1 Introduction

The possibility of obtaining direct usable images at the combined focus of a diluted interferometric array was long overlooked. Theoretical work on densified pupil interferometry (Labeyrie 1996; Labeyrie 1999), numerical simulations of its imaging properties (Boccaletti et al. 2000) and a new algorithm for co-phasing a diluted array of several apertures (Pedretti & Labeyrie 1999; Pedretti 1999) now open a new evolutionary track towards large optical arrays in space and on ground.

Fizeau optical arrays, the equivalent of a telescope carrying a multi-hole aperture mask, have imaging properties similar to telescopes. If Rayleigh's criterion is met, there is a peaked spread function which becomes convolved with features of the observed object. Field is infinite in principle, although limited by telescope aberrations. If the holes in the mask are much smaller than their spacing however, the spread function has a vast halo of diffracted light surrounding its central interference peak. This removes most energy from the peak and creates a useless continuous level in the image. The ensuing image degradation would become disastrous in the giant systems, spreading across kilometers, considered for space interferometry with metre-sized mirrors as aperture elements

Figure 1: Pupil densification with micro lenses. The converging beams coming from widely separated sub-apertures (EA) are collimated by lens L1. An image of the entrance aperture is formed, by field lens FL, on the micro-lenses array ML1 having a shorter focal length than thesecond micro-lens array ML2 located downstream. The focal length ratio of ML2 and ML1 is the pupil densification factor. With suitable phasing adjustments L2 forms a directly exploitable image on the detector C (©PASP)

Michelson (Michelson & Pease 1921), in his 20-feet beam, avoided the halo problem by densifying the exit pupil with his four-mirror periscopic arrangement, i.e giving it a higher sub-pupil size/spacing ratio than in the entrance aperture. This caused the spread function to lose its field invariance: fringes were moving across the diffractive halo if a point source moved. The convolution thus no longer applied. Extended versions of Michelson's beam, using many apertures instead of two, were considered since the 1970's, but the loss of the convolution relation appeared to make them incapable of forming direct images (Labeyrie 1985; Beckers 1986). The point was formalized into a "golden rule of imaging interferometry'' (Traub 1986).

Figure 2: Miniature hyper-telescope. A Fizeau mask is placed in front of an afocal diffracting telescope. Its image in the exit pupil has narrow sub-pupils which diffract light so that facing elements of the micro-lens array located downstream are each filled by the corresponding diffracted lobe. The pupil-array distance matches the focal distance of the micro-lenses in the array, so that the diverging diffracted beams are approximately re-collimated, thus achieving the desired pupil densification

Only much later did it appear that the rule can be evaded (Labeyrie 1996), with considerable benefit in terms of future applications. It was shown that a densified pupil such as Michelson's arrangement, but incorporating many elements, can provide direct images if the sub-pupil centres are preserved in terms of their relative locations. The image formation may then be described as a "pseudo-convolution'' (Labeyrie 1996), where two functions:

• a field-invariant "interference function", which is the Fourier transform of the sub-pupil centres array;
• a broad "diffraction function" which is the Fourier transform of a single sub-pupil, move at different speed when a point source moves away from the central axis of the instrument. The interference function moves faster than the diffraction function, the speed depending on the densification factor. If the densification factor is large then the movement of the diffraction function with respect to the interference function, becomes negligible.

The diffractive envelope can then be considered a static windowing function, limiting the angular span of the interference function. Inside the envelope, an ordinary convolution of the object with the interference function, takes place.

Systems other than the Michelson beam can be used to densify the exit pupil. Labeyrie proposed the system shown in Fig. 1 (Labeyrie 1999) which utilises 2 micro lens arrays to re-image the exit pupils as nearly contiguous wavefront segments.

Here we show the first images obtained on the double star  Gem from a miniature hyper-telescope proving that snapshot, high dynamic range images are obtainable from a Michelson type array.