4 Results

For each data set we produce a coronagraphic image of the binary star and the subtraction process obtained with a sampling parameter of 9.5 pixels/speckle area ($4\times 4$ pixels per group). In these samples, the optimal sampling is always smaller than the camera pixel sampling since the R/G value is less than 10 and therefore maximal SNR is actually achieved at 9.5 pixels/speckle area. However, for fainter companions (R/G>>10) a finer sampling would be required,together with more short exposures.

 

 

Table 3: Parameters measured on long exposure, (sample 2)

	$\Delta m$	G	$SNR_{\rm l}$	$SNR{\rm th_l}$	$SNR_{\rm g}$
Long exposure	7.3	172	55.9	39.4	21.3
1 photo-event

 

 

Table 4: Dark-speckle SNR, (sample 2)

Sampling	$SNR_{{\rm ds}}$	$SNR{\rm th_{ds}}$	$SNR{\rm sc_{ds}}$	$SNR_{\rm g}$
$1\times 1$ pix/group	3.1	16.5	37.3	20.7
$2\times 2$ pix/group	16.8	31.6	104.7	38.0
$4\times 4$ pix/group	62.2	54.1	196.5	48.5
Theoritical	-	71.9	-

Figure 4: Sample 3: 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 7.8 magnitudes

For each sample, we also give 2 tables. The 1 $^{{\rm st}}$ table presents the parameter infered from the long exposure, namely the brightness ratio, the AO gain (G), and compares the measured SNR ( $SNR_{\rm l}$) with expected value from Eq. (4) ( $SNR{\rm th_l}$). The 2 $^{{\rm nd}}$ table is then related to the dark-speckle process. The 1 $^{{\rm st}}$ row gives the SNR calculated on the negative "cleaned image'', for each available samplings. A theoretical estimate is presented in the 2 $^{{\rm nd}}$ row ( $SNR{\rm th_{ds}}$). The 3 $^{{\rm rd}}$ row gives the dark-speckle SNR after scaling ( $SNR{\rm sc_{ds}}$) for an accurate comparison of the long-exposure and dark-speckle images (Eq. (7)).

In addition, we have computed a more realistic signal to noise ratio including the global noise ( $SNR_{\rm g}$) on the subtracted images, namely the noise originating from the fixed speckle pattern. $SNR_{\rm g}$ has to be compared with $SNR_{\rm l}$ and $SNR{\rm sc_{ds}}$ respectively in tables related to the long exposure (Tables 1, 3, 5) and tables related to the dark-speckle exposure (Tables 2, 4, 6). The SNR obtained this way is considerably lower than other evaluated SNRs.

The samples presented hereafter have been obtained under moderately good seeing conditions: D/r₀=8.8 at $0.65~\mu$m and $\overline{v}=0.39$ m/s for a 100 mm pupil ( $\equiv 5.9$ m/s for a 1.5 m pupil). Depending on the room temperature regulation, r₀ ranges from 16 cm to 18 cm if scaled to a 1.5 m pupil.

In the following, we present the most relevant samples obtained in this laboratory simulation.

4.1 Sample 1

The first sample shows a companion 6.4 magnitudes fainter than the primary (Fig. 2). Binary star and reference star images were respectively reconstructed from 16143 and 6818 frames, each containing approximately 300 photon events. At this level, the brightest speckle can be saturated, but the companion, which is 2.4 times fainter than the average halo, is not affected. This saturation naturally decreases the halo/companion intensity ratio and may explain the discrepancy between measured and expected SNR in long exposures (Table 1). The $SNR_{\rm l}$ is indeed $63\%$ higher than the expected value. No such behaviour was observed in sample 3 where the photon rate is lower than 170 ph/frame. Also, since only the single photon events are exploited, the CP20 frames differ from a CCD long exposure which can contribute to this discrepancy. In addition, the measured and expected dark-speckle SNR (Table 2) are consistent, at least for the $4\times 4$ pixels binning ( $SNR_{{\rm ds}}=107.3$, $SNR{\rm th_{ds}}=97.4$), and it suggests that the discrepancy mentionned hereabove, does not result from an overestimated intensity ratio. Indeed, dark-speckle images are not affected by the saturation.

Once the dark-speckle image is scaled to the long exposure, the SNR obtained for both data with the finest sampling ($1\times 1$ pixel/group) are almost similar ( $SNR_{\rm l}=87.9$, $SNR{\rm sc_{ds}}=78$). Nevertheless, the best SNR calculated in the dark-speckle image with $4\times 4$ pixels/group, reaches 315, which is considerably higher than for the long exposure. A more realistic value ( $SNR_{\rm g}=80.8$) is computed, including the global residual speckle noise in the coronagraphic image.

 

 

Table 5: Parameters measured on long exposure, (sample 3)

	$\Delta m$	G	$SNR_{\rm l}$	$SNR{\rm th_l}$	$SNR_{\rm g}$
Long exposure	7.8	190	27.7	37.2	6.2
1 photo-event

 

 

Table 6: Dark-speckle SNR, (sample 3)

Sampling	$SNR_{{\rm ds}}$	$SNR{\rm th_{ds}}$	$SNR{\rm sc_{ds}}$	$SNR_{\rm g}$
$1\times 1$ pix/group	1.6	7.7	27.7	6.2
$2\times 2$ pix/group	4.6	15.0	37.9	9.5
$4\times 4$ pix/group	18.0	27.5	76.2	8.5
Theoritical	-	36.7	-

   
4.2 Sample 2

The sample 2 illustrates the detection of a fainter companion ( $\Delta m=7.3$) almost 5 times fainter than the average halo, after an integration of 38977 frames with a photon rate of 245 ph/frame (Fig. 3). On the dark-speckle image (Table 4) with $1\times 1$ pixel/group, the companion is almost undetectable and the measured SNR ( $SNR_{{\rm ds}}=3.1$) is lower than its theoretical value ( $SNR{\rm th_{ds}}=16.5$). After correct scaling of the dark-speckle image, the SNR (37.3) becomes closer to one obtained in the long exposure (55.9). Measured dark-speckle SNR approaches its expected value near the optimal sampling ( $SNR_{{\rm ds}}=62.2$, $SNR{\rm th_{ds}}=54.1$). When compared accurately through proper scaling, the dark-speckle SNR is then 3.5 times higher than the long-exposure SNR (Table 4, third column). From the analysis of the global residual fluctuations in the image, we derive an $SNR_{\rm g}=48.5$ for the largest sampling.

Figure 5: $\chi ^2$ map obtained with Fig. 4b and a Gaussian shape to enhance the contrast of the companion. Companion's location is given by the minimum of $\chi ^2$(arrow)

   
4.3 Sample 3

This is the faintest companion detected in these laboratory tests ( $\Delta m=7.8$). A large number of frames (65856), representing 22 minutes of simulated observing, was required to obtain significant SNR (Tables 5 and 6). Here again, the measured and theoretical dark-speckle SNR are more similar for a sampling of 9.5 pixels/speckle area. On the subtracted image (Fig. 4) the companion's Airy peak is still not distinguishable from the brighter residual speckles in the coronagraphic field. Nevertheless, as explained in Sect. 3.2.3, to accurately assess the model, the SNR was calculated in accordance with the local noise. We therefore derive an SNR of about 28 for the finest sampling which is relatively high with respect to the residuals in Fig. 4. Taking into account the global residual speckle noise leads to more realistic values ranging between 6.2 and 9.5 depending on the pixel sampling.

The long exposure SNR is close to the value expected from Eq. (4) ( $SNR_{\rm l}=27.7$, $SNR{\rm th_l}=37.2$). For $2\times 2$ pixels grouping, we have measured an $SNR_{{\rm ds}}=4.6$ in the dark-speckle image, but a better agreement with theoretical value is achieved with $4\times 4$ pixels/group ( $SNR_{{\rm ds}}=18$, $SNR{\rm th_{ds}}=27.5$). Nevertheless, the scaled dark-speckle image shows that the SNR is still higher ( $SNR{\rm sc_{ds}}=76.2$). It demonstrates that fainter companions could have been detected in these tests.

To accurately distinguish the companion's Airy peak from brighter residual speckles, a $\chi ^2$ test was carried out on the subtracted image of Fig. 4. At each pixel a $\chi ^2$ value is evaluated between the original data and a Gaussian shape, the width of which is constrained by the resolution. Fixed speckles are assumed differents in size and shape from a perfect Airy peak and are therefore efficiently removed with a $\chi ^2$ test which has been found better than a simple correlation. The gaussian's amplitude is chosen to minimize the global minimum of the $\chi ^2$ map indicating the location of the companion. Features such as companions close to a Gaussian shape and fainter that fixed speckles, are enhanced by the $\chi ^2$ test. The Fig. 5 shows the result of this process applied on the subtracted image presented in Fig. 4. This refinement is obviously less effective if one of the bright speckle is superimposed to the companion's Airy peak. Also the $\chi ^2$ test is restrained to point-like source.