1 Introduction

One of the most ambitious objectives of present astronomy is the direct detection of an Earth-like planet orbiting a nearby star, and the spectral analysis of its atmosphere to search for life. At first sight, the required angular resolution is not that great: seen at 10 parsec, the angular separation between Earth and Sun is 0.1 arcsec. There are about 100 to 200 target stars (F, G, K, M) within that distance (TPF Science Working Group 1999). The main difficulties come from the brightness ratio between the planet and its parent star. This ratio, of the order of 10⁹ in the visible, presents its most favorable value of about 10⁶ in the infrared near $10~\mu$m. The infrared range of wavelength is also favorable for the detection of molecular absorption bands such as ${\rm H_2O}$, ${\rm CO_2}$ and ${\rm O_3}$ (Léger et al. 1999).

The received flux from a planet located at 3 or 4 Airy discs from the star is about 10³ to 10⁶ times fainter than the average level of star-diffracted light and the photon noise will prevent the direct observation of the planet in realistic observing times.

Several techniques have been recently studied to cancel as best possible the star diffracted light (Roddier & Roddier 1997; Gay & Rabbia 1996; Baudoz et al. 2000a,2000b; Abe et al. 2001; Rouan et al. 2000). They can be roughly classified in three groups of methods, one referring to Nulling interferometry (Bracewell 1978; Mennesson & Mariotti 1997; Angel & Woolf 1997), the second to direct imaging using coronagraphy and the last to apodization (Nisenson & Papaliolios 2001). Nulling interferometry has lead to the Darwin project (Leger 1993; Leger et al. 1996), while coronagraphic techniques are presently under study for TPF (TPF Science Working Group 1999; Guyon & Roddier 2000; Boccaletti et al. 2000; Pedretti et al. 2000; Aime et al. 2001).

Coronagraphy, invented by Lyot (1930; Lyot 1939) for the observation of the solar corona, is the first recognized technique to provide direct imaging capabilities of exoplanets (Bonneau et al. 1975; Malbet 1996; Watson et al. 1991). The technique has been recently improved by the use of a $\pi$ Phase Mask (PM) instead of the Lyot's opaque mask. (Roddier & Roddier 1997; Guyon et al. 1999). To obtain the required reduction factors necessary for telluric planet detection, coronagraphy must be coupled with an apodization of the entrance aperture. In that case, reduction factors as large as 10⁷ have been reported for the monochromatic case (Guyon & Roddier 2000). This gain is obtained at the expense of a small loss of transmission.

Recently, Nisenson & Papaliolios (2001) proposed the alternative concept of Apodized Square Aperture (ASA), based on the elegant idea introduced by Watson et al. (1991) to take advantage of the rapid drop of the diffracted light along the diagonal direction for a square aperture. Provided that the adequate apodizer is realized, these authors have shown that an ASA can reach the required dynamic range for planet detection without coronagraphy. They describe transmission apertures with several shapes of apodizations (sonine, cosine, or squared cosine) for which the overall transmission factors are of the order of $15\%$.

The work proposed in this paper originates from the search of an apodization method based on an interference process which may be more effective than the classical one using amplitude transmission masks. Its principle and possible optical implementation are given in Sect. 2. The method makes it possible to obtain any power of cosine apodizations such as the ones considered for the ASA concept. For these apodizations, we give in Sect. 3 the analytical expressions for the PSF observed at the focus of the telescope.

The same kind of cosine apodization is analyzed for its use in stellar coronagraphy in Sect. 4: the cosine shape appears to be very efficient, provided that only a fraction of the cosine arch is used (larger for Lyot's mask than for R&R's mask). Section 5 examines the chromatic effets of the technique. Conclusion are given in Sect. 6. As we shall see, the proposed interferometric apodization technique appears to be a versatile solution.