5 Discussion

In this paper we have developed a theoretical study for coronagraphy of circular apertures with entrance pupil apodization, following the previous study for rectangular apertures (Aime et al. 2002). All the results for the rectangular configuration find their analog with circular apertures. The formalism is more complex, but the physical contents are similar. We have shown that the natural apodization functions for Lyot's coronagraphy and Roddier & Roddier's coronagraphy are the circular prolate spheroidal functions. We have proven that a total extinction of a monochromatic unresolved on-axis star can be obtained theoretically using a circular aperture apodized by circular prolate functions and Roddier & Roddier's phase mask coronagraphy.

The study we have made is for a monochromatic wave front. R&R's coronagraphy suffers from a double chromatism problem, the size chromatism and the phase shift chromatism compared to other phase shifting techniques, such as the wavelength-independent CIA (Gay & Rabbia 1996; Baudoz et al. 2000a; Baudoz et al. 2000b) or the less sensitive 4QC technique (Rouan et al. 2000), or the achromatic PKC (Abe et al. 2001, 2003). However, these techniques have other drawbacks: the CIA does not produces images, and the 4QC/PKC has a lower efficiency along the axis.

For Lyot coronagraphy, only a size chromatism exits: the mask size is adapted to the wavelength $\lambda_0$. At the wavelength $\lambda$, the diffraction pattern is not of the same size and the mask size is not correct. Equation (6) can be re-written, with the mask size:

$$a=a_0 \frac{\lambda_0}{\lambda},$$ (32)

where a₀ is the mask size at the wavelength $\lambda_0$. The kernel equation Eq. (8) may then be used for the computation. If the bandpass is not to large, the chromatic effect for Lyot is light. Numerical simulations of these chromatic effects can be found in Soummer et al. (2002a). However, to overcome this problem for large bandpass, one can use an achromatization system that produces a magnification of the focal image proportional to $1/\lambda$ over the bandpass (Wynne 1979). Then the aperture image must undergo the same procedure (in the reverse sense) for the Lyot stop, using a second Wynne corrector with opposite effect. Wynne correctors have been used experimentally to extend the bandwidth of speckle interferometry (Wynne 1979; Roddier et al. 1980), or for the Dark Speckles coronagraphy experiment (Boccaletti et al. 1998). A known limitation of Wynne correctors is their very small field of view. This may be acceptable for high contrast imaging applications. A detailed study of the optical design including the two Wynne correctors remains necessary to assess the performances of the whole experiment, over a given bandwidth.

For the achromatized Lyot technique, only a partial extinction of the star can be obtained, but there also, prolate apodizations give optimal results. Both the star and the planet appear in the final image with the same Point Spread Function (PSF) that is optimal in terms of concentration of light. The overall effect of Lyot's coronagraphy with prolate apodization is simply to reduce the magnitude difference between the planet and its parent star. Note that the multiple stage Lyot coronagraph proposed for rectangular aperture, is then also possible for circular aperture.

The reduction factor obtained with the prolate apodization is good enough to consider Lyot's technique for high contrast imaging, mainly because of its simplicity and its lower dependence on the deviation from the perfect case: chromatism, star diameter, telescope jitter.

The comparisons we made between circular and square aperture with the prolate coronagraphy technique are in favor of the circular aperture. For a perfect prolate apodized coronagraph using the Roddier & Roddier phase mask technique, the circular aperture provides a slightly better instrumental throughput for a smaller mask size. For Lyot coronagraphy (PALC) the results appear even more favorable to the circular aperture that gives a lower integrated residual energy in the diffraction wings, especially when a strong apodization is used. For a given throughput, the square aperture may however permit a better planet detection in four quadrants outside its main axial diffraction patterns. Because of its cross-like dead zone, delicate and time consuming 45 degrees telescope rotation will be needed to investigate the whole field. A circular aperture with a slightly lower throughput will probably be able to detect directly the presence of a planet in a shorter time. Moreover, the comparison we have done assume telescopes of same surface. For practical reason, it is probable that a square telescope would be obtained by diaphragming a circular one, which diameter equals to the diagonal of the square. In this case the circular aperture would be much more efficient. This comparison between a square and circular aperture does not apply directly to rectangular apertures, detailed in a previous study (Aime et al. 2002), since rectangular apertures present some specific interesting advantages (high angular resolution in one direction, possible $X,\lambda$ implementation for example)

This paper intended to present a theoretical and analytical approach for coronagraphy without atmosphere, using apodized apertures. Several issues have not been tackled in this paper and remain to be studied numerically. In particular, numerical simulations will be needed to evaluate the sensitivity of these formal solutions to the physical parameters: wavelength dependence, wavefront errors, mask positioning errors, angular diameter of the star, etc. It will be also interesting to check if the technique can give valuable results from the ground using an apodized aperture with adaptive optics. All these points will be studied in a more technical future work, and some of them are already presented in Soummer et al. (2002a). A comparison with the other coronagraphic techniques is also necessary; a major point will be to include signal to noise ratio as the main comparison criterion.

Acknowledgements

The authors would like to thank André Ferrari for helpful discussions.