Fig. 8.19 Mirror thickness geometries T(p) or T(p,6) for atmospheric field stabilization mirrors. (A): Moderately lightweight axisymmetric mirror with a conical edge optimized for a minimal axisymmetric flexure with a ring support. (B): Increased lightweight three-fold symmetry mirror with linear prismatic edge of the actuators and decreased in the most distant regions from them. For instance, the dimensionless thickness distribution with a linear prismatic outer edge is shown in Fig. 8.19B for ¡5 = b/a = 0.582 and v = 1/4.

• Case of a mirror with central hole: Some future large ELTs will be designed with the primary mirror pupil imaged at or near one or two mirrors of the main optical train. For instance, the pupil of the 42-m primary mirror of the Eso E-Elt project - a five-mirror design - is imaged by the concave mirror M3 on the flat mirror M4 which should be an adaptive optics mirror. Close to mirror M4 is the flat mirror M5, 2.3 x 2.75m2 in size, planned as an atmospheric field stabilization mirror. Because of the central obstruction of the secondary mirror and of its proximity to M4, the central zone of M5 is of no use; hence a holed mirror design will be appropriate for M5.

Let r = c be the radius of the hole where c < b < a. The mirror geometry must be determined such as the flexure due to the inertial tip-tilt motions is reduced to a negligible value (diffraction limited criterion). A first approach to solve this problem is to consider the gravity case. In this case, we showed in Sect. 8.8.2 that the flexure of a plain mirror supported at its center is minimal when its shape is a paraboloid. Now, for a holed mirror supported on a ring of radius r = b, an efficient minimization of the flexure in gravity will be obtained when the flexure shape in the region c < r < a is purely quadratic and shows identical deflections at edges, z(a) = z(c). One may assume that, for extremely small deflections, its quadratic flexure shape z(r) does differ from that of an arc of a circle taken in a radial section.

^ The minimal flexure z(r) of a holed axisymmetric mirror supported in gravity g along a circle of radius b is obtained when the flexure shape is a part of a torus. This surface is generated by revolving a horizontal circle arc, of ends r = c and r = a, about the z-axis, along a circle of radius (a + c)/2 = b', where generally b' = b.

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