cmax = ±102 x 105Pa at the substrate surfaces near the junction with the outer cylinder.

The Fizeau He-Ne 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 M3-interferogram (Fig. 6.20-Right).

• Tulip-form 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

Fig. 6.20 Left: Elasticity design and rear view of the double vase shell as MINITRUST M1-M3 mirror substrate. Right: Composite Fizeau He-Ne interferograms with respect to sphere references after spherical polishing without stress and in situ stressing (Loom)

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 hy-perbolization of the Minitrust secondary mirror. The mirror clear aperture area, defined by the radii rmin and rmax, is built-in 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 built-in condition dzFlex/dr = 0 at the inner clear aperture rmin entails, from the co-addition 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 non-significant 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 10-3 r2 + 0.09495 10-9 r4 + 0.0395 10-15 r6 + 0.205 10-22 r8

zFlex = 0.001874 10-3 r2 - 0.37538 10-9 r4 + 0.1555 10-15 r6 - 0.205 1 0-22 r8

ZSum = 0.458085 10-3 r2 - 0.28043 10-9 r4 + 0.1950 10-15 r6 + 0.000 1 0-22 r8

zopt = 0.458085 10-3 r2 - 0.28043 10-9 r4 + 0.1950 10-15 r6 + 0.000 10+00 r8

from where we satisfy the active optics co-addition 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 tulip-form design of M2 provides a very lightweight mirror. Although more difficult to elaborate accurately, this design avoids the outer cylinder weight of a vase-form geometry. The quasi-conical shape in the perimeter region shows high resistance to vibrations. Holed tulip-form 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 Minitrust-1 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

Fig. 6.21 Elasticity design and rear view of the tulip-form M2 mirror (Loom)
Fig. 6.22 MINITRUST on-axis beams, mirror substrate geometries, and baffles. The entrance pupil is on the secondary mirror

were (i) the orientation set up of M1-M3 with the telescope tube axis, and (ii) the centering and orientation set up of M2 with M1-M3. These alignments were carried out by cross-wire reticles and retroreflection of a He-Ne laser beam at the M3 vertex.

The telescope optical tests were realized on-axis 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 M1-M3 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 peak-to-valley (ptv) main residuals onto a He-Ne wavefront coming from infinity, i.e. in a single-pass or direct star wavefront,

Fig. 6.23 View of the modified-Rumsey telescope MINITRUST-1 under optical tests (Loom)
Fig. 6.24 MINITRUST-1 optical tests : He-Ne wavefronts from telescope double-pass (Loom)

The overall sum, including all order aberrations, is 0.280 XHe-Ne ptv corresponding to 0.048 XHe-Ne rms for Minitrust-1. This telescope presently stays at Loom. The second prototype telescope used the second mirror set and a mechanical mounting designed and built by Ias-Frascati [28].

The excellent imaging performances obtained with the active optics process of aspherization opens up the way to potential applications in 2-3 m class mirrors, for instance, for large wide-field sky-survey telescopes.

6.6.8 Mirror Aspherizations of a Large Modified-Rumsey 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 0.44-m to a 2.2-m 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 M1-M3 mirror substrate. The following optical design has a smaller back focal distance. Next, the M1-M3 substrate design provides decreased maximal stresses. A 2.2-m modified-Rumsey telescope is proposed hereafter by Lemaitre, Ferrari, Viotti and La Padula as a "three-reflection sky-survey"(Trss) telescope [7-9, 22].

• Optical design of a 2.2-m modified-Rumsey telescope - TRSS proposal: A

Three-Reflection Sky Survey (Trss) ground-based 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 modified-Rumsey 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 co-addition law associated to the above M1 normal thicknesses {tn} for a simply supported movable edge writes, in [mm], z1jSphe = 0.464455 10-4 r2 + 0.100192 10-12 r4 + 0.4322 10-21 r6 + 0.141 10-29 r8

Z1,Flex = 0.006355 10-4 r2 - 0.149236 10-12 r4 - 0.6528 10-21 r6 - 0.141 10-29 r8

z1jOpt = 0.470810 10-4 r2 - 0.049044 10-12 r4 - 0.2206 10-21 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 built-in 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 co-addition law for the optical surface associated to the above thickness distribution {tn} is the sum of three series which write, in [mm],

Table 6.14 Modified-Rumsey telescope design - Trss Proposal f/5-2° diagonal FOV - XX [3801,200 nm] - Efl=11 m. [Units: mm]






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