1 Introduction

Random aberrations on the pupil of the telescope due to atmospheric turbulence determine the angular resolution of ground-based telescopes. Adaptive optics systems perform a real time correction of the degradation introduced by the atmosphere (Wang & Markey 1978; Rigaut et al. 1991; Roggemann 1991). A wavefront sensor measures the instantaneous aberrations and, using this information, a deformable mirror changes its shape to compensate this distortion. One of the most interesting subjects in astronomy is the use of adaptive optics systems to detect extrasolar planets. A series of important problems appears in this kind of detection. The main limitation is the brightness difference between the star and the planet (about 10⁹ times). After compensation, the difference between the star halo and the peak of the planet is expected to be about 10⁴. In actual systems to reduce the star-planet luminosity ratio to this limit, it would be necessary to perform a compensation as complete as possible. The ideal high gain required for exoplanet detection is limited by a number of noises and technical difficulties. Papers of Angel (1994) and Stahl & Sandler (1995) show these problems and some possible solutions. Then it is necessary to reduce the scattered light of the star and suppress the diffracted light (Nakajima 1994). This compensation will reduce the long integration time necessary to increase the contrast between planet and background.

The improvement of the adaptive optics system on the brightness difference is represented by the system gain that is defined as the ratio between the peak and halo intensities. This parameter establishes a good connection between the system performances and the detection requirements (Labeyrie 1995). In this paper we introduce a theoretical expression for the system gain. The procedure we propose to obtain the gain is based on the image formation process (Cagigal & Canales 2000a; Goodman 1985). The wavefront is described through the structure function which provides two parameters: the correlation length and the residual phase variance in the compensated wavefront. From them it is easy to develop a model to describe the point spread function (PSF) of the system, so that, a clear relationship can be established between the PSF halo and the parameters obtained from the structure function. The gain expression we obtain resembles the approximated one proposed by Angel (1994). The main difference is that our expression has a more complex dependence on the residual phase variance, consequence of the procedure followed to obtain it. Under a simple approximation both expressions are equivalent since the correlation length and the actuator size equate for high compensation level. Hence, we propose a theoretical procedure based on the image formation process that allows an accurate estimate of the system gain using only two parameters: the correlation length of the compensated wavefront and the residual phase variance. Furthermore, we show an experimental method to estimate the actual residual phase variance in the compensated wavefront, which allows us to obtain a more realistic estimation of the system gain. Still there are some sources of errors that are not taken into account, as scintillation or residual temporal decorrelation errors, which would require a deeper analysis.