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Equation of mirrors: zi = (1/2Ri)r2 + A2,i r4 + A4,,r6. Axial separations: AS;. * Conic constant from third order: Ki = 8R3A4; — 1 and K = — 1 for a paraboloid.

Equation of mirrors: zi = (1/2Ri)r2 + A2,i r4 + A4,,r6. Axial separations: AS;. * Conic constant from third order: Ki = 8R3A4; — 1 and K = — 1 for a paraboloid.

is achieved from the flexure of the same polished spherical surface for both mirrors. The optical design in Table 6.9, which results from cross optimizations between optics and elasticity, allows one to obtain this common sphere for a Schott Zerodur substrate (E = 90.2 GPa, v = 0.243).

First, the elasticity design of M1 is determined as a full aperture vase shell mirror. The analytical shell theory in Sect. 6.3.4 with N=10 shell elements, including a simply supported movable base of the outer cylinder N+1, allows the iterative determination of the normal thickness distribution {tn} of the vase shell with successive ring-shell elements of tn/N radial width (Table 6.10).

Associated to this {tn} distribution is the active optics co-addition law of the surfaces which writes, in [mm],

Zi,Sphe = 0.223988 10-3 r2 + 0.11238 10-10 r4 + 0.1127 10-17 r6 + 0.141 10-24 r8 Zi,Flex = 0.002461 10-3 r2 - 0.17591 10-10 r4 - 0.1649 10-17 r6 - 0.141 10-24 r8 Z1,sum = 0.226449 10-3 r2 - 0.63530 10-11 r4 - 0.5217 10-18 r6 + 0.123 10-50 r8 Z1,Opt = 0.226449 10-3 r2 - 0.63530 10-11 r4 - 0.5217 10-18 r6 + 0.000 10+00 r8

from where the dimensionless radius p = r/rN for which dzFlex/dr = 0 is po =1.202. The total flexure sag in the range p e [0; 1 ] is Az1jFlex = 77.73 ¡m.

In a second stage, we similarly calculate for M3 an isolated built-in vase shell as-pherized by the same load q. Because the conic constants are in the ratio k3/k1 = 4.7994, this shows that the mean value of {tn} for M3 is smaller than that of M1. Since the M3 mirror is designed with clear aperture radius r3 max = 90 mm, thus at f/6.1, the thin plate theory applies. Hence, the mean thickness ratio is < {tn}M3/{tn}M1 > = (k1/k3)1/3 = 0.5928, which determines the mean thickness of M3 in accordance to the result from the shell analysis.

Now we depart from an M3 vase shell perfectly built-in by introducing an intermediate ring of thickness tir which links M3 to M1 and, thus, realizes a double vase shell. This ring extends in the narrow region from r3 max to r1min i.e. 90-110 mm. Its constant thickness to the normal must be determined such as the above flexural sag of M1 remains unchanged in this transformation; this value was set to tir = 30.220 mm. The transformation of a built-in condition into a continuity condition also implies a rotation of the M3 shell at r = r3,max. Hence, the co-addition law for this mirror must include a rotation component zRota which can be accurately

Table 6.10 Normal thickness distribution {tn} for the hyperbolization of MINITRUST vase shell M1 mirror f/2.5. Mirror clear aperture 440 mm. Substrate ext diam 480 mm. Load q=-80kPa. tn= 220mm, tx=30mm. Ropt= 2,208mm, Rsphe= 2,232.2mm, RFlex= 203,140mm, <R>=2,318mm. [Units: mm]

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