The interest in stellar coronagraphy is relatively recent (Bonneau et al. 1975; Smith & Terrile 1984; Malbet 1996; Beuzit et al. 1997; Mouillet et al. 1997), and nowadays mainly stimulated by the exoplanets searches. The direct imaging of an exoplanet is the most exciting objective for high contrast imaging techniques: it will provide valuable information such as its mass (removing the ambiguity due to the radial velocities technique), its albedo and its atmosphere chemical composition if spectral information can be obtained.
Several other scientific objectives, with different instrumental requirements, may also benefit from the techniques developed for high contrast imaging. Among them, the circumstellar disks for a better understanding of their dynamic and the physics of the grains (Mouillet 1997), the environment of evolved stars which could also include old planets, the Active Galactic Nucleis for a better imaging of the central jets. Since the first indirect exoplanet detection (Mayor & Queloz 1995), several techniques have been proposed to reach high contrast imaging (Gay & Rabbia 1996; Roddier & Roddier 1997; Rouan et al. 2000; Abe et al. 2001).
In a previous paper (Aime et al. 2002), we analyzed the theoretical aspects of the association of coronagraphy (Lyot et al.) with entrance aperture apodization and derived the integral equation for total coronagraphic extinction of the star light. This integral equation depends on the shape of the apodized aperture.
For the rectangular aperture case we analyzed (Aime et al. 2001), the apodization solution is given by the linear prolate spheroidal functions discovered by Slepian & Pollak (1961). These functions have already found many applications in optics, as well described by Frieden (1971). For coronagraphy, the relevant particularity is that these functions are invariant to a finite Fourier transform.
We showed that a theoretical total extinction can be obtained with a perfect Roddier & Roddier coronagraph. For Lyot coronagraphy, we showed that though a total extinction is not possible, very interesting solutions can be obtained with the prolate apodized coronagraph: much smaller masks than for the classical technique are required, and very high dynamic may be obtained, depending on the mask size and apodization. Although less powerful than the phase mask technique, the Prolate Apodized Lyot Coronagraph (PALC) was found to remain interesting because it is easier to implement and much less sensitive to chromatic effects in a wide band experiment. The technique for a square aperture was also compared to the Apodized Square Aperture concept that considers apodization alone (Nisenson & Papaliolios 2001; Soummer et al. 2002b).
The present paper generalizes the study to the case of coronagraphy with an apodized circular aperture. In this case, the solution for the apodization function is given by the circular prolate spheroidal functions developed by Slepian (1964) and Heurtley (1964). The complexity of the formalism was the occasion for us to go much deeper into the theoretical aspects of the problem, to find new analytical simplifications and bring a lot of interesting new results.
The present study is restricted to a perfect circular aperture, operating in space, for a monochromatic unresolved on-axis star. However, the equations considering a polychromatic source are also given and discussed. This is an aspect in which the PALC may perform better than the phase mask technique. The purpose of this study is to settle the theoretical background for circular apertures and give the ideal performance attainable for such a coronagraph. A more technical study of a realistic instrument will be realized in a future work and compared to the perfect case.
Copyright ESO 2003