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Therefore with a flat or quasi-flat VCM at rest, i.e. Q0 ~ this criterion for the determination of T20 means that the mirror is a paraboloid when at f/Q, while maximum spherical aberration residuals of the optical surface occur with opposite signs at ~f/2Qmin and f/Qmin.

2.6.3 Achievement of Boundary Conditions

Considering the VTD class, the boundary condition at the VCM perimeter is a free supported edge for Type 1 and Type 2, and free edge for Type 3. From (2.34), no radial bending moment and no radial tension must be applied,

With metal mirrors, taking into account that only the axial reaction acts at the edge of Type 1 and Type 2, these conditions can be approximated by using a small cylindric collaret that links the VCM to an outer thick ring (Fig. 2.12). Convenient axial length and radial thickness of the collaret can be defined with respect to the thickness T20 at the edge proximity, for instance from the value of T(0.99).

Because of the instable equilibrium of the acting forces in Type 3, the boundaries only require the VCM to be radially maintained into an outer ring. Since there is no reaction exerted to the edge, a possible solution is to use a very thin linking plate at the edge proximity and compensate for the sag by axial displacement of the zone of central contact during the deformation. Although these boundaries are difficult to fulfill, the comparison of the dimensionless thicknesses displayed by Fig. 2.2 shows

Fig. 2.12 Holosteric solutions linking active substrate and outer ring. Left: VTDs Type 1 and 2: Boundaries achieved by a thin cylinder. Right: VTD Type 3: Boundaries achieved by a thin plane plate that Type 3 VCMs are much thinner towards the edge. Therefore compared to Type 1 and 2, Type 3 VCMs are less sensitive to surface deviations from the paraboloid shape and should theoretically provide largest zoom ranges.

2.6.4 Design and Results with VTD Type 1 - Cycloid-Like Form

Type 1 VCMs have been built for optical telescope array interferometers and Fourier transform spectrometers (Ferrari et al. [21]). The following results are obtained with the design parameters of Fig. 2.4 and a thickness distribution T20 determined for f/Q = f/3.33 allows optimizing the zoom range [f/^-f/2.5]. An air pressure load generates convex shapes. The substrate is a stainless steel Fe87Cr13 quenched at a Brinell hardness BH = 330. The integration of system (2.32) provides the dimen-sionless thickness T2o and then thickness t. A small positive lens-like thickness is added to t in order to compensate for the plastic deformation resulting from the pre-stressing. The resulting thickness t*(r) was executed by a numeric command lathe on the rear face of the substrate (Table 2.1).

The prestressing was carried out by slightly overpassing the maximum curvature of the zoom-range (cf. next Section). After prestressing and final plane resurfacing, the mean aspect-ratio of these VCMs may be approximated by < t > /D ~ 1/60. He-Ne interferometric tests were carried out by use of Fizeau lenses of various curvatures and an accurate pressure gauge of resolution 10-4 (Fig. 2.13).

Table 2.1 Thickness t * (r) of a Type 1 VCM before plane surfacing. Eso Vlti and Jussieu Lpma: Zoom range [ f/^-f/2.5 ]. At VCM edge, the radial thickness of the cylinder-collaret is 25 ^m. t* is with a 18 ^m extra-thickness at center which includes a 14 ^m positive lens-like shape for plasticity correction (see Sect. 2.7) (Loom)

r 0 1 2 3 4 5 6 7 7.9 8+ 12 t* 318 316 311 301 286 265 235 188 150 5,000 5,000

Fig. 2.13 Optical tests of a Type 1 VCM with respect to concave calibers. Fizeau interferograms for a clear aperture 2a = 5 mm (Loom)

Fig. 2.13 Optical tests of a Type 1 VCM with respect to concave calibers. Fizeau interferograms for a clear aperture 2a = 5 mm (Loom)

2.6.5 Design and Results with a VTD Type 2 - Tulip-Like Form

Type 2 VCMs have been built for particular applications where discrete curvatures are preferred to a continuous variation of the curvature (Lemaitre [37]). The central force is generated by a motorized lead screw. These actuators are less complex to use than air pressure units with controller but may provide some vibrations during the zooming. Tulip-like VCMs are useful for pre-positioning the first pupil of telescope arrays such as with the Auxiliary telescopes of VLTI. Figure 2.6 shows that, for a given zoom range, the compared VTDs from large and small deformation theories present less deviations than for Type 2. In the following example with a VCM in quenched stainless steel Fe87Cr13. A positive central force generated convex shapes. The thickness t was derived by solving T20 in system (2.32) for the design value f/Q = f/6 and a zoom-range [f/^-f/4.5]. Before optical surfacing, the theoretical thickness t(r) was increased by a constant extra-thickness and the rear side of the mirror executed by a numeric command lathe (Table 2.2).

Table 2.2 Thickness t*(r) ofaType 2 VCM before plane surfacing. Zoom range [ f/~-f/4.5]. At the VCM edge, the radial thickness of the cylinder-collaret is 25 ym. t* includes an extra-thickness of 5 ym (Loom)

r 0.1 1 2 3 4 5 6 7 7.9 8+ 12 t * 515 410 359 320 286 253 217 171 135 5,000 5,000

Fig. 2.14 Optical tests of a Type 2 VCM with respect to concave calibers. Zoom range f/4.5. He-Ne patterns for full aperture 2a = 16 mm. The central force is varied from 0 to 5.85 daN (Loom)

For this zoom-range, the plastic deformation remains negligible and does not requires a compensation of the thickness distribution. After prestressing and plane surfacing, interferometric tests were carried out by use of Fizeau lenses of discrete curvatures and a ball-screw actuator driven by an encoded motorized system (Fig. 2.14).

2.7 Plasticity and Hysteresis

Because of the very large zoom range developed with "cycloid" type VCMs for 8 m aperture recombined telescopes (Vlti), it was found necessary to take under consideration the plastic deformation as well as the hysteresis deformation loop of the metal substrate. While plasticity goes back to the dawn of time, the discovery of hysteresis is due to J.A. Ewing in the 1880s.

A compensation of the plastic deformation and a hysteresis loop model can be determined in order to increase (i) the geometrical accuracy of the optical curvatures and (ii) the resolution of the curvature control. The plastic deformation error is corrected in the mirror figuring process, while hysteresis errors are compensated by the closed-loop control system.

2.7.1 Stress-Strain Linearization and Plasticity Compensation

For metallic alloys, the Ewing-Muir process [16] allows one to extend the linear range of the stress-strain relation. This plastic tightening, in French raidissement plastique, applies to the VCM substrates in quenched Fe87 Cr13 alloy. The process consists of prestressing the substrate at op.s slightly higher than the tensile maximum stress atmax which itself must be lower than the stress of rupture oult. Figure 2.15 displays the process in the case of an elongated rod. After applying a pre-stressing at Op.s, the new length at rest becomes permanently increased, but for next loadings laying under op.s the stress-strain law has been extended while remaining linear.

The prestressing applied to Type 1 VCMs (design in Table 2.1) was typically Op.s = 70daN.mm-2 corresponding to a loading q = 8.25 daN.cm-2. After final polishing, the VCMs operate up to a tensile maximum stress atmax = 68.4 daN.mm-2 corresponding to a loading qmax = 8.05 daN.cm-2 for the maximum permissible curvature Cmax.

Denoting C0 the initial curvature of a never previously stressed VCM, the final curvature at rest after prestressing becomes C*. The curvature difference due to plasticity is

He-Ne interferograms in Fig. 2.16 display the VCM shapes during the prestress-ing cycle of a previously unstressed VCM. The optical figures are recorded with

a

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