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To calculate the method's sensitivity in a more general way than done by Labeyrie
(1995), we first follow his derivation to the point where he calculates
the signaltonoise ratio.
The different photon distribution from the star and the planet defines the
signaltonoise ratio SNR, and according to the central limit theorem:
 
(1) 
where P_{*}(k_{*}) and P_{0}(k_{0}) are the probabilities to detect k_{*} and
k_{0} photons originating respectively from the star and
from the planet, per pixel in a short exposure, and where n is the total
number of short exposures. Replacing the values of P_{*}(0)
and P_{0}(0) in Eq. (1), we obtain:
 
(2) 
The calculation of signal to noise ratio given in Labeyrie 1995 assumed
, which is unnecessarily restrictive, and
unrealistic in some of the cases of interest. As suggested by one of us
(RR), a more general analysis can be made under the assumption
that and . Equation (2) then becomes:
 
(3) 
if j is the number of pixels per speckles, thus:
 
(4) 
Where and are the number of photons per
speckle in a short exposure, respectively for the planet and the star.
The variables used in [Labeyrie 1995] were:
n'=, = and
where T is the total integration time, t the short exposure time,
R the star/planet brightness ratio, G the gain of the adaptive optics
or the ratio between the Airy peak and the halo of speckles, N_{*} the total
number of photons per second detected from the star.
It provides a new expression for the SNR:
The sampling j should be fine enough to exploit the darkest parts of the
dark speckles, for a given threshold of detection , linking the
performance of the adaptive optics (G) and the brightness ratio (R).
The intensity across a dark speckle may be coarsely modelled as a cosine
function of the position by:
 
(5) 
where I_{h} is the mean intensity and the speckle size.
The intersection between and the line gives s, the
size of the pixel over which the light is integrated. It limits the minimal
measurable intensity (with ).
A detailed calculation gives .
We can assume that .Indeed, if I_{0} is the intensity of the Airy peak,
==
We also assume in the following that a planet can be seen if its own
intensity is higher than .
Now we are able to calculate a value of j:
 
(6) 
Figure:
Shape of a dark speckle according to Eq. (6).
If I_{h} is the mean intensity, integration on a pixel of size s yields an
intensity .

Recent numerical simulations ([Boccaletti 1998]) have shown that R/G should not
exceed 10^{3} to retain a reasonable value of the sampling parameter j.
The recorded level is I_{h}, but with dark speckles this residual level
decreases to , where depends mainly on the adaptive
optics performance.
With the value of j obtained, the SNR expression becomes:
 
(7) 
Solving for R, leads to a thirddegree equation:
 
(8) 
The Cardan method gives a single positive solution from which we derive a
final expression for the integration time.
 
(9) 
With the following values : R=10^{9}; D=8m; G=10^{6}; SNR=5; m_{v}=2.5;
q=0.2; and t=20ms, we find 2.7 hours of
integration, i.e. more than the result given by Labeyrie.
Angel's discussion (1994) of the longexposure method leads him to the following
expression:
 
(10) 
where is the optimum short exposure time. Using
again the same values, Eq.(11) gives a limiting brightness ratio of
1.4.10^{8} and the typical brightness ratio of 10^{9} would be reached in 140 hours.
The longexposure approach would in principle be more sensitive if
the adaptive optics and shadow pattern compensation could be made extremely
good. It is however less sensitive with current levels of adaptive performance.
In fact, the difference between the darkspeckle and directlongexposure
methods is more
subtle, and both work in different regimes. A critical value of the photon
stellar flux (N_{*c}) can be easily calculated by equalizing Eq.(10) and
Eq.(11) for the same value of the total integration time (T). It leads to a
second degree equation in N_{*}, with a single positive root:
 
(11) 
Therefore, if the photon flux is above N_{*c}, the darkspeckle method is
more efficient than the long exposure and, below this limit the direct imaging
is better. N_{*c} strongly depends on the adaptive optics. Let us take
a numerical exemple. If the goal is to reach a 10^{9} brightness ratio with a
gain of 10^{6} and 20ms exposures, the photon flux must be higher than
1.27.10^{9} photons/s, which can be achieved with a large aperture and wide
bandwidth (9.3.10^{9}ph/s for m_{v}=0, D=2.4m, and 20% efficiency).
However, for a space telescope with adaptive optics, the speckle
lifetime is under control with values of the order of 1s for exemple.
Thus, N_{*c} is decreased to 2.54.10^{7} photons/s, which is less restrictive.
However, the telescope should not be so large as to provide a partially
resolved image of the star, since it would fillin the dark speckles.
Next: Dark speckle lifetime
Up: Preliminary results of darkspeckle
Previous: Introduction
6/10/1998