5 Discussion

The use of a square grid aperture was dictated by the geometry of the micro lenses used, which permitted to easily achieve pupil densification. Periodic arrays, with either square or hexagonal pitch, have interference functions similarly having a periodic array of peaks of uniform height. Non-periodic arrays, composed, for example, of concentric rings of apertures providing little or no redundancyhave a speckled halo of side-lobes in their interference function. The average level of this halo is N times fainter than the interference peak, assumed perfectly phased.

We found that acquiring the image requires a pointing accuracy better than , corresponding to the distance of the first dispersed side lobes. The wavefront propagating from the star must be parallel to the main axis because an angle of the wavefront with the axis produces an increase of the OPD among the different apertures. As a result the white light central peak is shifted outside the magnified diffraction function. Since the array behaves as a bi-dimensional grating one of the dispersed peak may be found in the centre of the diffraction function. This will still form an image but dispersed and so elongated in one direction (Fig. 6).

A disadvantage of Michelson with respect to Fizeau arrays is the limited field of view. This is noticeable for an object moving off axis on the sky: the diffraction envelope from the sub-pupils moves with a lower speed than the interference peak forming the image, the speed proportional to the densification factor (Labeyrie 1996). From this observation, we define the field of view for a densified pupil interferometer as the angular extent on the sky over which the white image of stars remains inside the diffraction envelope of the sub-pupils. Owing to the pupil densification, the field thus defined is typically much narrower than the Airy radius of the sub-apertures on the sky. Following this definition, the field of view for a densified pupil interferometer depends on the amount of densification achieved. Maximum densification is achieved when the sub-pupils become adjacent. When this occurs the first dispersed side lobes coincide with the first zero of the diffraction function. In the practical case of our array we did not reach the maximum pupil densification since the first side lobes are still visible. The distance between 2 apertures is 8 mm; in the image plane the angular distance between the central white light peak and the first dispersed peak is then $\alpha =\lambda /s$, giving $\alpha$ slightly larger than $18^{\prime \prime }$($\lambda$ and s as previously defined).

The basic gain of densified-pupil imaging, with respect to Fizeau, is that the concentration of energy from a halo of many dispersed peaks into a single white central peak intensifies the main image, thus increasing its ratio to photon-noise. A more accurate device would contain 2 micro lens arrays working in an afocal configuration to meet the requirements of geometrical optics as sketched in Fig. 1. In the current device, shown in Fig. 2, the diffracted Airy patterns from the sub-apertures have their feet which fall upon neighbouring micro-lenses and this produces "ghost images'' in the focal plane shown in Fig. 4. The shape of the high resolution peak is not affected but its intensity is thus slightly reduced. Other losses in the system are caused by the imperfect alignment of the diffracted sub-pupils with the micro-lenses. This effect is seen as an asymmetric intensification of the side dispersed peaks with respect to the central white light peak. Most important perhaps, the phasing of the micro-lenses is less than perfect. This affects the intensity of the interference peak and creates a speckled halo around it.