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1. Introduction

In the early 90's the European Southern Observatory issued a call for proposal for a variable curvature mirror system serving for beam management purposes in the interferometric mode of the Very Large Telescope. While the VLTI is tracking an astronomical object, the delay-lines provide for the equalization of the optical paths of individual telescopes by varying positions in the interferometic tunnel. The optical system of a delay-line, represented at Fig. 1 (click here), is a cat's eye mounted on a carriage and whose location in the tunnel is carefully monitored in order to compensate optical path differences (OPD) up to 120 m between different telescopes. A parabolic primary mirror produces an image of the source on the secondary and after another reflexion on the primary, the beams are directed to the beam combiner telescope through the interferometric tunnel.

  figure193
Figure 1: Optical layout of the VLT Interferometric mode and location of the VCM (M14) in the delay line system

The variable curvature mirror, which is the secondary, is an important component for the achievement of the coherent combined focus. In the VLTI a large field of view (FOV) will be made available and this mode of operation requires a high quality imaging of the pupils. As demonstrated hereafter, the VCM system is critical for the wide FOV operation of the interferometer.

We consider an interferometer with a cat's eye system using a small plane mirror located at its focus as a secondary.

For paraxial rays, a tex2html_wrap_inline1001 displacement of the delay-line in the tunnel compensates for a tex2html_wrap_inline1003 optical path difference, but for off-axis rays (or "field rays'') coming with an angle i this OPD is equal to tex2html_wrap_inline1007 (see Fig. 2 (click here)). Then with a small plane mirror, when the OPD is compensated for paraxial rays, it is not for the rest of the field. But a cat's eye system has the great advantage that rays coming from different parts of the field are physically separated on the secondary mirror. Hence, bending the small mirror with tex2html_wrap_inline1009 at the distance R.i/2 from the axis compensates for the "field OPD'' tex2html_wrap_inline1013 and eliminates the difference between paraxial and off-axis rays (see Fig. 2 (click here)). This bending corresponds physically to make spherical the small secondary mirror, with a radius of curvature tex2html_wrap_inline1015 where R is the radius of curvature of the cat's eye primary. Then the OPD is compensated for all the FOV and the "field fringes'' are eliminated at the combined interferometric focus of the telescopes array. As the radius of curvature tex2html_wrap_inline1019 of the cat's eye secondary depends of the compensated axial OPD tex2html_wrap_inline1003, it is varying during the movement of the delay line system. This possible compensation for the OPD of "field rays'' in a cat's eye interferometer has been first noticed by P. Connes (Connes 1956; Connes et al. 1975).

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Figure 2: Field compensation with a Cat's eye system

The spherical deformation corresponds also to the one which maintains the center of symmetry O of the cat's eye system at the same location during the movement (see Fig. 2 (click here)), the variable focal length then permits to keep a pupil image at a precise location while the delay-line carriage is moving. In order to exactly remap homothetically the output pupil configuration of the telescopes array at the image beamcombiner, an accurate control of the variable curvature, with an error lower than tex2html_wrap_inline1023, is necessary.

Considering the FOV planned for the VLTI (3.5 or 8 arcsec) and the OPD to compensate for, the cat's eye secondary curvature must be continuous variable within a range from (84 mm)-1 to (2800 mm)-1. This has been thoroughly analysed and described by O. von der Lühe , (1992a, b) and other authors (Beckers 1991; Jörck et al. 1992), including further phase recoveries with the additional VLTI sub-array, 8 m Unit Telescopes and Auxiliary Telescopes.

The first approach to variable curvature mirrors belongs to G. Lemaıtre and the idea he used was to actively deforme a small metallic mirror with an uniform loading or a central force (Lemaıtre 1976). A prototype was developed for the European Synchrotron Radiation Facility and served to increase the field of view of a Fourier transform interferometer. The mirror was 20 mm diameter large and the maximal flexure achieved was 31tex2html_wrap_inline1029m, corresponding to a focal ratio varying from tex2html_wrap_inline1031. The major difference, between this first successful attempt and the ESO prototype, is the amplitude of the maximal deformation applied to the mirror. Due to the range of curvature needed in the delay-line, the simple theory of elasticity was not convenient. We had to extend the theory of elasticity to the large deformations where the total amplitude of the flexure can be of the same magnitude or greater than that of the plate thickness.


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