4 Experimental results

4.1 Observations

The theoretical estimation of OTF angular variation is compared with experimental data recorded in September 1997 with the ONERA adaptive optics bench installed on the 1.52 m telescope at the Observatoire de Haute Provence [Conan et al.1998a]. The Adaptive Optics Bench (BOA) is a 88 actuator system, with a Shack-Hartmann wave-front sensor (64 sub-apertures). Two images of two separated binary stars ($\xi$ Cephee and $\Theta$Orionis) were acquired at 850 nm, with a 10 second integration time, on the 29 September 1997 at 00 h 02 and 04 h 27 (UT). The visual magnitude of the two components are: 4.6, 6.5 for $\xi$ Chepee and 5.1, 6.7 for $\Theta$ Orionis. The separation of these binary stars is large enough (about 8 and 13 arcsec) to obtain two PSF's with no overlap (see Fig. 4). In each case, the brightest star is used for the wave-front sensing. During the same night at 21 h 21 (UT), a $C_n^2\$profile measurement was obtained by M. Azouit from Université de Nice (France) (see Fig. 5) using balloon probes [Vernin & Munoz-Tunón1992]. For the two images the observing conditions are summarized in Table 1. r₀ has been estimated using open-loop Shack Hartmann data recorded shortly before the corrected images and $\theta _0$ has been computed using r₀, measured $C_n^2\$profile and Eq. (16).

Figure 4: Images of the two components of: a) $\xi$ Cephee (separation $\simeq 8 \hbox{$^{\prime\prime}$ }$) b) $\Theta$ Orionis (separation $\simeq 13 \hbox{$^{\prime\prime}$ }$). 1 pixel =0.027 arcsec, image size = $64\time 64$ pixels

 

 

Table 1: Observing conditions for $\xi$ Cephee and $\Theta$ Orionis. In the two cases, the sampling frequency is equal to 264 Hz. $\theta _0$ is the isoplanatic angle defined with Eq. (16), L₀ is the outer scale, and SR the Strehl ratio given in percent

	r0 (in m)	L0 (in m)	D/r0
	(@500 nm)		(@850 nm)
$\xi$ Cephee	0.055	4	14.6
$\Theta$ Orionis	0.06	4	13.4
	$\theta _0$ (arcsec)	SR	SR
	(@850 nm)	PSF 1	PSF 2
$\xi$ Cephee	1.53	4%	2%
$\Theta$ Orionis	1.66	8%	3%

4.2 ATF computation

We have approximated this $C_n^2\$profile by 3 "equivalent layers'' (EL) [Fusco et al.1999a]. The true $C_n^2\$profile is divided in 3 slabs (0 to 1 km, 1 to 10 km and 10 to 27 km), for each slab an EL is located at an equivalent height defined as the weighted mean height of the slab: $h_{\rm eq} = \left (\int_{h_{\rm min}}^{h_{\rm max}} C^2_n(h) h {\rm d}h\right )/\left (\int_{h_{\rm min}}^{h_{\rm max}} C^2_n(h){\rm d}h\right )$ and with an associated strength given by the total strength of the slab $\left(\int_{h_{\rm min}}^{h_{\rm max}} C^2_n(h){\rm d}h\right)$. All the values are summarized in Table 2.

 

 

Table 2: Results of the equivalent layer calculations

	$h_{\rm min}$	$h_{\rm max}$	equivalent	equivalent
			height	strength
	(in km)	(in km)	(in km)	(in %)
layer 1	0	1	7.7	56.5
layer 2	1	10	3.3	33.5
layer 3	10	27	14.1	10

Because of the weak dependency of the angular decorrelation of the phase with the atmospheric profile, this $C_n^2\$approximation is enough to obtain a good estimate on the phase angular decorrelation. We checked that an increase of the layer number (three to one hundred) do not significantly change the results.

Figure 5: $C_n^2\$profile measurements obtained with balloon probes by M. Azouit [solid line] and position of the 3 equivalent layers [dashed line]

In order to estimate the OTF, the AO system is assumed to provide a perfect correction of a given number of Zernike polynomials. This number N is defined so that the Nfirst modes have a wavefront reconstruction SNR higher than 1. For this purpose, we use the theoretical behavior of the turbulent and of the noise variance [Rigaut & Gendron1992]. These variances are scaled using the measured values of r₀ and L₀ for the Zernike turbulent variance and by the propagation of the photon and detector noise in the wavefront sensor through the reconstruction matrix for the Zernike noise variance [Rousset1999].

With the observing conditions, the correction in terms of Zernike polynomials is efficient (SNR $\ge 1$) about up to the $10^{\rm th}$ Zernike for $\xi$ Cephee and the $15^{\rm th}$Zernike for $\Theta$ Orionis.

Now, we compute the OTF for each binary star component (Eq. 7): the theoretical ATF is given by Eqs. (12) and (8) using:

- the 3 EL's profile for the computation of the angular correlation of the Zernike;

- the first $10^{\rm th}$ (resp. $15^{\rm th}$) Zernike coefficients;

- the known separation between the two stars.

The on-axis OTF is given by the first component of each binary star (brightest star).

The comparison between the measured OTF and the computed OTF is shown in Fig. 6.

Figure 6: Comparison between the measured [dashed line] and the computed [solid line] off-axis OTF. The measured OTF for $\alpha =0$ (brightest star) [dotted-dashed line] is given for comparison: (top) $\xi$ Cephee and (bottom) $\Theta$ Orionis. The circular average of each OTF is plotted

In the two cases, a good accordance between measured and computed OTF is found although a large number of assumptions has been made to compute the ATF:

- $C_n^2\$constant during the whole night (3h 40mn and 8h between the $C_n^2\$measurement and the two observations). Even if we know that the sensitivity to this profile is low;

- r₀ and L₀ estimation from the WFS noisy data assumed to be perfect;

- efficient correction of respectively the first 10 and 15 Zernike coefficients, which is a crude approximation of the AO system.

These results are very encouraging and allow to be optimistic for large FOV image post-processing.

Now, let us use the theoretical model of the PSF degradation for the post-processing of AO wide FOV images of stellar fields.