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Hence we obtain lim{ lôïf} = + 2 <■- p2) + 3(1 - p')2 + -] i =

so the two types are asymptotically the same near the edge. For p = 0.85, the ratio in the above limit is ~ 1.05.

Since a Type 2 only requires a central force reacting at the edge, when superposing a Cv 1 mode with an Astm 3 mode the number of actuators is lower than for the case of a Type 1 which requires generating an edge-moment, a more difficult condition to achieve in practice. Hence, although theoretically less perfect than for a Type 1, the superposition of Cv 1 and Astm 3 modes with a Type 2 has been proposed by Hugot [24] for telescope integral field units (cf. Sect. 1.12.9). In this development, one may find a more appropriate VTD of the form T = a7%o T1 + (1 — a)T20 T2, where 0 < a < 1, which allows us to match the accuracy of both generated modes relative to the wavefront tolerances.

### 2.1.3 Optical Focal-Ratio Variation

From the three VTDs, we can determine the optical f-ratio variation, i.e. the zoomrange, generated by the Cv 1 deformation of the VCM. Assuming a flat mirror when in an unstressed state, let

be this f-ratio variation. After substitution, all three VTDs can be expressed by

For these distributions, the radial and tangential stresses are identical, arr = att. In practical applications, these stresses must be evidently lower than the tensile maximum stress OTmax of the mirror substrate. Therefore, the maximal value of the stresses orr or ott, derived fromEqs. (2.17a) and (2.17b), must satisfy

3 qE2

With the tulip-like VCMs, because of the point forces applied at the center, T20(0) ^ ^ in (2.19) and (2.21), and also for the stresses (2.24). In fact, the stem of the profile thickness is very narrow because of its logarithmic nature in ( — lnp2)1/3 that comes from the infinite pressure due to the central point-force. For practical applications it is always possible to limit the central thickness to a finite value. The stem truncation is done with respect to the Rayleigh quarter-wave criterion applied to the central area; the axial force is not applied on a point but on a small area, say of typical radius a/50.

With the cycloid-like VCM, |T20|max = ^20(0) = 1 from (2.16).

### 2.1.4 Buckling Instability

A self-buckling instability may happen during a curvature change. This is similar to the meniscus shell "jumping toy," in polymer material, which is manually brought, temporarily, to opposite curvature. Avoiding buckling instability requires taking into account the radial tension Nr existing at the middle surface and showing that the maximum compression value of Nr remains small compared to a critical value. This self-buckling instability is avoided by restricting curvatures to always having the same sign during zooming. Furthermore, all three VTDs T20 are decreasing to zero at the edge which also prevents from this instability.

### 2.2 Thin Plates and Large Deformation Theory - VTD

In the previous Section, the radial and tangential stresses in the middle surface of the plate have not been considered, so that the results are valid only if the sags a2/2R are small compared to the mean thickness < t > of the substrates. In order to design VCMs generating a large zoom range with the best accuracy, the analysis is deepened by taking into account the strain of the middle surface. As for constant thickness plates (cf. Timoshenko and Woinowsky-Krieger [58]) in the axisymmet-ric case, the displacement of a point of the middle surface can be resolved into two components: Assuming a plane middle surface before loading, let us denote z, u the axial and radial displacements (instead of using the notation w, u which should be more appropriate for a curved surface at rest). Then, considering the large deformation theory, the relative elongations, or strains, in the radial and tangential directions are defined by

where the second term of err takes into account the large deformation case, as can be compared with the low deformation strains in (1.109b). The corresponding radial and tangential tensile forces in the middle surface, Nr and Nt per unit length, are defined by

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