where p = 1 and p0 are the clear aperture radius and the null powered radius, respectively.

The algebraic balance of the first derivative extremals is provided by p0 = V3/2, and define the location of the least confusion focus. This is corresponding to slopes dZ/dp = ±1 at p = 1/2 or 1.

The algebraic balance of the second derivative extremals is provided by po = \J3/2, and is used as mirror best shape for the field balance of aberrations in all-reflective systems (see Chaps. 4 and 5). This corresponds to curvatures d2Z/dp2 = ± 6 at p = 0 or 1.

These particular shapes are displayed by level fringes in Fig. 3.8.

Somewhat similarly, the elastic deformation of a plate in basic boundary cases, such as clamped edge or simply supported edge, naturally provides a flexure combining the two modes z20 and z40.

Chapters 5, 6, and 7 include the study of several configurations providing combined flexure z20 and z40 modes by using the CTD class, quasi-CTD class or the hybrid rigidity class. We will see in these chapters that active optics configurations can be derived in an easier way than for the case where the Sphe 3 mode, z40, is pure (z40 ^ r4) as treated herafter.

3.3.5 Examples of Application

• Elasto-optical design parameters: Using the Schwarzschild notation, the representation of the quadric surface of an axisymmetric mirror is

1 2 1 + k 4 (1 + k)2 6 z = 2Rr + H+r r +L16R5L r6 + - , where k is the conic constant. Limiting to the development of the two first terms of this series, the deviation to the sphere of curvature 1 /R, i.e. the aspheric term, is the part Kr4/8R3 of the second right-hand term. The asphericity to realize by flexure is z = A40 r4; then, by identification, k k f

where Q is the mirror f-ratio and 2a its clear aperture.

Considering the scaling of the thickness t0 given by (3.26b) for a CTD, which is also the same for the three previous VTD classes (3.29b), (3.30b), and (3.31b), as for the hybrid class (3.36), the substitution of A40, leads in all cases to the aspect ratio t0 a

This ratio provides the set up of the execution conditions: The scaling thickness (t0) of the mirror is fully defined from the mirror semi-aperture (a), the optics (Q, k), the material (v,E), and the load (q).

• Metal mirror designed with VTD 2 configuration: The design and construction of a mirror aspherized by a central force in reaction at its simply supported edge has been carried out. Because of the edge support condition without bending moment, a metal mirror has been preferred. The design of the tulip-shaped mirror includes, in one piece, the thickness profile simply supported to an outer ring via a cylindric collar of thin radial thickness [16, 17].

A back support linked to the ring allows one to generate a central force F = na2q. A view of the mirror, the resulting interferogram of the Sphe 3 deformation obtained in-situ, and the design parameters derived from (3.38) are displayed by Fig. 3.9.

• Vitroceram mirror designed with VTD 3 configuration: The design and construction of a convex mirror hyperbolized by a uniform load in reaction at its center

Fig. 3.10 (Left) Minitrust flat-field three-reflection anastigmat telescope. (Right) View of the holed secondary mirror hyperbolized by stress polishing (Loom)

has been carried out. This mirror is the secondary of the flat field three reflection telescope Minitrust [23].

This modified-Rumsey optical system (Fig. 3.10) is a very compact anastigmat -four-times shorter than a Schmidt system - and should provide new capabilities with large format detectors in wide field astronomy for ground-based observations as well as for sky-surveys in space. During the stress polishing, the free edge condition - edge without bending moment or axial force - has been ensured by a soft paste non-soluble into water. Applied between the mirror edge and its surrounding removable ring, the paste is sucked into the gap by the partial vacuum, thus ensuring airtightness. The central hole, required for this mirror, provides support of convenient lateral stability for the stress polishing. At the telescope, other advantages of a mirror supported at its central hole are a substantial gain in weight, and a low deformation to own weight in the gravity field [15].

Because of the relatively large diameter of the central hole, the inner built-in condition is not exactly satisfied; this gives rise to a slight rotation of the inner ring, thus generating a small curvature Cv 1 added to Sphe 3. In addition, a Sphe 5 mode necessary for this application was simultaneously obtained from a slight change of the VTD towards the edge. So, the determination of the real VTD has been carried

Fig. 3.11 (Left) Tulip-form secondary mirror and stress polishing loads. (Right) He-Ne interfero-gram with respect to a sphere when under stressing. Design parameters: Zerodur-special vitroce-ram from Schott, a = 103 mm, K= —3.917 (hyperboloid), O = —8/3, v = 0.240, E/q = 8.89 1 05, a/t0 = 18.31 (Minitrust [23]) (Loom)

Fig. 3.11 (Left) Tulip-form secondary mirror and stress polishing loads. (Right) He-Ne interfero-gram with respect to a sphere when under stressing. Design parameters: Zerodur-special vitroce-ram from Schott, a = 103 mm, K= —3.917 (hyperboloid), O = —8/3, v = 0.240, E/q = 8.89 1 05, a/t0 = 18.31 (Minitrust [23]) (Loom)

out by analytical integration with the three modes Cv 1, Sphe 3, and Sphe 5. However, the effects of Cv 1 and Sphe 5 are small, and the final parameters are close to those directly derived from (3.38) for Sphe 3 alone. A diagram of the mirror with its applied loads, the resulting interferogram of the deformation after elastic relaxation, and the design parameters are displayed by Fig. 3.11.

3.4 Active Optics and Third-Order Coma

The third-order coma, Coma 3 mode, is defined by n = 3 and m = 1. This asymmetric wavefront function, or mirror shape, is represented by

The solutions at of this mode are derived from (3.14) by stating that the load q is null. For the Coma 3 mode, the substitution of n, m leads to zero for the coefficient of the first term in r-1 ln r cos Q whatever A0 is, while the coefficient in r-1 cos Q of the second term is -4(3 + v)A0 which requires A0 = 0 only if q = 0. Therefore, after simplification, the Coma 3 configurations are all generated by the third and remaining term of (3.11a), (3.16), and (3.14). These terms are j

Vr = - 2A3i E [5 — v — (3+v) a] At r-ai cos Q, (3.40b)

j q = 2(3 + v)A3i E (a - 2) a At r-1-ai cos Q. (3.40c)

Since a 2-D prismatic load of the form q = q0r cos Q is to be rejected as extremely difficult to achieve practically, the roots to retain are those corresponding to a null uniform load q = 0 ^ ai = 2 and a2 = 0.

Substituting the at in (3.9), the distributions of flexural rigidity are contained in the representation

With the notation of (3.22) and (3.23), let us write the rigidity and a dimensionless thickness as

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