Simulated images from a diluted array

 We considered the case of densified-pupil interferometers ([Labeyrie 1996]), a new class of instruments where beams are combined like in Michelson's 20- and 50-feet systems, but with more than 2 beams. The densified-pupil concept evades the ``golden rule'', once thought inescapable, which stated that imaging interferometers had to be Fizeau-like  ([Labeyrie 1983,Traub 1986,Beckers 1986]). Fizeau-type interferometers having many apertures can produce snapshot images, but not if the aperture is highly diluted since most energy then goes in the numerous side-lobes rather than in the central interference peak. Densifying the exit pupil (Fig. 1), by extending to many apertures the kind of ``periscopic'' arrangement originally used by Michelson, can concentrate the energy in the interference peak, but appeared to destroy the isoplanaticity needed to observe resolved objects.

 

Figure 1: Image formation with a densified-pupil array. The array's exit pupil is the convolution of an array of Dirac peaks a-, with a sub-pupil. The Fourier transform of the peak array c-, called the interference function, here has a central peak surrounded by 54 lower peaks. The broad multiplicative envelope, called the diffraction function, shown in d- is the Fourier transform of the sub-pupil, made proportionally larger in the exit pupil than in the entrance aperture. The image of a non-resolved star is obtained in e- as the product of the patterns c- and d-. With a resolved star, only the interference function is becoming convolved with the star function if the pupil is highly densified. 

The solution recently found uses a restricted class of Michelson-type arrangements where:

• the pattern of sub-aperture centres is identical in the entrance and the exit pupils.
• all sub-pupils are of homogeneous size and shape.

It can provide full energy concentration in the combined high-resolution image, but the field and object size are restricted to about $N \times N$ resolved elements, if N is the number of apertures.

If the exit pupil is highly densified, as is necessarily the case in large space instruments such as the proposed Exo-Earth Discoverer ([Boccaletti et al. 1999]), the Terrestrial Planet Finder ([Angel & Woolf 1997,Woolf 1997]) and the Exo-Earth Imager ([Labeyrie 1998]), then a valid approximation for the monochromatic intensity distribution B(x, y), in the combined focal image was shown to be ([Labeyrie 1996]):  

$$B_{(x,y)} \approx {\gamma_{d}}^{-2} A_{(x,y)} \left [O_{(\frac{x}{\gamma_d}, \frac{y}{\gamma_d})}\otimes I_{(x, y)} \right],$$ (1)

where O_(x,y) is the intensity distribution of the object. I_(x,y) is the interference function of the interferometer, i.e. the Fourier transform of an array of Dirac peaks located at the sub-aperture centres, A_{(x, y)} the diffraction pattern from one sub-aperture and $\otimes$ is the convolution symbol. Finally, x and y are the coordinates in the focal plane. The quantity $\gamma_{d}=(d_o/D_o)/(d_i/D_i)$ describes the amount of pupil densification, D_i and d_i, being the pupil and sub-pupil sizes at the entrance, while D_o and d_o are the corresponding quantities at the exit pupil.

Eq. 1 shows that the interference function I_(x,y) (Fig 1.c) is convolved with the object's intensity distribution and then multiplied by the wide diffraction pattern of the sub-pupil (Fig. 1.d). For the large values of the pupil densification this latter pattern is considered insensitive to object position and simply behaves, where Eq. 1 is valid, as a windowing function which multiplies the convolution.

I_(x,y) generally has a central peak if the sub apertures are phased, a desirable situation which improves greatly the quality of the snapshot image B_(x,y) (without phasing, speckle interferometry can still produce images). In monochromatic light, the phasing is required modulo $2 \pi$ radians, but reaching the phasing condition at numerous wavelengths simultaneously requires all optical paths to be equal, as is the case for a perfect giant telescope. In our simulation we used 3 wavelengths to reduce the path differences from $10 \mu m$ to nearly zero. The example discussed here involves 27 identical apertures, arrayed along a circle, but the algorithm can be extended to any hierarchy of triplets arbitrarily arrayed.