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4 Experiment at the TIRGO telescope

  \begin{figure}\par\begin{tabular}{\vert c\vert}
\psfig{figure=ds1757fig6.epsi,height=4.75cm}\\ \hline
\end{tabular}\par\end{figure} Figure 6: Description of the interferometric experiment with the TIRGO telescope. Representation of the mask simulating the LBT pupil, and how it was mounted on the TIRGO secondary mirror. The ratio b/d, where b is the center-to-center baseline and d the diameter of each aperture, is equal to that of LBT, i.e. b/d$\sim 1.76 $. The two values of d used during the run are mentioned, as well as the resulting fraction of collecting area


Table 1: Observation conditions and resulting value of parameter D/r0 (J-band). D is the projection of d on the primary. Median seeing estimates of each night was provided by averaging 3 long exposure FWHM measures taken without the mask and spread over the night
d (cm) D (cm) seeing ( $\hbox{$^{\prime\prime}$ }$) r0 (cm) D/r0
0.72 5.4 $ 4.0 \pm 0.5 $ $ 6.4 \pm 0.8 $ $ 0.84 \pm 0.11 $
0.57 4.3 $ 3.5 \pm 0.5 $ $ 7.4 \pm 1.1 $ $ 0.58 \pm 0.08 $

In early 1998, we started an experiment at the 1.5m TIRGO infrared telescope, with a mask simulating the pupil of the LBT telescope. The main idea was to record realistic LBT-like data by simulating the level of adaptive optics (AO) correction expected for LBT by the choice of the ratio D/r0 (where D is the diameter of one of the mask apertures projected on the primary and r0 the Fried parameter), in order to investigate the properties of the atmospheric parameters and study the process of image formation and reconstruction (Fig.6).

An observing run, on the nights of 21 and 22 March 1998, permitted to collect a serie of measurements of one point-like and one binary star under two different values of the ratio D/r0 (Table1). Actually, it can be noticed that the reduction in D/r0 obtained with the use of a mask (i.e., by making D smaller) is not equivalent to that expected from the use of adaptive optics on the real LBT (i.e., by making r0 larger). In fact, in our case the mask holes produce a rescaling of the frequencies in the turbulence power spectrum. The effect of correction by adaptive optics, on the other hand, is not equal at all frequencies. But, for the aim of this experiment, these two corrections present enough similarities to be considered equal in first approximation. In order to simulate the rotation of the LBT-like pupil function, i.e. the aperture synthesis by earth rotation, we recorded object interferograms at four nearly equidistant mask orientation angles by rotating the secondary mirror. The data were recorded in a broad band J filter at the 1.5m TIRGO infrared telescope using the ARNICA camera (Lisi et al. [1996]) which is equipped with a $ 256 \times 256 $ pixels NICMOS3 detector and presents a pixel size of $ 0\,\hbox{$.\!\!^{\prime\prime}$ }98 $. During the run, we made use of the fast read-out mode of a $ 32
\times 32 $ pixels sub-array, developed for lunar occultations, which allows typical integration times of around 20ms. A brief description of this mode can be found in Richichi et al. ([1996]). We present below a summary of some of the data reduction results.

4.1 Measurement of atmospheric parameters

\psfig{figure=ds1757fig7.epsi,height=13cm}\end{tabular}\par\end{figure} Figure 7: Measure of atmospheric parameters obtained by fitting PSF data of the experiment. Evolution of fringe contrast (top row) and of differential piston (bottom row) in the data set for D/r0 = 0.84 (integration time of 100ms) and D/r0 = 0.57 (integration time of 150ms). The dashed lines represents rms intervals

\psfig{figure=ds1757fig8.epsi,height=6.5cm}\end{tabular}\par\end{figure} Figure 8: Variation of relative fringe contrast with integration time, in relative value, obtained by coadding, for each integration time, several series of consecutive PSF interferograms

A least-square PSF fitting program was developed and applied to our data in order to extract some atmospheric parameters of interest. It allowed us to retrieve the evolution with time of the fringe contrast, and of the random average optical path difference between the two apertures (that is the differential piston). This was done for the two values of the ratio D/r0, on a 30s period, and for an unique baseline orientation (Fig.7), and we found consistent results in terms of average contrast and piston root-mean-square (rms) values. Indeed the case of major atmospheric degradation leads to a smaller average contrast: for D/r0= 0.84 and D/r0= 0.57 we found respectively an average contrast of 0.26 and 0.60. Accordingly, we found a piston rms value of respectively 0.11 and 0.06 in unit wavelength.

In addition, combining several consecutive short exposure PSF interferograms, we studied the variation of contrast with integration time under the same two conditions of turbulence degradation (Fig.8) and defined a coherence time $t_{\rm c}$ as the integration time which corresponds to a loss of a certain percentage of fringe contrast. We deduced the value of $t_{\rm c}$ with $ 10 \% $ loss for our experiment and attempted to extrapolate the result to the LBT case (Correia [1998]).

4.2 Application of the reconstruction technique to real data: $\gamma $ Leo

  \begin{figure}\par\begin{tabular}{\vert c\vert c\vert\vert c\vert c\vert}
...\psfig{,height=4cm}\\ \hline
\end{tabular}\par\end{figure} Figure 9: Left panel: Set of the $ 64 \times 64 $ pixels pre-processed interferograms patterns of the binary $\gamma $Leo corresponding to each orientation angle. Each corresponds to an integration time of 100ms ( D/r0 = 0.84, J-band). Right panel: Corresponding set of SAA PSF estimates used for the deconvolution process. The pixel size is $
0\hbox{$.\!\!^{\prime\prime}$ }98 / 3 \sim 0\hbox{$.\!\!^{\prime\prime}$ }33 $, North is right, East is down (images are displayed on a linear scale)

\psfig{figure=ds1757fig10.epsi,height=6cm}\end{tabular}\par\end{figure} Figure 10: Reconstructed image of the binary $\gamma $ Leo obtained using the Lucy-Richardson algorithm adapted for multiple deconvolution of simultaneous data frames (1000 iterations). The number of data frames per orientation angle used here is 200, and each frame corresponds to 100ms integration time. The pixel size is $
0\hbox{$.\!\!^{\prime\prime}$ }98 / 3 \sim 0\hbox{$.\!\!^{\prime\prime}$ }33 $. Contour levels are from $5\%$ to $100\%$ in steps of $5\%$

The aperture size of the mask and the seeing conditions led, for this observation, to D/r0=0.84. In the following, we used mainly a data set of 100ms integration time exposures. This data set is composed of a total of 1200 interferograms of the binary $\gamma $Leo (Algieba), a sequence of 300 interferograms for each of the four different mask orientation angles ( $ 0\hbox{$^\circ$ }$, $ \simeq
46\hbox{$^\circ$ }$, $ \simeq 86\hbox{$^\circ$ }$, $ \simeq 126\hbox{$^\circ$ }$). The object is a binary star (ADS 7724) composed of a K1III main component and a G7III companion. According to the Hipparcos catalogue([ESA 1997]) and to calibrated V-J colors, the main component magnitude and the magnitude difference are respectively mJ = 0.21 and $ \Delta m_J= 1.39$. The angular separation is $ 4\,\hbox{$.\!\!^{\prime\prime}$ }58 $, almost two times the diffraction limit of the simulated interferometer. In addition, 300 interferograms of an unresolved bright star (BS 5589, a M4.5III star with $ m_J \simeq 0 $) were recorded for the first mask orientation. These reference data are used for the system response (optics + atmosphere), the so-called PSF, in the deconvolution process. In this run, the total collecting aera was limited by the fixed pixel size of the camera. For this reason, only $ 0.26
\% $ of the mirror area was used, leading for this integration time to a relatively poor signal-to-noise (S/N) ratio on the frames. Assuming frozen turbulence driven by the wind velocityv, the characteristic evolution time is of the order of B/v, where B is the projection of b on the primary i.e. here B=9.5 cm. Taking v=10 m/s leads to a typical evolution time of the order of 10ms. Consequently, the level of atmospheric degradation conditions (which is evaluated from the D/r0 ratio to a $ 46 \% $ Strehl ratio) allowed us to assume that the blurring arising from high order atmospheric turbulence of evolution time scale inferior to 100ms remains important. Note that, in order to minimise variations in seeing conditions during data acquisition, all frames of both binary and point-like stars have been recorded during consecutive periods of the night.

In the reconstruction, we used 200 interferograms of the binary star per orientation angle. A total of 800 interferogram patterns were pre-processed before applying the reconstruction method. Pre-processing consisted in subtracting the average sky background, a bilinear upsampling from $ 30 \times
30 $ pixels to $ 90 \times 90 $ pixels and then the extraction of a $ 64 \times 64 $ pixels frame centered on the photocenter fringe pattern (see Fig.9). This oversampling technique allowed a sub-pixel centering of each original 30 $\times$ 30 pixels interferogram, i.e. to remove the atmospherically induced image motion.

Moreover, in order to obtain an accurate PSF estimate needed for the LR algorithm, a Shift-and-Add (SAA) process was computed over the whole PSF data set. Then we used the same pre-processing of the object interferograms to obtain the PSF estimate (see Fig.9). Unfortunately, the PSF was only recorded for the first orientation angle. The PSF corresponding to other orientation angles were obtained by computer rotation using a bilinear interpolation. Note that this rotation was performed on the oversampled frames, allowing to conserve more information in the PSF shape. However, it is unlikely that the responses of the system for each orientation angle are identical to the rotated responses. Therefore the rotated PSF's obtained in this way are only an approximation.

The companion is clearly visible in the reconstructed image of Fig.10, in spite of numerous limitations in this data set. Indeed, in addition to the low S/N ratio, and the approximation of the rotated PSF, the original data were also slightly under sampled ( $ \sim \,1.8 $ pixels per fringe FWHM).

\par\end{figure} Figure 11: Reconstruction of the binary $\gamma $ Leo according to the number of data frames per orientation angle processed, using the adapted LR algorithm (1000 iterations). The pixel size is $ 0\,\hbox{$.\!\!^{\prime\prime}$ }98 / 3 \sim 0\,\hbox{$.\!\!^{\prime\prime}$ }33 $. Contour levels are from $5\%$ to $100\%$ in steps of $5\%$

\psfig{figure=ds1757fig12.epsi,height=6cm}\end{tabular}\par\end{figure} Figure 12: Profile of the reconstructed binary along the direction of separation, recovered using different number of frames per angle

\psfig{figure=ds1757fig13.epsi,height=6cm}\end{tabular}\par\end{figure} Figure 13: Reconstructed image of the binary $\gamma $ Leo obtained deconvolving simultaneously 50 frames per orientation angle of 50 ms integration time each (1000 iterations). The pixel size is $
0\hbox{$.\!\!^{\prime\prime}$ }98 / 3 \sim 0\hbox{$.\!\!^{\prime\prime}$ }33 $. Contour levels are from $5\%$ to $100\%$ in steps of $5\%$

We further analysed the effect of the number of frames on the reduced data. Figures 11 and 12 show the evolution with this parameter of, respectively, the reconstructed object, and the profile of the reconstructed binary along the direction of separation. It can be seen that the gain obtained in the resulting reconstruction using multiframes deconvolution is important for a small number of data frames, and tends to converge rapidly. The variation of the FWHM of the main component is consistent with the fact that deconvolving more frames simultaneously improves the quality of the reconstructed object, i.e. the sharpness of the reconstruction. In fact, for a number of 5, 10, 50 and 200 frames per angle used, the FWHM of the main component obtained is respectively $ 2\,\hbox{$.\!\!^{\prime\prime}$ }47 $, $ 2\,\hbox{$.\!\!^{\prime\prime}$ }34 $, $ 2\,\hbox{$.\!\!^{\prime\prime}$ }31 $ and $ 2\,\hbox{$.\!\!^{\prime\prime}$ }21 $. It is interesting to note that we approached the theoretical diffraction limit for our mask of $ 1\,\hbox{$.\!\!^{\prime\prime}$ }73 $, but we were unable to reach it because of the limitations previously mentioned. It is however important to stress that this $ 2\,\hbox{$.\!\!^{\prime\prime}$ }11 $ interferometric resolution does allow us to resolve $\gamma $ Leo, whereas the $ 5\,\hbox{$.\!\!^{\prime\prime}$ }83 $ resolution of only one aperture would not permit it. We noticed moreover that this simultaneous multiframes deconvolution leads to a better result than deconvolving simultaneously the sum of the frames for all baseline orientation.


Table 2: Measured binary parameters retrieved from the deconvoluted images compared to the Hipparcos parameters of $\gamma $Leo (HIP 50583). Measurements are both on 50ms (50 frames/angle) and 100ms (200 frames/angle) integration time deconvolution results and on data (PSF-fitting)
  $ \rho $( $\hbox{$^{\prime\prime}$ }$) $ \theta (\hbox{$^\circ$ }) $ $ \Delta m_J $
Hipparcos catalogue 4.58 124.3 1.39
      ap. phot. PSF-fitting
Measured (100ms) 4.66 124.1 2.16-1.97 $ 1.58 \pm 0.33 $
Measured (50ms) 5.05 126.5 2.15-2.11 $ 1.56 \pm 0.11 $

A data set composed of 50ms integration time frames was reduced in the same way and the result obtained is presented in Fig.13. This shorter integration time "freezes'' better the atmospheric distorted fringe pattern and the result is a more detached binary image. Note that the spurious signal present on the main component sides probably came from a non perfect centering of each fringe patterns due to the poor S/N ratio.

We tested the validity of these reconstructed images by performing the comparison of the retrieved binary parameter values with those of the Hipparcos catalogue (included in the CHARA catalogue, Hartkopf et al. [1998]). Centroid calculations performed on $ 5 \times 5 $ pixel boxes centered on each maximum intensity pixel of the two star brightness distributions led to the relative location of the binary. Two methods were used in order to measure the magnitude difference $ \Delta m_J $. Firstly, relative aperture photometry was performed with aperture radii from 3 pixels to 7 pixels, and we deduced therefore a range of retrieved $ \Delta m_J $. Secondly, we obtained photometry by PSF-fitting of the non-deconvolved frames, using the location retrieved by means of deconvolution as an input fixed parameter. This PSF-fitting is based on a least-square fitting routine identical to that used in Sect.4.1 and was applied to a same number of coadded frames for each orientation angle. Results reported in Table2 are an average over the four orientation angles, weighting with the $ \chi^2
$ in order to take into account the error fitting of the other free parameters. From the values reported in Table2, one can see that we are able to retrieve both the position angle $ \theta $ and the angular separation $ \rho $ of $\gamma $Leo, with a better accuracy in the 100ms case. Concerning the relative magnitude retrieved by aperture photometry, $ \Delta m_J $ is larger than the catalogue value by an amount of 0.77 to 0.58mag for respectively 3 to 7 pixels aperture radius and 100ms integration time. This discrepancy is probably due to the poor S/N ratio present in the recorded frames, and seems to be a common characteristic of most of the non-linear image restoration techniques when applied to strongly noise-contaminated data (Lindler et al. [1994]; Christou et al. [1998]). On the other hand, this interpretation is confirmed by the simulations performed in Sect.3. Photometry measurements obtained by means of PSF-fitting lead to more accurate $ \Delta m_J $ values and have the advantage to give error estimates.

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