1 Introduction

The concept of off-axis adaptive astronomical systems has become quite popular in recent years because it gives a principal possibility to override certain difficulties arising when the real-time correction has to be applied for weak stars (or for extended objects). This approach assumes that some star separated by a small angle from the weak star of interest is used as a reference one. If the reference star is bright enough to perform real-time wavefront measurements, the information obtained can be used for improvement of the weak star image. However in order to get an improvement, the separation angle has to be chosen in such a way that the wavefront distortions associated with both stars are correlated; otherwise the effect will be negative. On the other hand, one needs some criterion to estimate a resulting quality of correction. The long-exposure Strehl ratio to be obtained for the weak star after correction is an observable and meaningful criterion, so it is of practical interest to calculate this parameter as a function of the separation angle, propagation and turbulence conditions, adaptive system performances, etc.

In this paper we restrict our consideration to the case of perfect adaptive correction that allows us to concentrate the attention on some principal aspects of off-axis correction and on the calculations with recently obtained experimental data. The problem under treatment is a part of more general one: the so-called isoplanatic problem. As for the theoretical development related to the last problem, many approaches based on different considerations were suggested (Wang 1975; Shapiro 1976; Fried 1978, 1982; Roddier 1981; Chassat 1989; Welsh & Gardner 1991; Sasiela & Shelton 1993). However, as for the weak-turbulence conditions that is the case for astronomical applications, one can get the straightforward solution. The result of interest follows directly if one applies the Rytov method for solution of the parabolic equation (Tatarski 1968) which describes the wave propagation through the atmospheric turbulence. We refer and outline this approach in Sect. 1. Based on this solution, we calculate the residual structure function of the phase fluctuations associated with the on-axis star. Then we introduce an isotropic approximation for this function that simplifies significantly the calculation of Strehl ratio. In Sect. 2, we present the results of the calculations for the Strehl ratio with both the theoretical model of C_n² profile and with the experimental profile recently measured in San Pedro Mártir (SPM) observatory (Mexico). The results obtained are compared and discussed at the end of that section.