2 Principle

The principle of densified-pupil multi-aperture imaging interferometers, called hypertelescopes, was previously described (Labeyrie 1996; Gillet et al. 2001; Riaud et al. 2002).

Figure 1: Hypertelescope principle. Positive (left) and negative (right) pupil densification, using respectively a diverging and a converging lens array, followed in each case by a longer-focus array of converging lenses. A tilted incoming wavefront from an off-axis star is densified with two confocal lens arrays (ML1 and ML2), thus producing a densified wavefront which focuses to a single narrow interference peak within the broader diffraction pattern from each sub-pupil. It becomes stair-shaped since the slope of each wavefront segment is reduced by the pair of micro-lenses. The stair's average shape remains flat and parallel to the incoming wavefront since all pairs of micro-lenses introduce identical propagation delays. But each wavefront segment, arriving with a slope angle $\alpha$, emerges with an angle $\alpha *f1/f2$ where f1and f2 are the micro-lens's focal lengths of ML1 and ML2. The resulting wavefront is finally a stair-shaped wavefront with the same average slope as at the entrance of the densifier.

Let us consider light beams coming from N small mirrors arrayed periodically in two dimensions with period s. Beams from these mirrors are initially combined according to a Fizeau interferometer geometry. At the combined focus, the image has a central peak surrounded by many secondary dispersed peaks. Densifying the entrance pupil, shrinks and intensifies the combined image, thus attenuating or suppressing the secondary peaks. The densified pupil increases the limiting magnitude but limits the field of view to a non-aliased field of the interferometer.

The white central peak, in the densified-pupil image, appears only for stars within a small region of the sky. Following Gillet et al. (2001), we call ZOF (Zero Order Field) this narrow usable field and HOF (High Order Field) the peripheral sky field of size $\lambda/d$ where d is the size of one sub-aperture (1 mm):

$$% ZOF({\rm sky})=\frac{\lambda}{s}=1.22\frac{\lambda}{d\cdot\gamma_{\rm D}}$$ (1)

where $\gamma_{\rm D}$ the densification ratio. This densification ratio is defined by:

$$% \gamma_{{\rm D}}\equiv\frac{\left( \frac{B}{d}\right)_{i}}{... ...{{\rm o}}}=\frac{d_{\rm o}}{d_i}\ $\ if\ $\space B_i=B_{\rm o}$$ (2)

where d_i and $d_{\rm o}$ are the entrance mirror and exit sub-pupil diameters respectively, and B_i and $B_{\rm o}$ the entrance and exit baselines in the pupil densifier. In the configuration considered here (Fizeau interferometer), both baselines are equal. But in case of a Michelson interferometer, the entrance baseline B_i is different from the output baseline $B_{\rm o}$.

The number of resolution elements in the ZOF is given by:

$$% N_{{\rm resels}}=\left( B/(d.\gamma_{\rm D}) \right)^2$$ (3)

B is the baseline of the interferometer (here 10 cm).