6 Conclusion

We have proposed in this paper a simple apodization technique for rectangular or square apertures that can be implemented using classical optical devices. It consists of splitting the focal image in two parts that are shifted and added coherently, so that maxima and minima of the oscillatory part of the diffraction pattern cancel each other. This is equivalent to the use of an aperture with a cosine apodization that can be obtained using a Michelson or a Mach-Zender interferometer operated with a small angle between the mirrors. By using several interferometers, any integer power of cosine functions can be obtained. The apodization is done in one direction (along a side of the rectangle). Two interferometers set in orthogonal directions are needed to apodize the rectangular aperture in the two directions.

The technique is then studied for application to the ASA concept recently proposed by Nisenson & Papaliolios (2001). Our approach allowed us to give a simple analytic expression for the cosine diffraction patterns considered by these authors. For a cosine to the power N apodization, the resulting focal plane amplitude is simply written as the sum of N+1weighted and shifted sine cardinal functions. The weights are given by Pascal's triangle. The shifts must be equal to 1/L (L being the length of the aperture in units of wavelenth) for a full cosine-arch apodized-aperture (maximum at the center and zero at the edges). In that case, very simple analytical expressions can be worked out. Making the shifts b times smaller, one obtains a partially apodized aperture (central part 1/b of the cosine-arch).

We have then analyzed the effects of apodizations on coronagraphy. Here also we have been able to derive analytic expressions for the residual amplitude left in an image of the aperture after the coronagraphic experiment. We found that the full apodized aperture (b=1) is not suitable for coronagraphy. On the other hand, very good results can be obtained with partial cosine apodizations. For R&R's phase mask technique a rejection of $1.1 \times10^{-6}$ is obtained using a simple cosine apodization with b=2.160, and a rejection of $3.3 \times10^{-8}$ is obtained using a cosine squared apodization with b=2.995.

The computation was made for a square aperture, but it could also apply to a rectangular aperture. For Lyot's technique, the cosine apodization is not very good, and a cosine squared is needed (10^-5 rejection with b=1.254).

It was beyond the scope of this paper to give the relative factors of merit of the ASA technique to the Lyot and R&R phase mask coronagraphic techniques; a major point will probably be the technical possibility of realization of each of them. The behavior depends also on the distance of the planet to the star, coronagraphic techniques being more effective for very close objects. If this distance is large, the disparities between the techniques is reduced. The proposed interferometric apodization technique appears to be versatile and can be adjusted to produce the partial or full cosine arch apodizations required by these techniques. The effect of its chromatism is not too severe for a reasonable bandwidth.