The generalized theoretical analysis of correlation functions leads to an
accurate modelling of anisoplanatism. The residual wavefront error derivation
presented in Sect. 2 and 4 provides a useful analysis to evaluate the effects
of anisoplanatism upon the performance of solar adaptive optics systems. The
theoretical analysis uses the wavefront expansion on the Zernike polynomials
which is also used in Sect. 3 to analyze the problem of the wavefront slope measurements on the granulation image when compensating with a tiltable mirror.
A detailed evaluation of the anisoplanatism is deduced from the Zernike polynomial
angular correlations, derived using the Mellin transform technique. In
Sect. 5, the optical transfer functions are evaluated for a Zernike expansion following
the Wang and Markey approach. The calculation of OTF to cases when the observing
source is extended is developed in order to evaluate the degradation in the
field of view of the image due to anisoplanatism after adaptive optics
compensation.
We compute the isoplanatic angle corresponding to a 
/5 maximum wavefront
error in the field of view for T.H.E.M.I.S. after compensation by the image stabilizer
optical system and different observation wavelengths. Depending on the level
of the strength of the turbulence, the accessible wavelength range is limited.
We compare the performances of the actual image stabilizer optical system and
a project of 10 controlable modes of correction (tilt, focusing, astigmatism
and coma). We demonstrate the capacity of 10 controlable corrected modes for
high quality compensation but to the price of a small field of view while the
capacity of the image stabilizer optical system is for medium quality compensation
and large field of view observations.
The study may be useful for the design of future adaptive optics system for
solar observations. Depending of the specific goals, large or small field of
view and medium or high image quality, the system may be totally different.
The authors are grateful to R. Muller for granulation images of the sun from
the Pic-du-midi Observatory.