The goal of this study is to compare the performance of the image stabilizer optical system and adaptive optics system which will correct aberrations as tilt, focusing, astigmatism and coma considering not only the image resolution but also the degradation of the compensation by the system in the field of view. Note that noise in the wavefront measurement will not be considered and the bandwidth of the servo-loop will be considered infinite.
In order to determine the isoplanatic domain surrounding the target structure
on-axis, we have to use a criterion to evaluate the quality of the restored
image. This criterion is based on the residual phase variance, defined in
Eq. (15 (click here)) for the compensation of first aberrated modes and in
Eq. (25 (click here)) for image stabilizer system, which must be lower than a given
value 
(in ra
). An adaptive optics system compensating for
turbulence disturbances up to the maximum radial degree N (N = 1
for image
stabilizer system) is characterized by the residual phase variance:
where 
is deduced from Eq. (15 (click here)) in the case of compensation
of aberrated modes up to the maximum radial degree N and from Eq. (25 (click here))
in the case of image stabilizer system. The isoplanatic
angle 
(half of the field of view) is given by the equality:
Let us notice that the standard deviation 
may be expressed in wavelength unit:
(rms).
Figure 3: Residual phase variance after compensation versus
the angular separation 
. For comparison: residual phase variances
after correction of J = 3 polynomials (N=1), J = 6 (N=2),
J = 10 (N=3) and
after correction of pure tilts from wavefront
slope measurements (dark points).
Curves are calculated with the modelised profile at Iza
a site
(D/r0 = 4)
In Fig. 3 (click here), the residual phase variances after compensation are
plotted for T.H.E.M.I.S. telescope 
and for the modelised profile at
Izaa site defined by Fig. 2 (click here) (r0 = 22 cm).
The curves correspond to different adaptive optics systems compensating for
turbulence disturbances up to the maximum aberration modes J (i.e. maximum
radial degree N) compared to the image stabilizer system. These configurations
are possible cases which could be considered for practical implementation
(Kupke et al. 1994). Such type of curves have been first plotted by Chassat in
order to manage optimal correction in terms of angular decorrelation by tuning
up the number of corrected modes.
Notice that the Cn2 profile, chosen to illustrate the derivations corresponds
to a ratio D/r0 = 4. In the three others modelised profiles of
Fig. 2 (click here), the contribution of the turbulence near the ground increases
or decreases both with the high altitude layers at 4000 m above the telescope
altitude (and probably generated by the Pico del Teide). Therefore, the function
) in Eq. (26 (click here)) remains identical when changing the ratio D/r0.
In such situations, the residual variance is given by a shift along the y axis
of Fig. 3 (click here) corresponding to the new value (D/r0)5/3.
First, let us underline the significant reduction of the residual phase variance
on-axis 
when increasing the number of corrected modes. The results
shown by the curves (J=3, 6 and 10) are in perfect agreement with the residual
variance given by Noll (Noll 1976).
The image motions compensation curve ( = 0) reveals
the contribution of the error made when correcting pure wavefront tilts from
wavefront slope measurements.
Secondly, these results allow to understand that for small field of view observations,
increasing the number of corrected mode is the best choice in terms of image
quality (i.e. small phase variance) keeping in mind the fast degradation of this
quality in the field. The obtained residual phase variance is very small after
correction of 10 modes (point at = 0). On the contrary, the on-axis correction
after image stabilisation is relatively poor, but the residual phase variance
does not vary rapidly with the field angle. At very large angle (
10 arcsec), the correction after image stabilisation is even better than one
obtained after correction of 10 aberrated modes, because of the low wavefront
slope angular decorrelation. This demonstrates the importance of the decorrelation
of the higher wavefront deformation modes in the field.
Figure 4: Isoplanatic domain (twice as isoplanatic angle
) corresponding to an image quality criterion of
, versus wavelength, for the
T.H.E.M.I.S. telescope aperture
after compensation of J = 3 first
aberrated modes, J = 10 modes
and after image stabilisation (dark points).
Curves are calculated with the
modelised profile at Izaa site ( D/r0 = 4)
Using the relation between the wavelength and the residual phase variance,
Eq. (27 (click here)) can be written:
where r0 is the Fried parameter
calculated at the wavelength 
= 0.5
.
Figure 4 (click here) presents the isoplanatic angle for
(corresponding to /5, i.e. a Strehl ratio around
20%) calculated by Eq. (28 (click here)). This value of residual error is
somewhat arbitrary and rather a poor performance for an adaptive optics system but
can be sufficient for many astronomical observations. As expected, the isoplanatic
angle increases with the increase of the observation wavelength.
Note that after image stabilisation is larger than
after compensation of 10 first aberrated modes when observing at wavelength
larger than 0.7 . Let us remind that for any field point at an angular
distance to the target structure on-axis smaller or equal
to , the residual error is lower than 
/5. Depending on
the choice of a maximum residual phase, the considered observation wavelength
domain is not totally accessible (bandwidth from 0.5 up to
0.9
with T.H.E.M.I.S.).
This is shown in Fig. 5 (click here) which present the
isoplanatic domains after image stabilisation for three different values of
Fried parameter r0 obtained by the profile models (Fig. 2 (click here)). For instance, with a Fried parameter r0 = 13 cm,
the observation wavelength must be larger than 
in order to achieve a residual error better than /5. For r0 = 22
cm, 
is 
.
Figure 5: Isoplanatic domain (twice as isoplanatic angle
) corresponding to an image quality criterion of , versus wavelength, for the T.H.E.M.I.S. telescope aperture
after image stabilisation. Curves are calculated with three
modelised profiles from experimental measurements at Izaa site by
Arcetri University (Barletti et al. 1973)