4 Expected detection capabilities

We consider the case of a companion off-axis by
with respect
to an on-axis source, both sources being unresolved. We note *No* and *Nc* for
the number of collected photo-events respectively for on-axis source
(magnitude *mo*) and companion (magnitude
). Using
the
quantum efficiency,
the optical throughtput, (4.*RT*) the
beamsplitting factor and
the reference flux (zero magnitude
at wavelength ), *S* the collecting area,
the working
spectral bandwidth centered at ,
and
the exposure-time we have
for a single frame:

(29) |

(30) |

The coronagraph gives two twin-images of the companion. For a given , the shape of each image is roughly given by the central lobe of the normalized Airy distribution weighted by the spatial response of the coronagraph with (Eq. 5).

Taking into account the two twin-images, the "signal'' for *M*exposures is given by:

(31) |

where means sum over the central lobe (that is a disk of radius ).

The noise originates in the fluctuations of
the unwanted illumination in the image plane comprising residual light from
incomplete extinction of the on-axis source (-dependant) and the
contribution of the background. Adding the detector noise contribution we have
for a given pixel at

(32) |

where and are the variances of the number of photo-events counted by the pixel at , from the residual halo and from the background respectively. Detector contribution is represented by the variance ron

The background illumination is from a grey body of emissivity and temperature *T* seen within a solid angle
by a pixel. The
corresponding variance of background induced photoevents is:

(33) |

where is the Planck's function and

The noise from the residual illumination is a double stochastic process based
on Poisson statistics and Rayleigh statistics. From the Mandel's formulae
(Goodman [1985]) the associated variance includes two terms: one reflects the
Poisson noise at a given level of light
the other reflects the
fluctuations of this level (speckle noise). So we have:

(34) |

For ground based observations the speckle noise is the dominant one, except at a sufficiently low level of light.

The distribution is given by: , where describes the normalized residual energy (Eq. 26).

To evaluate the total noise we have to consider the sum of variances for the pixels covered by the image of the companion. We can only do that directly, if the fluctuations do not correlate from one pixel to another one, which is questionable considering the speckle noise contribution. Indeed, the exposure-time for a single frame is typically much larger than the coherence-time of the atmosphere, so that a correlation between adjacent pixels is likely to be destroyed. Moreover working at large bandwidth leads to an additional decorrelation.

Therefore the SNR for a companion off-axis by
with respect to an
unresolved on-axis source can be written as (
is a vector and *b*stands for
):

(35) |

The dependance on for the noise stands in the summation area, that is a disk of radius centered at . This drives us to insert into the integrands the weighting function (pil-box) . Figure 8 and Fig. 9 show examples of expectable detection capabilities (at for

Working wavelength
m, bandwidth
m,
reference flux
10^{-14} W cm
,
telescope diameter *D*=3.6 m, optical throughput
,
exposure time (single frame)
s, number of frames *M*=5000, Fried's parameter *r*_{0}=30 cm
(that is an
m) of 140 cm), *R*=*T*=0.5, quantum efficiency
,
ron=20 e
pix^{-1} exposure^{-1},
background temperature *T*=280 K, background emissivity ,
field of view
corresponding
to the central lobe of the Airy pattern of the collecting aperture.

Figure 9:
Similar to Fig. 8 but here in the case of INCOMPLETE CORRECTIONS up
to radial modes 1, 3, 5, 7. Values used are the same as for Fig. 8. Note the
change in vertical scale (0 to 10) |

From these profiles, it is apparent that even in the case of incomplete corrections (at a level likely to be nowadays negative) detection capabilities of our Achromatic Interfero Coronagraph are worth considering in the study of the stellar environment and similar topics. For example, clear detection of a companion as close as half the Airy radius, exhibiting a magnitude difference of the order 6 appears as a reasonable goal in rather ordinary observation conditions and with limited integration time (500 s here). Let us point out that the given estimates correspond to raw data and do not include the significant increase of performance usually brought about by appropriate data processing. Moreover, improvements relating to the correction capabilities of the adaptive optics devices (cut-off frequencies, number of actuators, etc.) are still in progress and the relevant profiles are likely to be close to the ones of the full correction case. The major limitation encountered in the correction process comes from the tilt effect, however there are serious hopes for much better capabilities. Thus, owing to the result of the present theoretical analysis it is conceivable that companions with (roughly speaking) up to 12 would be detectable from raw data, as suggested by profiles in Fig. 8. Moreover the reported values of can be increased by observing simultaneously (on the same camera) in two separate and adjacent spectral bands covering the total bandwidth of work, which relaxes the problem of the reference source and the possible random variations of the point spread function (Racine et al. [1999]). Of course an additional gain is achievable by using a longer integration time (number of frames) and by using a longer exposure-time for each single frame when allowed by atmospheric seeing and adaptive optics capabilities.

The large appearing within the Airy radius in the case of complete corrections (Fig. 8) reflects the behaviour of the residual halo, whose central part becomes larger and larger and darker and darker as the quality of the correction becomes better and better, the ultimate limitation to detection being then essentially photon noise. The bump appearing in the profiles about one Airy radius reflects the bump in the field response to extinction (Fig. 3).

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