1 Introduction

Astronomical observations on earth are strongly affected by turbulent air motions in the atmosphere. They deform randomly the wavefront of the incoming light of an astronomical source, so that randomly distributed speckles are usually present in a short-exposure image. In a long-exposure image these speckles sum up to form a blurred image. Atmospheric turbulence set therefore a severe limit to the angular resolution which, in the optical wavelength range, is rarely better than the theoretical angular resolution of a 10 to 20cm aperture.

Adaptive optics (AO) has been developed to overcome this problem. It corrects the wavefront deformation in real time by means of a deformable mirror (DM) and a wavefront sensor (WFS). Due to incomplete measurements, noise and time-lag errors, the correction is often only partial and the image of a point-like source consists of a diffraction-limited peak superimposed on a large diffuse halo. It is therefore important to deconvolve the astronomical images in order to do accurate photometry or to detect faint structures without ambiguities. For this, the point spread function (PSF) of the system "atmosphere-telescope'' has to be known.

The classical approach consists in calibrating the PSF by taking an image of a point-like source just before and/or after the object acquisition. The quality of the wavefront correction, however, and hence the PSF too, depends on the atmospheric conditions and the magnitude of the reference star. Due to variations of the atmospheric seeing and differences in the properties of the reference sources, the observed PSF may not correspond accurately enough to the PSF at the time of the observation of the object.

In theory, this problem can be overcome when estimating the long-exposure PSF from the WFS measurements saved during the object acquisition. In this approach, however, we have to assume that the high-frequency part of the turbulent phase follows the Kolmogorov model. A method which reconstructs the PSF from WFS measurements has been developed by Véran et al. (1997) and successfully applied to the AO system PUEO installed at the Canada-France-Hawaii Telescope (CFHT). We have adapted their method to ADONIS, the AO system at the ESO 3.6 m telescope, which is based on a Shack-Hartmann device instead of a curvature sensor as used in the PUEO system.

The reader should keep in mind, that the reconstructed PSF will always correspond to the on-axis PSF, i.e. the PSF of the reference source. If the separation between the object source and the reference source is so large that anisoplanetism effects cannot be neglected anymore, then the reconstructed PSF will not be exactly the same as the object PSF. We investigated, if it was possible to reconstruct the on-axis PSF from the WFS measurements, and did not include anisoplanetism effects.

Section 2 recalls the concept of an AO system and analyzes the different elements taking as example the ADONIS system. We give a short description of the PSF reconstruction algorithm and refer the interested reader to Véran et al. (1997a) or Véran (1997) for a more detailed description of the method. In Sect. 3 we present the experimental approach of the reconstruction method to ADONIS. In particular, we discuss the methods of determining the measurement noise and Fried's parameter r₀. Section 4 presents the results of our PSF reconstruction obtained for different observing conditions. Finally, Sect. 5 discusses the limits of the PSF reconstruction and the photometric precision one can expect when deconvolving images with the reconstructed PSF.