where the product V0A22 is positive.

This VTD configuration is displayed in Fig. 3.20-Right. The radial and tangential maximum stress or and ot must be lower than the ultimate stress cult of the material. The radial maximum stress or = 6Mr/t2, derived with the scaling thickness t0 and the definition of the rigidity, is ar = 3 ay° Sr with Sr = T22 cos2O, t02

where the dimensionless maximum stress Sr in (x, z) section, i.e. O = 0, appears to be identical to the thickness T22 (Fig. 3.21).

Fig. 3.21 VTD class: Thickness and stresses of Astm 3 flexure mode

Fig. 3.21 VTD class: Thickness and stresses of Astm 3 flexure mode

One may notice that VTD solutions for Astm 3 mode and Cv 1 mode type 1 (Sect. 2.1.2) are identical.

3.5.3 Hybrid Configurations

Similarly as for Sphe 3 and Coma 3 modes, hybrid configurations can be derived for an Astm 3 mode providing this deflection over the full mirror diameter. With a CTD as the central zone and a VTD for the outer zone a < r < b, we can link the two distributions at r = a, and for instance, assume that their thicknesses are locally equal by writing t(a) |CTD = t(a) |VTD. However, two features must be taken into account:

- the VTD T22 is of finite thickness at the center for the Astm 3 case,

- the external force applied to the edge of a VTD mirror is self-reacting.

From these properties, hybrid configurations will not provide interesting simplifications for generating Astm 3 deformation modes.

3.5.4 Balance with a Curvature Mode and Cylindric Deformations

The co-addition of an Astm3 mode z = A22 r2 cos29 with particular values of the curvature mode Cv 1 can generate cylindric surfaces z ^ x2 or z x y2 (level fringes in Fig. 3.22). This can be achieved as well with a CTD or a VTD but the boundaries are simplest with a VTD since the application of bending moments at the edge is not required.

Noticing that the Cv 1 mode generated by a uniform load in reaction to the edge leads to the VTD (2.16) which is fully identical to the VTD providing the Astm 3 mode, the superposition of these modes can apply. Therefore, we can generate cylindrical mirrors by deformation of a thickness profile T22 = T20 =(1 - r2/a2)1/3. Let us write

Fig. 3.22 Co-addition of Astm 3 mode with varying Cv 1 mode. The curvature modes are respectively Z20 = -A22r2, 0, + A22r2

the co-addition of Cv 1 with Astm 3 . If we consider curvature amplitudes A20 of the same sags as those of astigmatism amplitudes at 9 = 0 and n/2, i.e. ±A22, the cylindric deformations are represented by

Zcyi = A22 r2( ±1 + cos 29) with A2o = ±A22. The associated loads are derived from (2.16) and (3.57b). From the first,

and from the second, to to a


By equalizing and setting A20 = ±A22, we obtain the coupling relation of the net shearing force with the uniform load

1 - v a so that q and the perimeter external force

A22 EtQ 3(1 + v)a fully define each of the two cases providing cylindric deformations.

3.5.5 Sagittal and Tangential Ray Fans in Mirror Imaging

• Conjugation relations in the general case: Let us consider the general case of an astigmatic incident beam reflected by a bi-axisymmetric surface of curvatures 1 /Rs and 1/Rt in its sagittal and tangential sections respectively [4, 30a]. Let s and t be the distances of an astigmatic object from the surface vertex to the sagittal and tangential focii and, s' and t' the distances from the vertex to the conjugated focal images. The relations of conjugate imaging are

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