In this subsection we denote as the generic name 
one of the
functions 
, 
or 
. The same is
for their Fourier transforms: 
. These functions are the
CC and the CS of a high-light level zero-mean speckle pattern.
Since is a slice of the triple correlation
of O(x), it is possible to take advantage of the
calculus of the bias terms made by Aime et al.
(1992) in the photodetected triple correlation. Equation (2.18)
of this last paper leads to the following expression for the
photodetected cross-correlation 
of the zero-mean
is a photon bias term whose expression is
where 
is the average number of photons per image,
is the correlation function of the zero-mean high-light
level speckle pattern (standing for 
, 
and
) and m is its mean. The bias terms are not as
simple as for the photodetected AC (Aime et al. 1992) where
it is just a Dirac delta function at the origin.
The photodetected CS 
is biased by frequency-dependent
terms
where W(u) is the power spectrum, Fourier transform of .
It is remarkable to notice that bias terms are real. The imaginary
part of the photodetected cross-spectrum is unbiased. Its expression
is
For a bright enough reference star, the detection at high light level
of the specklegrams S(x) allows to compute the high-light level
zero-mean cross-spectrum 
. We assume that
the specklegrams I(x) are detected in photon-counting
mode. We denote as 
the ratio between the
photodetected cross-spectrum of I(x) and the cross-spectrum
of S(x)
Even in the case of a well resolved double star where the
convolution relation may be applied,
is not a good estimator of the double star
cross-spectrum because of the complicated bias terms. Its real
and imaginary parts are
where NI is the average number of photons per image in the
specklegrams of I(x). Here again it appears that the imaginary
part of is unbiased. This may be interesting if
we remember that this imaginary part contains the information
on the relative position of the stars in O(x).
Figure 5: Simulation of photodetected CS for different number of
photons per image. The upper figures are for 2000 photons/frame,
the middle are for 40 photons/frame and the lower ones are for 15
photons/frame. The computation was made on two sets of 5000
images with the same parameters as in Fig. 3. The pictures on the
left are typical specklegrams. Curves are real (middle) and imaginary
(right) parts of the biased object's cross spectrum
estimated as indicated in the text. Notice that
even at the lowest light level (15 photons/frame) it is possible to
predict the relative position of the stars using the sign of the
slope at the origin 
of the imaginary part
In this subsection we suppose that both I(x) and S(x) are
photodetected. We denote as NS the average number of photons
per image in the specklegrams of S(x). We shall see that the
information on the relative position of the stars is still present.
This information is contained in the slope of the imaginary part
of 
at the origin (see Fig. 2 (click here)). For
a high-light level detection where is estimated as
written in Eq. (20 (click here)), the slope s, defined
in Eq. (7 (click here)), can be written as (after a few algebra)
where we use the fact that 
. The sign
of s is that of 
.
We denote as the slope at the origin of
defined as the ratio between the photodetected cross-spectra of
I(x) and S(x). The expression of is similar to the previous
equation
and from equation 25 (click here) expresses as
Taking the expressions given in Eq. (24 (click here)) for 
,
in the case of 
this relation may be approximated by
Figure 6: Simulation of 10000 photodetected images of a double star of
separation d=10 pixels and intensity ratio 
. The average
number of photons per image is 50. The parameters of the simulation are
2.60 m for the telescope diameter, r0=30 cm
and wavelength 
Å. 10000 images of a single
reference star have also been synthesized with the same
conditions. Curves on the left are the real part of the
photon-biased and unbiased CS 
of the
double star's images. Curves on the right are the biased
and unbiased object's
CC . The ratio of the two peaks is 0.90 for the
biased data and 0.73 for the unbiased ones
Figure 7: a) is the triple correlation 
of clipped photon counting specklegrams of a double star. It is a
plot in the 
plane in the case where 
,
and d are collinear. The double star has a separation
of 10 pixels and an intensity ratio . b) is a
schematic representation of a) where the relevant peaks are drawn
as filled circles with values of the TC indicated. Curves c),
d) and e) are the CC, the function 
and the
function 
. These curves correspond to slices of the TC
along the directions indicated by the vertical arrows.
As expected the CC does not show any asymmetry. The
function looks similar to the unclipped
CC with a slight asymmetry
between the peaks (no photon bias correction has been
applied here). The function presents four
peaks. The two external ones are due to photon bias. The
two central ones contain information about the relative
position. Their asymmetry is in
the opposite sense than those of the secondary peaks of
the CC
The signs of s and are the same because the denominator
of Eq. (31) is positive. The relative position of the stars can
then be retrieved in spite of the photon bias. Results of a
simulation are shown in Fig. 5 (click here).
The frequency-dependent bias terms in the expression of
can easily be removed by subtracting the
photodetected power spectrum 
, whose expression
is derived from Aime et al. (1992) and is valid in
the case where the high-light
level mean is zero
From Eq. (24 (click here)) it appears that
This bias is quite easy to remove when processing real data.
and are computed directly from the data,
then subtracted. The remaining bias is the constant 
and can be estimated beyond the cutoff frequency.
The efficiency of this bias subtraction is shown in Fig. 6. It is a simulation of 10000 photon-counting specklegrams (50 photons/image) of a double star with a separation of 10 pixels and an intensity ratio of 0.5. As expected, the major improvement of the bias subtraction is to restore the asymmetry of the cross-correlation's secondary two peaks, thus allowing a better diagnostic of the relative position of the two stars.
Some photon-counting detectors have centroiding electronics which compute in real time the photon coordinates and cannot distinguish between one photon and more photons which have come onto a given pixel during the integration time. Intensities on the specklegrams are then thresholded to "1'' and this is what we call "clipping''. Such images are then equal to their square and the CC is equal to the AC. The asymmetry is lost.
We propose computing alternative quantities to solve this problem.
The first one is:
If 
is small compared to the speckle size, but larger than
the "centreur hole'' size (Foy 1987), I(x) and 
are
correlated enough to provide with the same properties than
the CC.
The following function may also be computed but it requires the
knowledge of the star separation d:
Figure 8a-d): Simulation of clipped photon-counting specklegrams.
1000 frames have been simulated for a double star of separation 10
pixels and intensity ratio 0.5. 1000 frames of a point source have
also been simulated. Parameters of the simulation
are: telescope diameter: 2.60m, Fried parameter:
30cm, wavelength: 500 nm
and number of photons per frame: 200 (each frame is 
pixels). a) is the function of the double
star divided by those of the reference star. Curve b)
is a slice along the 
axis. c) and d) are the
same for the function . The two peaks (1) and (2)
give the information about the relative position of the stars
(brighter star on the left in this simulation). The other peaks
of 
are ghosts due to photon bias
Figure 9: Simulation of 100 specklegrams of a
double star with the same parameters than Fig. 3. The image size is
. Left: a typical specklegram. the white square on the left
figure demarcates a sub-image of 
centered on the photocenter
of the
speckle pattern. The full-width-middle height of the speckle pattern
shape is 
pixels. Right: a cut of the object's CC
after deconvolution by a reference star's CC
computed from 100 specklegrams. Dashed line: computation
on the whole images, full line: computation on the sub-images. In
both cases, the mean of the specklegrams has been estimated as the
average of the intensity on the image, then subtracted. The CC
computed on the whole images gives an intensity ratio 
.
For the sub-images, the statistical mean is almost constant:
the CC is almost unbiased and the ratio of the two secondary
peaks gives 
(actual value is 0.5)
These two functions correspond to slices of the triple correlation:
and 
. In order
to understand the behavior of these quantities, we have computed the
triple correlation of clipped
photon-counting specklegrams of a double star, for fully-developed
speckle patterns. In that case the complex amplitude at the focal
plane is a Gaussian random variable and analytical expressions
can be obtained for the clipped TC (Aristidi et al. 1995).
Figure 7 (click here) shows the TC and the functions
and for the Gaussian hypothesis. The
function has the same behavior than the unclipped
CC: two asymmetrical peaks giving the couple orientation. The
function is a bit more complicated. It should
present two asymmetrical peaks separated d (asymmetry is the
opposite of those of the CC) but there are also two "ghosts''
at spatial lags 
caused by photon bias. Simulations have
been
performed on clipped photon-counting specklegrams. The results,
presented in Fig. 8 agree with the analytical
model.