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cmax = ±102 x 105Pa at the substrate surfaces near the junction with the outer cylinder.
The Fizeau HeNe interferometric tests of the mirrors under stress showed accurate hyperboloid figures. For each mirror M1 and M2, the autocollimation tests with respect to a sphere were carried out at each Kerber zone, i.e. at V3/2 of the clear aperture radii r1max and r3 max respectively; hence the source is moved ~13.3 mm towards the shell to pass from the Mi  to the M3interferogram (Fig. 6.20Right).
• Tulipform secondary mirror  Stress figuring: We have seen, in Sect. 6.6.5, that the active optics parabolization and hyperbolization of a convex mirror can be readily realized by use of a vase shell and stress figuring under uniform load. However, in the case of a convex mirror with a noticeable central hole  as for the present design  another alternative with a uniform load consists of using a variable thickness distribution (VTD) surrounding a rigid ring. Because this VTD is infinite
at the center and becomes zero at the mirror edge, we have called it a "tulip form" (cf. Sect. 2.1.2).
A holed tulip form secondary mirror was investigated and developed for the hyperbolization of the Minitrust secondary mirror. The mirror clear aperture area, defined by the radii rmin and rmax, is builtin at rmin into a thick ring whilst the free outer edge of radius rext is slightly larger than rmax. A waterproof paste at the free external edge allows application of the bending load q by partial vacuum on the rear area of the VTD (see Fig. 3.11 in Chap. 3). The hyperbolization of the M2 mirror is achieved by elastic relaxation after spherical and stress figuring where the axial reaction is absorbed by the thick ring at r ~ rmin. The radial shearing force corresponding to the uniform load q is
Assuming that the inner thick ring is strictly undeformable, the builtin condition dzFlex/dr = 0 at the inner clear aperture rmin entails, from the coaddition law zOpt = zSphe + zFlex, that the slopes for the figuring sphere zSphe and optical surface zOpt are identical at this radius,
This fully determines the radius of curvature RSphe of the figuring tool and the elastic deformation to generate zFlex.
Since the mirror is convex and the VTD decreases from center to edge, the middle surface of the substrate can be accurately assumed as slightly departing from a plane. Hence the elasticity theory of thin plates applies to the determination of the M2 VTD. The unknown rigidity D(r) = Et3(r)/[12(1  v2)] is determined by integration of the derivative equation where z = zFlex. Using a Schott Zerodur vitroceram (see E, v in Table 1.10) and a load q = 0.8 x 105Pa, the numerical integration is carried out from rmin toward increasing radii with a small increment dr, starting with a provisional thickness value t(rmin). The starting thickness is modified and the integration process repeated up to obtaining t(rmax) = 0. The final VTD is determined by use of a small enough increment dr providing nonsignificant change in the integration result (Table 6.13).
From the result of iterations with the thin plate theory, the sphere and the flexure write, in [mm],
Table 6.13 Axial thickness distribution {tn} for the hyperbolization of MINITRUST tulip form M2 mirror f/5.4. Mirror clear aperture radii rmin = 50 mm and rmax = 100 mm. rext = 103 mm. Load q =—80kPa. ROp,t = 1,091.5mm, RSphe = 1,096.0mm, RPiex = 266,812mm, <R>~ [Units: mm]
r 30+ 50 50+ 60 70 80 90 95 100 103* tz 32.00 32.27 14.300 9.962 7.083 4.878 2.988 2.062 1.042 0.350
* Avoiding zero for a practical realization, this thickness was actually set from the tangent at t (r max).
ZSphe = 0.456211 103 r2 + 0.09495 109 r4 + 0.0395 1015 r6 + 0.205 1022 r8
zFlex = 0.001874 103 r2  0.37538 109 r4 + 0.1555 1015 r6  0.205 1 022 r8
ZSum = 0.458085 103 r2  0.28043 109 r4 + 0.1950 1015 r6 + 0.000 1 022 r8
zopt = 0.458085 103 r2  0.28043 109 r4 + 0.1950 1015 r6 + 0.000 10+00 r8
from where we satisfy the active optics coaddition law zSphe + zFlex = zOpt. During stress figuring the maximum radial stresses arise at r = 70 mm with values or = ±64 x 105 Pa at the substrate surfaces. Three substrate samples were built whose rear profiles were obtained by computer control diamond turning (Fig. 6.21).
The tulipform design of M2 provides a very lightweight mirror. Although more difficult to elaborate accurately, this design avoids the outer cylinder weight of a vaseform geometry. The quasiconical shape in the perimeter region shows high resistance to vibrations. Holed tulipform mirrors supported in their inner region are interesting lightweight configurations for space telescopes [21].
• Telescope optical tests in the laboratory: Two samples of the telescope optics were built for Minitrust1 and 2. The optics schematic including the shape of the two substrates for the three mirrors, and the baffles, is shown by Fig. 6.22.
The tube of the first prototype telescope was designed as a classical Serrurier truss with a central square frame which allows maintaining the mirrors coaxially whatever the gravity orientation. Since the Mi and M3 mirrors are de facto perfectly coaxial as belonging to the double shell substrate, the only necessary alignments
were (i) the orientation set up of M1M3 with the telescope tube axis, and (ii) the centering and orientation set up of M2 with M1M3. These alignments were carried out by crosswire reticles and retroreflection of a HeNe laser beam at the M3 vertex.
The telescope optical tests were realized onaxis by autocollimation with a Fizeau interferometer  Minifiz from Phase Shift Technology  imaging a point source at the telescope focus. After passing a first time in the telescope, the output beam was reflected by a plane mirror and was passed a second time in the telescope (Fig. 6.23).
From data reductions of the wavefronts issued from a double pass through the telescope, the optimal in situ load for M1M3 was q = 0.794 x 105 Pa; the theoretical value was 0.8 x 105Pa. After a preliminary alignment sequence, the first doublepass interferogram displayed a dominating Coma 3 due to a residual decentering error of M2. In the final alignment phase, this aberration was reduced to a negligible value (Fig. 6.24).
The results from final alignments and data reductions gave the following peaktovalley (ptv) main residuals onto a HeNe wavefront coming from infinity, i.e. in a singlepass or direct star wavefront,
The overall sum, including all order aberrations, is 0.280 XHeNe ptv corresponding to 0.048 XHeNe rms for Minitrust1. This telescope presently stays at Loom. The second prototype telescope used the second mirror set and a mechanical mounting designed and built by IasFrascati [28].
The excellent imaging performances obtained with the active optics process of aspherization opens up the way to potential applications in 23 m class mirrors, for instance, for large widefield skysurvey telescopes.
6.6.8 Mirror Aspherizations of a Large ModifiedRumsey Telescope
From the optical and elasticity designs of telescopes and mirror substrates in the previous sections, all the geometries can be scaled up or down, thus straightforwardly obtaining the optical and elasticity design parameters of any larger or smaller instrument. Doing so, all the mirror individual and resulting telescope focal ratios remain unchanged, as are the maximum stresses ar,max ^ (r2/t2) q provided the load q is unchanged and the elastic constants (E, v) of the material are the same. However, a significant scale up in the size of a mirror requires introducing a somewhat larger safety factor to avoid a glass rupture from the active optics aspherization.
Passing from a 0.44m to a 2.2m clear aperture telescope (scale factor of 5) is done hereafter by decreasing the load to q = 0.5 x 105Pa instead of q = 0.8 x 105 Pa in Sect. 6.6.7. Hence, because of the load change, the results in the latter section cannot be scaled up for the double vase shell as the M1M3 mirror substrate. The following optical design has a smaller back focal distance. Next, the M1M3 substrate design provides decreased maximal stresses. A 2.2m modifiedRumsey telescope is proposed hereafter by Lemaitre, Ferrari, Viotti and La Padula as a "threereflection skysurvey"(Trss) telescope [79, 22].
• Optical design of a 2.2m modifiedRumsey telescope  TRSS proposal: A
ThreeReflection Sky Survey (Trss) groundbased telescope has been proposed for wide spectral range observations extending from the UV band to IR bands. As in the previous section, its design is a modifiedRumsey form. The elasticity continuity conditions for the slope and sag variations in the area of the intermediate ring linking M1 to M3 lead to cross optimizations between optical and elasticity designs. Setting the Trss input pupil at M2 provides an optimal balance of the M1 and M3 clear aperture areas (Table 6.14 and Fig. 6.25).
• Double vase shell primary and tertiary mirrors  In situ stressing: The elasticity design of a double vase shell as the common substrate of M1 and M3 mirrors is carried out by similar iteration processes to those in the previous section. This determines the common spherical surface for simultaneously hyperbolizing both mirrors. In order to reduce the stress level for large mirrors, the in situ stressing of a Schott Zerodur substrate is hereafter achieved with the reduced load q = 0.50 x 105 Pa (~ 0.5 atm). Hence, the substrate thickness distributions are thinner than those in Sect. 6.6.7 when scaled up by a factor of five, and the maximum stress level does not exceed cmax = ±8MPa at the surface of the double vase shell (Tables 6.15 and 6.16).
The active optics coaddition law associated to the above M1 normal thicknesses {tn} for a simply supported movable edge writes, in [mm], z1jSphe = 0.464455 104 r2 + 0.100192 1012 r4 + 0.4322 1021 r6 + 0.141 1029 r8
Z1,Flex = 0.006355 104 r2  0.149236 1012 r4  0.6528 1021 r6  0.141 1029 r8
z1jOpt = 0.470810 104 r2  0.049044 1012 r4  0.2206 1021 r6 + 0.000 10+00 r8
For mirror M3, the total radius of curvature RFlex of the flexure is deduced from the A2 terms of the sum zFlex + zRota of a perfectly builtin edge plus the tangential edge rotation of the intermediate ring assumed equal to that of mirror M1 at r = r3,max. We have seen that this latter flexure can be assimilated as a single term curvature mode [cf. (7.79)]. Hence, the coaddition law for the optical surface associated to the above thickness distribution {tn} is the sum of three series which write, in [mm],
i 
Surf. 
Ri 
ASi 
A4,i 

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