Resolved imaging of extra-solar planets with future 10-100 km optical interferometric arrays

ASTRONOMY & ASTROPHYSICS SEPTEMBER 1996, PAGE 517 SUPPLEMENT SERIES Astron. Astrophys. Suppl. Ser. 118, 517-524 (1996) Resolved imaging of extra-solar planets with future 10−100 km optical interferometric arrays A. Labeyrie Collège de France & Observatoire de Haute Provence (CNRS), F-04870 Saint Michel l’Observatoire, France E-mail Received January 3; accepted January 30, 1996 Abstract. — In the recent years, interferometric arrays of optical telescopes have reached sizes of the order of 100 m, but they have yet to produce high-resolution images. The analysis of image formation now shows that such images are theoretically obtainable directly in the recombined focal plane, if there are enough telescopes. Resolved images of extra-solar planets are in principle obtainable with 10 km ground-based arrays. Key words: techniques: interferometric planetary systems 1. Introduction The recent spectroscopic detection of periodic velocity changes on the star 51 Peg provides likely evidence for the presence of a planet (Mayor & Queloz 1995). Images showing such extra-solar planets as unresolved dots near their parent stars will perhaps be obtained within a decade, using coronographic telescopes in space or on Earth (Bonneau et al. 1975; Ken Knight 1977; Bracewell 1979; Burke 1986; Brown 1990; Angel 1994; Labeyrie 1995; Malbet et al. 1994). A subsequent step will be the formation of resolved images showing some detail of these planets. At optical wavelengths, this will require apertures in the size range from 1 to 100 km, to be achieved in the form of multi-element interferometric arrays. Ever since light from separate telescopes could be recombined and made to interfere (Labeyrie 1975), larger systems became used and provided high-resolution data (Mourard et al. 1994; Mariotti 1992), although not in the form of images. Interferometrically coupled telescopes became then considered for the major new projects in optical astronomy, such as ESO’s four 8 m telescopes and the pair of 10 m Keck telescopes. For “snapshot” imaging, i.e. the formation of a usable high-resolution image in a single exposure lasting minutes, plans are also made for a dedicated interferometric system of many 1.5 m telescopes, the Optical Very Large Array or OVLA (Labeyrie et al.1992). The initial design involves a 600 m ring with 27 telescopes, but later expansion to sizes beyond one or ten kilometers is considered. The corresponding angular resolution will increase from 10−4 to 10−5 arcsec. In space, 100 km ar- rays with superior sensitivity and 10−6 arcsec resolution are probably feasible. This article outlines a theory of image formation in multi-element optical arrays. It shows that resolved images of extra-solar planets are in principle obtainable from the ground, as well as in space. 2. Imaging properties of diluted optical arrays Fizeau’s 1868 proposal to install a mask with two holes, or sub-apertures, on top of a telescope allowed Michelson to resolve the satellites of Jupiter. For improved resolution, he subsequently increased the baseline span beyond the size of the largest telescope then available by installing on top of it, “in periscopic fashion”, a 20 feet (6.5 m) beam carrying four flat mirrors. Such optical configurations can be extrapolated towards large arrays of many elements, all of which may be considered as sub-apertures of a single giant optical aperture. The sub-apertures used by Michelson were not much larger than the size of atmospheric turbulence cells. The following discussion assumes sub-apertures of arbitrary size, but equipped with adaptive optics to provide diffraction-limited “sub-images” (i.e. images formed by each sub-aperture), all of which are recombined in a single image where interference occurs (see Fig. 1). For fully constructive interference, in the presence of instrumental and atmospheric phase shifts, each sub-image is also adaptively phased, for example by translating a mirror with a fast actuator. 518 A. Labeyrie: Feasibility of resolved images Fig. 1. Principle of Fizeau and Michelson configurations for a large multi-element interferometer (A,B). A is equivalent to Michelson’s periscopic train, while zoom lenses Z on each beam of the telescope-like array B provide adjustable conversion from Fizeau to Michelson geometries. The zoom lenses are assumed to preserve the image focus while changing the image scale. They can be adjusted from a neutral position, providing the 1x image magnification corresponding to the Fizeau geometry (C), towards increasingly demagnified images providing Michelson’s wider sub-pupils (D). The variable focal ratio in the sub-images leaves the array’s global focal ratio 1/α nearly invariant, thus not affecting the scale of the fine interference structure. When zooming however, the image’s Airy envelope varies in size since it is the diffraction pattern of the sub-apertures When utilized to construct such multi-aperture arrays, the “periscopic” principle of Michelson provides a somewhat un-natural imaging configuration since the multielement exit pupil is re-arranged with respect to the entrance pupil, the relative size of the sub-pupils being increased at the exit (see Fig. 1). One can also arrange to have the sub-pupil centers displaced with respect to their arrangement in the entrance pupil, however such arrays do not have the field-invariant interference function considered in this article. The following discussion is restricted to the class of Michelson arrays, hereafter called “conformal”, where the pattern of sub-pupil centers is identical in the entrance and exit pupils. With the increased relative size of the sub-pupils, the sub-images are shrunk with respect to the interference patterns which they form when becoming superposed at the common focus. When the source moves off-axis, the fringes move faster than the sub-images. Tallon & Tallon-Bosc (1992) have shown that the ensuing image degradation can be corrected post-detection. I show now that conformal Michelson arrays can provide directly usable high- resolution images at their focus, with a significant gain in signal/noise ratio. Figure 1 sketches two equivalent configurations for a large multi-element Michelson interferometer. Each is adjustable, in terms of the “pupil concentration coefficient”. γd = [do /Do ]/[di/Di ] where Di and di are the entrance pupil diameters, respectively for the array and the subapertures, while Do and do are the corresponding diameters in the exit pupil. In the Fizeau mode γd = 1. For an “extreme Michelson array” having its exit pupil completely filled according to a square or hexagonal grid pattern of sub-pupils, γd = N −1/2 Di di−1 . With a 10 km array of 100 elements, each 1.5 m in size, arranged on a square grid so as to provide a 10×10 filled array in the exit pupil, γd = 666. A comparable value is obtained with a ring-shaped array of 27 elements, other things equal. According to classical diffractive optics, the monochromatic imag (...truncated)