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Subsections

  
3 PSF reconstruction: Experimental approach

To test the PSF reconstruction algorithm, we measured the residual wavefront on a bright point source which is simultaneously imaged on the scientific detector. The reconstructed PSF from the WFS measurements can then be directly compared to the long-exposure image. To simulate reference sources of different magnitude, we put appropriated neutral densities in front of the WFS. For each neutral density, we took measurements alternatively in open loop (OL) and in close loop (CL), to compare the derived r0 and noise values (see below).

To reconstruct the PSF, we need to know: i) the measurement noise (noise hereafter) in order to estimate accurately the residual phase covariance \( {\cal C}_{\epsilon \epsilon } \), and ii) r0 in order to estimate \( D_{\phi _{{\rm a}_{\perp }}} \) and \( {\cal C}_{rr} \). Let us first discuss the methods we use to estimate these two quantities.


  \begin{figure}\includegraphics[width=9cm]{ds9074f4.eps}\end{figure} Figure 4: Noise estimated on one set of data as a function of the noise estimated on another set of data. The rms deviation with respect to the dotted straight line (noise equality) is about 7-8%

3.1 Evaluation of the noise


  \begin{figure}\includegraphics{ds9074f5.eps}\end{figure} Figure 5: Noise estimated on OL data versus the noise estimated on CL data, both averaged over all subapertures for a) good seeing (0.8 arcsec) and b) bad seeing (1.5-2 arcsec)

 The noise for a Shack-Hartmann device is generally a superposition of photon noise and read-out noise. Although theoretical expressions can be found in the literature (Rousset 1994 e.g.), it is always preferable to estimate the noise directly on the experimental data. The noise which is supposed to be white can be found from the measured wavefront slopes, either analyzing their temporal autocorrelation or their temporal spectral density. Since one is the Fourier transform of the other, both methods are in principle equivalent. In practice, we generally use the autocorrelation method which consists in deriving the noise from the bias on the zero point of the autocorrelation function by fitting e.g. a parabola near its origin (Gendron & Léna 1995). We checked that the noise derived this way is consistent with the noise extracted from the high frequency tail of the temporal spectral density. In order to test the accuracy of the method, we estimated the noise on four consecutive samples of 5120 measurements each. Figure 4 compares, for the set of useful subpupils and the x and y directions, the noise estimated in one sample to the noise found on another sample. The rms deviation is typically of the order of 7 to 8%.

It is straightforward to estimate the noise on OL data, but it is usually very difficult to do it on CL data when the correction is good. The reason is that the consecutive slope measurements are less correlated due to the wavefront correction, making difficult to extrapolate the autocorrelation function to its origin. We estimate the noise on CL data whenever it is possible, otherwise we do it on the OL data. It is therefore important to know, if both derivations are compatible. In Fig. 5, we compare, for different set of data, the noise evaluated in OL to the noise evaluated in CL (each averaged over all subapertures), when this was possible, i.e. for small AO loop gains. Clearly, both derivations are consistent for good seeing ( \( 0\hbox{$.\!\!^{\prime\prime}$ }9 \)). It is not quite the case for bad seeing( \( 1\hbox{$.\!\!^{\prime\prime}$ }5-2\hbox{$.\!\!^{\prime\prime}$ }0 \)), for which the OL noise tends to be larger than the CL noise, probably due to the finite size of each sub-aperture. In fact, the optical field of one sub-aperture which is needed to image a focal spot accurately is of the order of \( 2\, \mathrm{FWHM}+6\, \sigma _{\alpha} \), where FWHM is the full width at half maximum of the spot and \( \sigma _{\alpha }^{2} \)the variance of the angle-of-arrival. Taking the above seeing values, we find a necessary field of \( \approx 7\hbox{$.\!\!^{\prime\prime}$ }0 \) in the case of very bad seeing and \( \approx 3\hbox{$.\!\!^{\prime\prime}$ }0 \) in the case of good seeing. Since the field of view of the ADONIS wavefront sensor is \( 6\hbox{$.\!\!^{\prime\prime}$ }0 \) (EBCCD), the image can be truncated when the seeing is bad, leading to a bad estimate of its center of gravity.

3.2 Evaluation of Fried's parameter \( r_{0}\protect \)

 We have to know r0 in order to calibrate the high-frequency phase structure function \( \bar{D}_{\phi _{\perp }}(\vec{\rho }) \) and the covariance of the remaining error \( {\cal C}_{rr} \). We now discuss the different methods to estimate r0.

In OL, we can derive r0 either from the variance of the absolute angle-of-arrival \( \alpha \) or from the variance of the differential angle-of-arrival \( \Delta \alpha _{{\rm d}}=\alpha (x)-\alpha (x+d) \), where d is the distance between the centers of two subapertures (Ziad 1993). The advantage of using the differential angle-of-arrival is that it does not depend on telescope vibrations which could bias the r0 evaluation when using the absolute angle-of-arrival. Experimentally, we find, however, that both methods give the same results within 5%.

In CL the variance of the angle-of-arrival is reduced due to the wavefront correction and hence, it cannot be directly linked to r0. In this case, we estimate r0 from the variance of the mirror commands by means of the following equation (Véran et al. 1997a):

\begin{displaymath}\sigma _{{\rm m}_{i}}^{2}=\left( \sigma _{{\rm a}_{0_{i}}}^{2... ...t \left\vert H_{{\rm n}}(g_{i,}\nu )\right\vert \, {\rm d}\nu, \end{displaymath} (29)

where \( \sigma _{{\rm a}_{0_{i}}}^{2} \) and \( \sigma _{{\rm r}_{0_{i}}}^{2} \) are the theoretical variances of the turbulent modal coefficients and the remaining error for D/r0=1, respectively (assuming Kolmogorov turbulence). The last term is the modal noise filtered by the noise transfer function. Its influence on the calculation of r0 is negligible. Also, \( \sigma _{\rm r_{0}}^{2} \)is small compared to \( \sigma _{{\rm a}_{0}}^{2} \). The above equation is valid, if the bandwidth of the AO system is higher than the highest cut-off frequency of the temporal spectra of ai and ri. If this is not true, then we will overestimate r0. The presence of an outer turbulence scale L0 will decrease the variance of the first modes, particularly the tip/tilt modes. For this reason, we exclude these two modes. We also exclude the higher modes for which the AO bandwidth is low (due to small gains) and, in any case, much lower than their cut-off frequency. Thus, we restrict the estimation of r0 to the modes 3 to 25.


  \begin{figure}\includegraphics[width=9cm]{ds9074f6.eps}\end{figure} Figure 6: \( r_{0}\protect \) estimated from the OL data (angle-of-arrival) versus the \( r_{0}\protect \) computed from the CL data (mirror commands). The two smallest \( r_{0}\protect \)values (triangles) calculated in OL are probably affected by large biases. For the larger \( r_{0}\protect \) (black circles), the values calculated from the angle-of-arrival are in good agreement with those derived from a seeing monitor located outside of the telescope dome

Figure 6 compares the values of r0 obtained from the variance of the angle-of-arrival in OL to those obtained from the variance of the mirror commands in CL. r0 estimated from the OL data is always larger than r0 computed from the CL data. The two smallest r0values (triangles) calculated in OL are probably affected by large biases. For the larger r0 (black circles), the values calculated from the angle-of-arrival are in good agreement with those derived from a seeing monitor located outside of the telescope dome. Theses values are in turn in good agreement with those derived from the mirror commands, if we restrict the evaluation of r0to the modes 3 to 5. This indicates the possible presence of dome or mirror seeing (see Sect. 5.2).

For our PSF reconstruction, we use in practice the r0 value obtained from the variance of the mirror commands on the modes 3 to 25.


 

 
Table 2: Experimental conditions and results of the PSF reconstruction. The sampling frequency of the WFS was 200Hz in all cases and its effective wavelength about 0.6 \( \mu{\rm m}\protect \). We show \( r_{0}\protect \)estimated on the modes 3 to 25 and, in parentheses, \( r_{0}\protect \)estimated from the variance of the angle-of-arrival in OL
               
  Ex. A1 Ex. A2 Ex. B1 Ex. B2 Ex. C1 Ex. C2 Ex. C3
MV (magnitudes) 7.4 7.7 10.3 10.3 10.6 11.6 8.5
Mean flux ( \( {\rm e}^{-} \)/ ssp / frame) 80 48 40 39 7 4.5 25
r0 at 0.55 \( \mu {\rm m}\) (cm) 10.8 (13.3) 9.1 (10.6) 6.5 (10.0) 5.3 (10.0) 12.3 (14.7) 14.7 (18.3) 12.7 (16.9)
WFS camera RETICON RETICON EBCCD EBCCD EBCCD EBCCD EBCCD
IR wavelength 2.12 \( \mu {\rm m}\) 2.12 \( \mu {\rm m}\) 2.15 \( \mu {\rm m}\) 2.15 \( \mu {\rm m}\) 2.15 \( \mu {\rm m}\) 2.15 \( \mu {\rm m}\) 2.15 \( \mu {\rm m}\)
Total integration time 20 s 20 s 50 s 50 s 100 s 100 s 100 s
average gain 0.56 0.46 0.19 0.49 0.09 0.08 0.27
gain min - gain max 0.30 - 1.0 0.33 - 0.69 0.04 - 0.51 0.09 - 0.89 0.03 - 0.3 0.1 - 0.26 0.1 - 0.44
               
Residual phase \( \sigma _{\epsilon }^{2} \) at 0.55 \( \mu {\rm m}\) 22.8 \( {\rm rd}^{2} \) 21.1 \( {\rm rd}^{2} \) 40.6 \( {\rm rd}^{2} \) 47.4 \( {\rm rd}^{2} \) 23.1 \( {\rm rd}^{2} \) 20.9 \( {\rm rd}^{2} \) 20.8 \( {\rm rd}^{2} \)
\( \hookrightarrow \) Low-order residual phase \( \sigma _{\epsilon _{\parallel }}^{2} \) 18.6 \( {\rm rd}^{2} \) 15.6 \( {\rm rd}^{2} \) 30.8 \( {\rm rd}^{2} \) 33.7 \( {\rm rd}^{2} \) 19.9 \( {\rm rd}^{2} \) 18.4 \( {\rm rd}^{2} \) 17.6 \( {\rm rd}^{2} \)
\( \hookrightarrow \) High-order phase \( \sigma _{\phi _{{\rm a}_{\perp }}}^{2} \) 4.2 \( {\rm rd}^{2} \) 5.5 \( {\rm rd}^{2} \) 9.8 \( {\rm rd}^{2} \) 13.7 \( {\rm rd}^{2} \) 3.2 \( {\rm rd}^{2} \) 2.5 \( {\rm rd}^{2} \) 3.2 \( {\rm rd}^{2} \)
Modal measurement error \( \sigma _{{\rm n}}^{2} \) 17.5 \( {\rm rd}^{2} \) 28.6 \( {\rm rd}^{2} \) 32.7 \( {\rm rd}^{2} \) 40.7 \( {\rm rd}^{2} \) 128.1 \( {\rm rd}^{2} \) 179.8 \( {\rm rd}^{2} \) 33.0 \( {\rm rd}^{2} \)
Remaining error \( \sigma _{{\rm r}}^{2} \) 1.2 \( {\rm rd}^{2} \) 1.5 \( {\rm rd}^{2} \) 2.2 \( {\rm rd}^{2} \) 3.1 \( {\rm rd}^{2} \) 0.7 \( {\rm rd}^{2} \) 0.6 \( {\rm rd}^{2} \) 0.7 \( {\rm rd}^{2} \)
               
Strehl ratio of observed PSF 25.5% 24.4% 10.1% 7.6% 21.6% 19.9% 26.3%
Strehl ratio of reconstructed PSF 26.7% 28.6% 9.4% 6.2% 21.6% 24.7% 27.1%
FWHM of observed PSF \( 0\hbox{$.\!\!^{\prime\prime}$ }126 \) \( 0\hbox{$.\!\!^{\prime\prime}$ }135 \) \( 0\hbox{$.\!\!^{\prime\prime}$ }187 \) \( 0\hbox{$.\!\!^{\prime\prime}$ }199 \) \( 0\hbox{$.\!\!^{\prime\prime}$ }169 \) \( 0\hbox{$.\!\!^{\prime\prime}$ }178 \) \( 0\hbox{$.\!\!^{\prime\prime}$ }159 \)
FWHM of reconstructed PSF \( 0\hbox{$.\!\!^{\prime\prime}$ }129 \) \( 0\hbox{$.\!\!^{\prime\prime}$ }126 \) \( 0\hbox{$.\!\!^{\prime\prime}$ }174 \) \( 0\hbox{$.\!\!^{\prime\prime}$ }170 \) \( 0\hbox{$.\!\!^{\prime\prime}$ }160 \) \( 0\hbox{$.\!\!^{\prime\prime}$ }156 \) \( 0\hbox{$.\!\!^{\prime\prime}$ }162 \)


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