3 Data reduction

3.1 Data processing

A code, written in C language to analyse the photon coordinates delivered by the camera, generates two kinds of data set:

• the long exposure image, obtained by co-adding the frames (mapping of single photon events);
• the dark-speckle image, obtained by mapping the zero-photon events on each frame before co-adding them to generate a negative "cleaned image''.

The zero-photon occurrence denotes a lack of light, and therefore the "cleaned image'' is presented in negative form.

Labeyrie (1995) and Boccaletti et al. (1998a, 1998b) have explained how the dark-speckle method improves the detection of faint structures. It requires fine sampling to achieve very low light level detection with an optimal value given in Boccaletti et al. (1998a):

$$j=0.62{R\over G}$$ (1)

where j is the number of pixel per speckle area, R the star/planet brightness ratio, G the star/halo brightness ratio, referred to as the adaptive optics gain by Angel (1994). The ratio R/G is therefore the relevant parameter for imaging faint companions. For a companion 10 times fainter than the average speckled halo, a sampling of 6.2 pixels/speckle area is needed. The fainter the companion, the finer the sampling needed.

To adjust the sampling with respect to the ratio R/G, the zero-photon events counting can be made on "big'' pixels containing $2\times 2$ or $4\times 4$ physical pixels. A zero-photon event is recorded if the photon count is strictly zero in this group of physical pixels. This process is more efficient than box averaging in this context, and thus enables to change the pixel sampling on dark-speckle images. Since the detector has near zero read-out noise, the signal to noise ratio is not degraded through this operation.

The initial sampling on the camera was therefore set fine enough to allow such grouping of pixels. Diluting the photon rate also prevents the camera saturation. To avoid a significant loss of resolution, the groups are kept smaller than $4\times 4$ pixels. With $1\times 1$, $2\times 2$ and $4\times 4$ pixels/group, the sampling is then respectively 153, 38.3 and 9.5 pixels/speckle area. The latter is very close to the Shannon sampling.

The final images are still dominated by the fixed residual speckles, as mentioned in Boccaletti et al. (1998b). These static defects originate in fixed aberrations and from the residual amplitude pattern on the pupil (spider for instance). The coronagraphic images of the binary and the reference stars are recentered and scaled in intensity for subtracting this residual speckle pattern, on both the long-exposure and the dark-speckle image. The map subtraction is quite efficient far from the central star but becomes less effective near the mask, where speckles can be brighter than the hidden companion.

No attempt was made to correct flat-field and dark-current, since they can be neglected at first order, with such a photon-counting camera.

3.2 Data interpretation

We describe hereafter, how estimates of the primary/secondary intensity ratio, the AO gain, and signal-to-noise ratios were derived.

3.2.1 The intensity ratio

The intensity ratio was measured on the long exposure by integrating the pixel intensities on an area S encircling the $1^{{\rm st}}$ bright ring of the Airy pattern, as follows:

$$R = \int_{S}{I_*(x-x_0,y-y_0)\over{I_*(x-x_{\rm c},y-y_{\rm c})-I_{{\rm ref}}(x-x_{\rm c},y-y_{\rm c})}}{\rm d}x{\rm d}y$$ (2)

where I_* is the intensity distribution of the binary star with (x₀,y₀) and $(x_{\rm c},y_{\rm c})$ the Airy peak coordinates of respectively the primary star and the companion. $I_{{\rm ref}}$ is the intensity distribution of the reference star (scaled in intensity to the binary star) and describes the contribution of the star's diffracted halo at the companion location.

I_*(x-x₀,y-y₀) is measured on off-axis image where the primary star Airy peak remains visible while $I_*(x-x_{\rm c},y-y_{\rm c})$ and $I_{{\rm ref}}(x-x_{\rm c},y-y_{\rm c})$ are measured on the on-axis coronagraphic images.

The subtraction of the PSF halo ( $I_{{\rm ref}}$), measured on the reference star, is needed for accurate brightness ratio measurement, since in most cases the star halo intensity is brighter than the companion itself.

The photometric accuracy with the photon-counting camera is rather poor, owing to the imperfect photon-centroiding (Thiébaut 1994) achieved by the electronics. The photometric accuracy achieved this way is about 0.2 magnitude.

   
3.2.2 The AO gain

The AO gain (G) is defined by Angel (1994) as the ratio of the star's Airy peak intensity to the speckled halo intensity. G is thus an estimator of the residual scattered light, and improves with the AO performance. Since the halo is smoothed on the long exposure, an accurate estimate of the average gain can be obtained at the companion's location using the expression:

$$G = \int_{S'}{I_*(x-x_0,y-y_0)\over I_{{\rm ref}}(x-x_{\rm c},y-y_{\rm c})}{\rm d}x{\rm d}y.$$ (3)

Here, the integration is performed over speckle area (S').

With the BOA adaptive optics, under good seeing conditions and with moderate pupil diameter (see Sect. 4), it is therefore possible to reach a gain of about 100-200 at the companion location (7.6$\lambda/D$ from the Airy peak). A companion about 5 magnitudes fainter than the star will thus be easily detected in a long exposure on which the speckled halo is smoothed to this intensity level (G). A dark-speckle analysis imaging is therefore required to improve the detection of fainter companions ( $\Delta m>5$).

Figure 1: Coronagraphic image profiles showing the peak (respectively the dip) of the companion at position x=124. Long exposure (solid line) and dark-speckle image with $1\times 1$ pixel/group (dashed line) are stricly equivalent but displayed differently, as indicated in Sect. 3.2.3. The long exposure has positive value with 0 minima, while the dark-speckle exposure is presented in negative form with a maximum value equal to the total number of frames. The dotted line shows similar dark-speckle analysis processed with a $2\times 2$ binning. The SNR is significantly improved

   
3.2.3 The signal-to-noise ratio

Figure 2: Sample 1: Coronagraphic image of the binary a) and subtraction with a reference star b) for the sampling of $4\times 4$ pixels/group, maximizing the SNR. The brightness ratio between the primary and the companion (arrow) is 6.4 magnitudes. Here, the subtraction is almost perfect

Theoretical expressions of the signal-to-noise ratio SNR are:

$$SNR_{\rm l}={G\over R}\sqrt{T\over t}$$ (4)

for the long exposure (Angel 1994), and:

$$SNR_{{\rm ds}-j}={N_*\over R}\sqrt{tT\over j+{tN_*\over G}}$$ (5)

for the dark-speckle exposure (Boccaletti et al. 1998a). Where T is the total integration time, t the short exposure time or the speckle lifetime in second unit, j the sampling parameter (pixels/speckle area) and N_* the photon rate of the star (photons/s).

Once the pixels are grouped in $1\times 1$, $2\times 2$ or $4\times 4$ pixels on the dark-speckle exposure, 3 different $SNR_{{\rm ds}}$ can be evaluated.

 

 

Table 1: Parameters measured on long exposure, (sample 1)

	$\Delta m$	G	$SNR_{\rm l}$	$SNR{\rm th_l}$	$SNR_{\rm g}$
Long exposure	6.4	155	87.9	53.9	27.7
1 photo-event

 

 

Table 2: Dark-speckle SNR, (sample 1)

Sampling	$SNR_{{\rm ds}}$	$SNR{\rm th_{ds}}$	$SNR{\rm sc_{ds}}$	$SNR_{\rm g}$
$1\times 1$ pix/group	6.8	31.7	78.0	28.9
$2\times 2$ pix/group	30.2	59.4	175.3	51.7
$4\times 4$ pix/group	107.3	97.4	314.8	80.8
Theoritical	-	134.6	-

Figure 3: Sample 2: Coronagraphic image of the binary a) and subtraction with a reference star b) for the sampling of $4\times 4$ pixels/group, maximizing the SNR. The brightness ratio between the primary and the companion (arrow) is 7.3 magnitudes

The goal of these laboratory tests was to assess the validity of the dark-speckle model (Boccaletti et al. 1998a), and therefore only the local noise was considered to derive the SNR. Since it does not take into account the residual speckle noise due to static aberrations, this process leads obviously to an optimistic SNR, but more consistent with theoretical expressions.

At the companion location, the SNR is measured over a speckle area (S'), on both the long exposure and the dark-speckle image, according to the following relation:

$$SNR\!=\!\!\int_{S'}\!\!\!{\left\vert{I_{{\rm ref}}(x\!-\!x_{\... ...{I_{{\rm ref}}(x\!-x_{\rm c},y\!-y_{\rm c})}}}{\rm d}x{\rm d}y$$ (6)

$I_*(x-x_{\rm c},y-y_{\rm c})$ and $I_{{\rm ref}}(x-x_{\rm c},y-y_{\rm c})$ represent the 1 or 0-photon events count respectively with and without companion.

As explained hereabove, the intensity of the companion's Airy peak on the binary star, is compared to the residual intensity at the same location on the reference star. The noise originating from static aberrations is thus not accounted in the SNR calculation, and a realistic comparison with the model becomes possible.

At this point, it is possible to compare measured (Eq. 6) and expected SNR (Eqs. 4 and 5), separately for the long exposure and the dark-speckle exposure. But the direct comparison of the dark-speckle and the long exposure efficiency is not straightforward. The long exposure has positive values and 0 minima while the dark-speckle image is presented in negative form with a maximum value equal to the total number of frames (Fig. 1). Therefore, dark-speckle images ( $I_{{\rm ds}}$ in Eq. (7)) have to be scaled to the long exposure, with the following expression:

$${\rm Iscaled_{ds}}={\rm max}(I_{{\rm ds}})-I_{{\rm ds}}$$ (7)

Once $SNR_{{\rm ds}}$ is re-calculated on the dark-speckle scaled image, both $SNR_{\rm l}$ and $SNR_{{\rm ds}}$ can be accurately compared (Sect. 4). As mentioned above, the photon-counting camera provides only the 0 and 1-photon events. The long exposure $SNR_{\rm l}$ and the dark-speckle $SNR_{{\rm ds-1}}$, obtained with the finest sampling, are therefore strictly equivalent when properly scaled. However, the dark-speckle imaging outperforms long exposure for pixel sampling of 38.3 and 9.5 pixels/speckle.