2 Principle and specific features of the concept

Basically our coronagraph is a Michelson interferometer modified by inserting in one arm an achromatic -dephasing and a pupil -rotation in its plane. Both effects are obtained by replacing in one arm the flat mirror by a cat's eye. The coronagraph is placed in a collimated beam from the telescope and the beams from the two arms are recombined at a common pupil plane where a focusing element forms an image on a camera. The cat's eye is designed so as to match the pupil planes for each of the two arms. When the source is point-like and set on-axis, our coronagraph delivers a point spread function (PSF in the following) whose maximal intensity is weighted at zero when the Optical Path Difference (OPD in the following) is made zero, so that image plane is then utterly dark. This nulling would affect the same companion, but because of the pupil rotation the light from the companion is unaffected by the destructive interference process. In that case, the off-axis source yields two twin-images (half power each) in the focal plane while the image of the "blinding'' central on-axis source has been removed thanks to the nulling process. In Fig. 1, the generic set-up is schematically described and in Fig. 2 the nulling process is described in terms of wavefronts and complex amplitude of the fields. Actually a real set-up would use instead separated beams in each arm with symmetrical double pass of the beamsplitter.

Achromaticity of the dephasing results from the crossing of an additional focus. This property is described in various references (Born & Wolf [1970]; Gouy [1890]; Boyd [1980]; Baudoz et al. [1997]).

A point to stress is that the interferometric process does not result in a "dark zone'' at the center of image plane, but rather suppresses in the whole image plane all light coming from an on-axis unresolved source. In other words, the system provides the extinction of the central source contribution by weighting it down to zero when OPD is made zero.

Achromaticity is a specific feature of our coronagraph and makes it quite flexible regarding the choice of the working wavelength and of the spectral bandwidth as well. Also it makes it easy to work with two adjacent spectral bands, each providing images on opposite quadrants of a detector. An optimization of the optical coatings (beamsplitter and mirrors) is recommended when going from visible domain to infrared domain so as to keep performance at its best.

The other specific feature of our coronagraph relates to close-sensing capabilities since it allows sensing as close as a fraction (0.5 to 0.3) of the Airy radius, which is better than the diffraction limit set by the aperture and unreached by other coronagraphs. For those interesting features to be fully efficient, two conditions specific to our device must be satisfied:

- 1.
- the complex amplitude of the interfering fields must remain fully centro-symmetric up to the recombination stage (to keep effectively "flat'' the darkened flat-tint in the recombining pupil plane);
- 2.
- the OPD between the two arms of the interferometer must be kept as close to zero as possible (to achieve weighting of PSF at zero).

Few equations are given here since they will be used later in this
paper. In image planes, vector coordinates are referred to as
or
(meaning *x*/*F*, where *x* is the linear vector coordinate and *F*the focal length of the output focusing component), while in pupil planes
(aperture and exit pupil) vector coordinates are noted ,
meaning
length/.
Since we use circular apertures, polar coordinates will be
used:
in pupil planes and
in image planes.
Notation
stands for the Fourier Transform of the distribution *X*(FT in the following). The complex amplitude of the incoming field is noted
and the complex transmission of the collecting pupil is noted
up to the recombining pupil plane at exit. OPD between the
interfering fields is noted "*d*''. The beamsplitter reflexion factor and
transmission factor for energy respectively are *R* and *T* (typically 0.5each). The amplitude in the recombining pupil plane is:

Unless specified, we will work henceforth with OPD at zero.

The complex transmission in pupil planes is given by
where
describes the transmission of the perfect pupil
of diameter *D* (
is unity within the aperture and zero outside). The
phase term
takes into account all phase corrugations arising up to
the recombining pupil plane, which includes corrugations from permanent
aberrations met in each arm.

Let us consider a couple of point-like sources (binary star). One component is
on-axis (intensity
), the other (intensity )
is off-axis
by the angle ,
the intensity distribution is described by

(2) |

and the incoming field is expressed as:

(3) |

with = and = .

The intensity we have (Re means "real part''):

(4) |

where is the intensity Point Spread Function normalized to unity, given by the square modulus of and where denotes complex conjugate, notation < > means statistical average.

The centro-symmetry condition has totally removed the contribution of the on-axis component. We find two twin-images ( and ) of the off-axis source (displayed symetrically with respect to the origin of coordinates) and a mixed term containing the product of complex amplitude distributions, one for each interferometer arm.

As soon as the off-axis angle is large enough, the amplitude distributions do not overlap and the mixed term cancels. This fact raises the question: how can we decide whether the separation is large enough to cancel the mixed term? In other words: what is the response of the coronagraph versus field coordinate?

This response, the extinction profile versus field coordinate is obtained by
integrating over the image plane the energy from an off-axis running point-like
source. We introduce
,
which is the energy collected
when the coronagraphic effect does not apply. Then, the expression of the
integrated intensity normalised to unity at origin is a circular distribution
whose radial profile is given by:

where

This extinction profile , that we consider as an "extinction lobe as seen from the sky'', is narrower than the Airy lobe of the telescope, from here originates the close-sensing capabilities of our coronagraph, thus enabling it to go beyond the diffraction limit. Let us note that the response remains practically stable once the companion stands outsides the "sky hole''.

Even in an ideal situation, limitations apply to the extinction efficiency:

- 1.
- The light from a source partially resolved by the telescope is not utterly removed from the image plane, because the extinction profile is zero only at the central point of the field (part of the source covers a non-utterly-darkened field area);
- 2.
- When observing from the ground, the radiation travelling along the axis is spread by atmospheric differential dispersion and thus prevents a point-like source to be seen as such (part of the radiation seems to come off-axis). Let us note that this limitation affects any coronagraphic device.

For a partially resolved source, described here by
we find the residual energy by integrating
(Eq. 5) up to the angular radius
of the source.
Setting
the expression of the residual integrated energy in
image plane is:

(6) |

Figure 4 shows the behaviour of

Figure 4:
Integrated residual energy (normalised on the available energy) versus the angular diameter of the partially resolved source (expressed in fraction of
) |

Because of the chromatism of the air optical index, the image of a point-like
source is slightly dispersed. Therefore, working with a finite bandwidth, the
extinction can be complete only for the central wavelength while the other
wavelengths yield point-like sources apparently off-axis. Thus we find a finite
residual energy in the image plane, according to the field response of the
coronagraph given in Eq. (5). This effect depends on the zenithal distance *z*of the source, the working central wavelength
(or the wavenumber
)
and the spectral bandwidth
(or
). Let us consider a point-like source at a true zenithal
distance *z*, atmospheric refraction shifts the image by an angle d*z* whose
expression is:

(7) |

Assuming that the source is correctly set on-axis for , its image at will be shifted off-axis by the differential angle , and some unwanted energy appears in the image plane, according to the spatial response of the coronagraph. We define a residual energy factor

(8) |

Again we have assumed a uniform spectral distribution for the source. From Allen ([1973]) we have where

(9) |

Figure 5 shows curves of the residual energy factor

This effect becomes more severe when the wavelength decreases, when the
bandwidth increases and when the aperture is enlarged. Note that this
energy appears spread along the direction of dispersion only. This effect is
deterministic and can be removed by observing a suitable reference star.
Another way to remove this effect is rely on an Atmospheric Dispersion
Compensator (using Risley prisms) in order to balance the atmospheric
dispersion at a given *z* (Breckinridge et al. [1979]). It must be noted that
this kind of limitation applies to any ground-based coronagraph.

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