rid of the turbulence problem. However, alternative techniques have been developed to restore the original resolution of ground based telescopes. Adaptive optics (AO) is one of them and probably the most promising one. Operational AO systems have already produced astronomical results (Léna 1994; Roddier et al. 1995). An adaptive optics system analyzes the disturbed wavefront and applies in real time corresponding commands to a deformable mirror in order to cancel the phase perturbation (see e.g. Roddier 1994 for a description). Depending on the quality of the correction achieved, a partially or fully corrected image is obtained. The so-called guide source corresponds to the source used by the wavefront sensor (WFS) to analyse the perturbed wavefront. At the moment, most of current adaptive optics experiments or projects only provide the natural guide star (NGS) mode. The phase perturbation depends on the viewing direction. So, any distance between the guide source and the science object will add an anisoplanatic error in the AO correction. From this point of view, the best choice for the guide source is the science object itself.

One can find in the literature (Ellerbroek 1994;
Wilson & Jenkins 1996 for recent examples) calculations of
the mean correction achieved
by an AO system running in some given turbulence conditions.
The atmospheric coherence time *t*_{0} and the atmospheric
coherence length *r*_{0} are two parameters used to describe these conditions.
Both are wavelength-dependent () and function of the airmass.
As the turbulence effects get worst or one goes to
shorter wavelengths (faster turbulence and/or smaller *r*_{0}),
the correction is poorer. Adaptive optics works in many cases
in the partial correction regime.

If we ignore anisoplanatism, the AO image of a science object is the science object intensity distribution convolved with the point spread function. The point spread function (psf) is defined as the AO image obtained with a point source and then corresponds to the instrumental response function. So, the psf directly defines the quality of the image. In the full correction case, the psf correspond to the theoretical diffraction limit which is the Airy pattern for a clear circular aperture telescope. In the partial correction case (the general one), the psf is degraded by the power of the uncorrected or non fully corrected terms. Qualitatively, a superimposed halo (the so-called seeing halo) and a widening of the psf are observed (Rigaut et al. 1991).

Some scalar parameters have been generally used to describe the
long or short exposure psfs.
The Strehl ratio (Sr) is used to quantify the degree of correction
achieved. Appendix A shows how to extract this parameter from
point source AO images. A Strehl ratio equal to one means a full correction.
In case of no correction, the Strehl ratio of the
seeing limited image is about
where *D* is the telescope aperture diameter.
The full width half maximum (fwhm) of the psf is an indication of the image
sharpness and tells about the spatial resolution achieved. For
two-dimensional psfs, this parameter is actually the fwhm of the
azimuthally averaged profile. The fwhm of the diffraction-limited psf
is while that of the seeing-limited psf
(the seeing angle) is .
A psf could have a sharp core but widely spread out. The 50% energy
radius (*r*50) tells within which radius is concentrated 50% of the
energy. As for the fwhm, this parameter is computed on the
azimuthally averaged profile.

It is quite difficult to predict from simulation and/or analytic
calculation all the AO servo system and turbulence statistics induced effects.
The different wavefront realizations and other noise sources make
the AO system correction a statistical process.
The long-exposure psf is the instantaneous psf averaged over a time long
enough to have a good signal to noise ratio. In the partial
correction regime,
the turbulence statistics are here considered as the predominant process,
the integration time should be much larger than *t*_{0}.
Typically, this means one minute and 10 seconds in the infrared and in the
visible respectively. Equivalently, it is the time
to smooth the speckle pattern produced by the non-coherent energy
present in a partially corrected image which is responsible for the seeing
halo. By contrast, the central part of the psf
gets its typical shape in a few coherence times.
Unfortunately, the atmospheric turbulence is often a non-stationary
process even over for such short times and is also function of the airmass.
These points will be illustrated in this paper.

This work is found on natural guide star AO data taken with two operational A0 instruments based on a Shack-Hartmann wavefront and a piezo-actuated deformable mirror. This paper outlines adaptive optics performance and the limitations induced by the atmosphere to the AO system in the partial correction regime. The anisoplanatism issue and other sources of limitations due to the photon, background or detector noises are not considered here. Most previous works have concentrated on comparison between prediction and observed AO correction in term of Strehl ratio. However, the shape of the point spread function and its stability for various correction regime are important for an astronomical use. Practical problems like psf calibration, deconvolution of AO image or post-processing, which are complex issues (Tessier 1995), are therefore adressed.

Instruments used to produce the data are described in Sect. 2. In Sect. 3, the temporal and geometrical characteristics of the point spread function are shown for various correction regime. Especially, the stability of the psf against the changing turbulence conditions is tackled, some relationships between psf scalars are shown. Image quality degrades dramatically below a certain Strehl ratio value. Some comparison with theoretical predictions is done. Post-processing techniques like image deconvolution require some specific observational and calibration procedures which are presented in Sect. 4. Overall performances of AO data as a function of the correction are given in Sect. 5. In Sect. 6, we see that calibration and blind deconvolution techniques can be mixed to try and get an improved near-sighted deconvolution process. Section 7 shows how to combine speckle techniques with AO to improve the performance in the low correction regime. Conclusions are drawn in Sect. 8, from these processed NGS AO data, AO are shown to push down the limits compare to the seeing-limited imaging or the speckle imaging.

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