2 Principle and specific features of the concept

The principle has been described elsewhere (Gay & Rabbia [1996]; Gay et al. [1997]) and is recalled here for the reader's convenience.

Basically our coronagraph is a Michelson interferometer modified by inserting in one arm an achromatic $\pi$-dephasing and a pupil $\pi$-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.

Figure 1: Schematic of the Generic Set-up of our coronagraph

Figure 2: Left: collected wavefronts, one from the central source the other (tilted) from a companion. Center: wavefronts on the recombiner lens. Right: amplitudes and resulting intensity in image plane

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 $\alpha,\beta$or $\mu$ (meaning x/F, where x is the linear vector coordinate and Fthe focal length of the output focusing component), while in pupil planes (aperture and exit pupil) vector coordinates are noted $\xi$, meaning length/$\lambda$. Since we use circular apertures, polar coordinates will be used: $(q,\phi)$ in pupil planes and $(\rho,\theta)$ in image planes. Notation $\hat{X}$ stands for the Fourier Transform of the distribution X(FT in the following). The complex amplitude of the incoming field is noted $\psi(\xi)$ and the complex transmission of the collecting pupil is noted $P(\xi)$ 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:

 

$$Q_{\rm d}(\xi) =\sqrt{R.T}.\Biggl[~\psi(\xi).P(\xi)$$

$$\textrm{ \ \ \ \ \ }+\exp(i.\pi).\psi(-\xi).P(-\xi).\exp\left(i.2\pi.\frac {d}{\lambda}\right)\Biggr]\cdot$$ (1)

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

The complex transmission in pupil planes is given by $% \begin{array}[c]{c}% P(\xi)=\Pi(\frac{\xi}{D}).\exp(i.\Phi(\xi)) \end{array}$ where $\Pi(\frac{\xi}{D})$ describes the transmission of the perfect pupil of diameter D $\$($\Pi$ is unity within the aperture and zero outside). The phase term $\Phi(\xi)$ 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 $\Omega_{0}$), the other (intensity $\Omega$) is off-axis by the angle $\mu$, the intensity distribution is described by

$$\Omega(\alpha,\mu)=\Omega_{0}.\delta(\alpha)+\Omega.\delta(\alpha-\mu)$$ (2)

and the incoming field is expressed as:

$$\psi(\xi,\mu)=\omega_{0}+\omega.\exp(-i.2\pi.\xi.\mu))$$ (3)

with $<\vert\omega_{0}\vert^{2}>$ = $\Omega_{0}$ and $<\vert\omega\vert^{2}>$ = $\Omega$.
The intensity $I(\beta,\mu)$we have (Re means "real part''):

$$I(\beta,\mu) =R.T.\Omega.[~A(\beta-\mu)+~A(\beta+\mu)-$$

$$\qquad -2.% {\rm Re}(\hat{P}(\beta-\mu).\hat {P}^{\ast}(\beta+\mu))]$$ (4)

where $A(\beta)$ is the intensity Point Spread Function normalized to unity, given by the square modulus of $\hat{P}$ and where $^{\ast}$ denotes complex conjugate, notation < > means statistical average.

The centro-symmetry condition $P(\xi)=P(-\xi)$ has totally removed the contribution $\Omega_{0\textrm{ }}$ of the on-axis component. We find two twin-images ( $A(\beta-\mu)$ and $A(\beta+\mu)$) 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 $\mu$ 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 $w_{0}=4.R.T.\Omega.S$, 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:

$$w(\rho)=\frac{w_{0}}{2}.\left[~1-\frac{2.J_{1}(2.\pi.D.\rho)}{2.\pi.D.\rho})\right]$$ (5)

where J₁is the Bessel function of order 1 and D is the diameter of the telescope. Recalling that $\frac{2.J_{1}(2.\pi.D.\rho)}{2.\pi.D.\rho}$is the radial profile of the Airy amplitude distribution for a telescope of diameter 2.D, we see that the extinction profile corresponds to the diffraction limit of a telescope which would have twice the diameter of the one effectively used. In other words, a point-like source located as close as a fraction of the first dark ring of the effective angular Airy pattern can be considered as off-axis and thus can be detected. This is illustrated in Fig. 3, where for comparison the Airy energy profile for a telescope of diameter D also appears.

This extinction profile $w(\rho)$, 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''.

Figure 3: Radial profile of the response of our coronagraph (response normalized to w0) with respect to the sky coordinate, counted in Airy radius $1.22\frac{\lambda}{D}.$ The normalized Airy profileof the aperture is given for comparison. As close as a fraction of the first zero of the Airy profile of the aperture, the response of the coronagraph has reached its maximum value. At $\lambda=2.2~\mu$m with an aperture diameter of 3.6 m this correspond to close-sensing down to 0.07 arcsec. Closer sensing is possible depending on detection capabilities in terms of the Signal to Noise Ratio

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 $\Omega(\rho)=\Omega .\Pi(\frac{\rho}{\Theta})$ we find the residual energy by integrating $w(\rho)$ (Eq. 5) up to the angular radius $\ \Theta/2$ of the source. Setting $y=\pi.D.\Theta$ the expression of the residual integrated energy in image plane is:

$${w}(y)=\frac{{w}_{0}}{2}.\left[1-\frac{1-J_{0}(y)}{\frac{y^{2}}{4}}\right]\cdot$$ (6)

Figure 4 shows the behaviour of w versus the angular diameter $\Theta$ counted in Airy angular radius. For example, in an ideal situation, observing with a 2.4 meter telescope at $2.2~\mu$m a "large'' source with $\Theta=22$m arcsec (a tenth of the Airy radius), would lead to a residual integrated energy which amounts to 4.5 10^-3 of the total collected energy.

Figure 4: Integrated residual energy (normalised on the available energy) versus the angular diameter of the partially resolved source (expressed in fraction of $1.22\frac{\lambda}{D}$)

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 zof the source, the working central wavelength $\lambda_{0}$ (or the wavenumber $\sigma_{0}=\lambda_{0}^{-1}$) and the spectral bandwidth $\Delta\lambda$ (or $\Delta\sigma$). Let us consider a point-like source at a true zenithal distance z, atmospheric refraction shifts the image by an angle dz whose expression is:

$${\rm d}z(\sigma,z)=(n(\sigma)-1).\tan(z).$$ (7)

Assuming that the source is correctly set on-axis for $\sigma_{0}$, its image at $h=\sigma-\sigma_{0}$ will be shifted off-axis by the differential angle $\beta(h,z)={\rm d}z(\sigma,z)-{\rm d}z(\sigma_{0},z)$, and some unwanted energy appears in the image plane, according to the spatial response of the coronagraph. We define a residual energy factor g(h,z):

$$g(h,z)=\frac{1}{2}-\frac{1}{2}.\frac{2.J_{1}(2.\pi.D.\sigma_{0}.\beta(h,z))}{2.\pi.D.\sigma _{0}.\beta(h,z)}\cdot%$$ (8)

Again we have assumed a uniform spectral distribution for the source. From Allen ([1973]) we have $n(\sigma)=1+A+B.\sigma^{2}+....$ where A=3 10^-4and $\nolinebreak B=1.6$ 10^-6 $\mu {\rm m}^{2}$, so that we have $\beta (h,z)\approx2.B.h.\sigma_{0}$. The residual energy factor G at a given zis the integral of g(h,z) over the whole bandwidth. Using $y=\frac {h}{\Delta\sigma}$ and $% \begin{array}[c]{c}% m=4\pi.D.\sigma_{0}^{2}.B.\Delta\sigma.\tan(z). \end{array}$ we have:

$$G=\frac{1}{2}-\frac{1}{2}.\int\nolimits_{-\frac{1}{2}}^{+\frac{1}{2}}\frac{2.J_{1}(m.y)}{m.y}.{\rm d}y.$$ (9)

Figure 5 shows curves of the residual energy factor G versus z. They refer to observations at $\lambda _{0}=2.2$ $\mu$m with $\Delta \lambda =0.4$ $\mu$m with telescope apertures of 1.5 m, 3.6 m and 8 m respectively.

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.

Figure 5: Integrated residual energy G(z) in the image plane caused by atmospheric differential dispersion at zenithal distance z for apertures of diameter 1.5 m, 3.6 m and 8 m. The full bandwidth of K-band is used ( $\lambda _{0}=2.2$ $\mu$m, $\Delta \lambda =0.4$ $\mu$m)