## Theory of Shells and Aspherization of Axisymmetric Mirrors Meniscus Vase and Closed Forms

6.1 Active Optics Aspherization of Fast f-Ratio Mirrors

We consider in this Chapter axisymmetric mirror substrates characterized by highly curved surfaces i.e. surfaces with a significant cambrure. The elasticity theory of thin plates basically assumes a plane middle surface. Thus, its validity field remains limited to mirrors of moderate f-ratio.

For mirrors faster than f/4 or f/3, the theory of shallow spherical shells, in French "coques surbaissees," allows taking into account the significant curvature of the middle surface and the "in-plane" radial and tangential tensions appearing on it during the flexure. This theory elaborated by E. Reissner [25,26] in the 1940s brings a remarkable accuracy in the elasticity analysis and is of utmost importance for the active aspherization of telescope mirrors in astronomy.

In its general form, this theory also concerns shallow shells loaded in a non-axisymmetric manner. We only consider hereafter the rotational symmetry case where a uniform load is applied all over the surface. In this Chapter, a free parameter is the determination of a slightly variable thickness distribution - Vdt - of a shell made of successive and continuous shell rings in order to generate from a sphere -the mirror surface - and a flexure the resulting shape required by the optics. The results presented hereafter show that the shallow shell theory has proved accurate for the flexure calculation of mirror f-ratios up to f/1.7 and is probably still valid for faster mirrors.

Various boundary conditions occur at the shell contour for the radial displacement and tangential rotation. With respect to these conditions, we consider several geometrical configurations: (i) a meniscus form with a simply supported and radially free edge, (ii) a vase form semi-built-in with an outer cylinder, and (iii) a closed form made of two shells built-in together via an outer cylinder.

### 6.2 Theory of Shallow Spherical Shells

Let us consider a shallow spherical shell and denote <R > the radius of curvature of its spherical middle surface. In a cylindrical coordinate system (z, r, 9), the shape z(r) of the middle surface can be represented by

G.R. Lemaitre, Astronomical Optics and Elasticity Theory, Astronomy 313

and Astrophysics Library, DOI 10.1007/978-3-540-68905-8.6, @ Springer-Verlag Berlin Heidelberg 2009

V if

V if

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