3 Departing from the ideal case

Since the effect of non-zero OPD is a non-zero weighting of the total energy
from the on-axis source, we readily have a specification of the tolerance
regarding OPD. Using *w* for the residual energy and *w*_{0} for the energy
available when no extinction occurs, we have, using Eq. (1) and the
centro-symmetry condition:

(10) |

Selecting a given maximum value

A value of *M*=10^{-4} maintains
<
.
This precision implies that servo-loop of OPD sould be performed. A rms
precision of
is currently achieved in the visible
(Reynaud & Delaire [1993]) and corresponds to better than
at
m. In addition, monitoring OPD variations during the recording of
images with suitable exposure-time allows a selection of "best exposures'' in
terms of lower values of *M*.

In this situation the condition no longer applies. Therefore complex amplitudes at the recombination step no longer mutually destroy (Eq. 1), and some energy from the central source is able to reach the image plane.

Departure related to modulus of the complex transmission relates to essentially obscuration and vignetting in the pupil plane (telescope spider for example), and it is ruled out by using a suitable mask in an intermediate pupil plane, restoring a centro-symmetric distribution for the modulus.

Departure related to phase of the complex transmission results from phase distorsions accumulated up to the recombination pupil plane. Phase distribution conveys both deterministic and stable distorsions (aberrations, misadjustment, etc.) and rapidly changing random distortions (atmospheric turbulence). It must be noted that phase distortions described by an even function do not prevent the removal of the on-axis source. This means for example that a defocus and aberrations such as spherical aberration, astigmatism and chromatic aberration still allow the extinction process to work fully. However, they tend to degrade images of the companion, which decreases detection capabilities. On the contrary, such "odd'' aberrations as coma and distorsion lead to an incomplete destructive interference for the on-axis source. This problem also concerns aberrations which could be different in each arm of the device. Indeed, aberrations and misadjustments can be reduced to an acceptable level by proper optical design and control. Thus we will limit our analysis to the case of turbulence, which for ground-based operation is much more severe than the residual instrumental aberrations. This analysis is based on the description of the phase distorsions over the aperture in terms of a weighted sum of Zernike polynomials (Born &Wolf [1970]; Noll [1976]; Roddier [1990]).

Any "odd'' distortion generates a spurious halo, a residual volcano-shaped light distribution around the center of image plane. Extension of this halo remains roughly the one of the focal patch. Note that the trouble is not the halo itself but fluctuations of its shape, whether they result from photon noise or from instabilities of the intensity distribution (speckle noise), this latter being largely dominant for ground-based observations. Noise is not uniformly distributed in the focal plane, since it depends on the local illumination level within the halo.

We consider the case of an on-axis point-like source, with intensity
leading to the incident field
with
=
.
Our starting point is the following Equation
describing the complex amplitude in recombining pupil plane with OPD set to
zero:

where describes the phase distortions in the recombination pupil plane. Those turbulence-induced distorsions are random and gaussian (central limit theorem). To evaluate the unwanted integrated residual energy we have to work with Thanks to the Gaussian statistics and using the phase structure function = (Roddier [1981]), the normalized integrated energy versus the parameter the parameter

(12) |

with as stated previously (Eq. 5).

This
expression shows that the use of real time compensation of wavefront
distorsions is mandatory, since even for *p*=1 we have a residual energy
larger than
0.3 *w*.

In the following analysis we consider the use of real-time wavefront compensation by adaptive optics for both the integrated residual energy and the residual energy distribution. The first provides only a useful numerical figure to summarize the performance relating to extinction and serves to compare various situations. The second is needed to evaluate the Signal to Noise Ratio (SNR) pertaining to the detection of a companion, since this SNR depends on the location of the companion in the image plane via the level of residual light at this location. So, we need to express analytically the distribution in the image plane and not only its integral. A convenient way to express analytically this distribution is to use the small phase distortions hypothesis, leading to . The validity of this assumption will be checked later, but initially, we can refer to the fact that the largest part of the phase distortion effects comes from the tip-tilt of the wavefront (Roddier [1990]). Since this defect is the first which is compensated for in any adaptive optics device, we believe our assumption is relevant.

We first describe the amplitude at the recombination step in the pupil plane, then we evaluate the integrated residual energy and finally we express the residual light distribution in the image plane.

To describe the corrected amplitude at the recombination pupil plane, our
starting point is Eq. (11) but since we consider real-time correction of
phase distorsions we now deal with a residual phase distribution
where
is the phase function and
represents the
applied corrections. Assuming that
is small leads to a new
expression for the amplitude :

(13) |

showing that only the odd part of is to be taken into account. The even part is automatically corrected "by design''. Introducing for odd residual phase distorsions over the pupil of diameter

(14) |

Thus, describing we still need to describe . It is usual to describe a phase distribution over a pupil of radius by a weighted sum of Zernike polynomials (Noll [1976]; Roddier [1990]; Conan [1995])

Here we use the notation introduced by Noll, where

the

(16) |

Using the pupil distribution giving when and else zero, the following orthogonality relation holds (where we use =

(17) |

For a given phase distribution , the weighting coefficients

(18) |

Applying corrections up to

(19) |

This expression is equivalent to a sum of the

Let us point out that restricting the summation to the odd part of
does not mean
that we restrict the sum to odd values of *j*, but rather that we restrict it
to odd values of *n*, the radial polynomial order (and therefore to odd values
of *k*). This, for example, automatically eliminates such defects as defocus,
3rd order astigmatism, 3rd order spherical, etc. as already mentioned.

If we want to take into account the case of incomplete correction we write:

(20) |

where we have introduced the weighting parameter

When correction is performed up to index *J*, the residual energy *w*_{J} in
image plane can also be found by integrating the intensity in pupil plane:

(21) |

Let us disregard for a while the automatic cancellation of even phase distortions, and let us introduce a coefficient (effectiveness of the correction) defined by . Taking into account the weighted Zernike sum and the orthogonality relation we have:

(22) |

For a correction device working up to

(23) |

This expression is quite similar to the one derived by Noll ([1976]) for the residual variance of the phase over the pupil, when correction is achieved up to (and including)

So, for a 150 cm telescope diameter and a *r*_{0} of 10 cm (leading to
53 cm at 2.2 m) we have
at 2.2 m,
which corresponds to a rejection by
magnitudes of the unwanted energy,
integrated over the whole image plane.

In the more realistic case of incompletely achieved corrections, we use values
of
taken from numerical simulations based on measured
performance of a real servo-correction device (Conan [1995]). Values for
are shown in Table 1 (for even orders *n*, efficiencies
are automatically set to unity, they are not shown
beyond *n*=7 that is *j*=36)

j |
1 | 2, 3 | 4, 5, 6 | 7-10 | 11-15 | 16-21 | 22-28 | 29-36 |

n |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

1 | 0.995 | 1 | 0.9 | 1 | 0.7 | 1 | 0.6 |

In this case, we define a figure *g*_{J}^{u} similar to *g*_{J} (u
for incomplete) and we have
In the same situation as above we have now an integrated rejection of
roughly 2.6 magnitudes.

It must be kept in mind that these numerical figures pertain to the total unwanted energy in the image plane, not to the one found at the location of an image of companion. So they must not be interpreted as the limiting magnitude difference that our coronagraph is able to work with. A more relevant evaluation of detection capabilities needs the energy distribution in the image plane to be expressed.

It is more convenient here to come back to the expression
where *p*_{j} traces the presence of the defect associated with *j* and where
we have an infinite sum.

The residual energy distribution
in the image plane (polar coordinates
and )
is given by the square
modulus of the FT of
and we have (using
for the FT of
*Z*):

(24) |

and the residual energy normalized on the collected energy is:

(25) |

which is more conveniently expressed in using

To evaluate we use for the expression (Born & Wolf [1970]):

(27) |

and for < according to Noll ([1976]) as above, we use:

(28) |

where it is apparent that only remain terms with .

Figure 7 shows examples of radial profiles of the residual halo when full correction is applied up to radial orders 3, 5, 7, 9 successively (the higher the order the fainter the halo). Taking into account corrections beyond order 9 would induce fainter and fainter contributions of the halo, raising it further and further from the center of the field.

This behaviour is interesting not only for reducing the level of unwanted energy but also for cleaning progressively from the center, which improves the close-sensing capabilities as early as the lower order is corrected.

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