Fig. 6.8 Left: Calculated vase shell. Right: Edge-modified vase shell nr i h.n+l i i

Fig. 6.8 Left: Calculated vase shell. Right: Edge-modified vase shell

Fig. 6.9 Flexure tests of plane samples bent by moments

-F F

The samples were made of quenched Fe87 Cr13 steel alloy well known for its stressstrain linearity. Their geometries were given the same central thickness t but various curvatures Rj at the junctions to rigid ends. The bending moments generated by forces F and —F acting at large distances from the junctions allowed determining the deviation angle a. From these measures, one shows that there exists only one RJ/t-value for which parameters t and a are in accordance with the results of the beam theory: bending moments applied at the beam ends, via square-corner junctions, provide a constant curvature flexure. For larger RJ/t-ratios, the a-value which gives the same flexure as that of the beam theory becomes smaller than the semiseparation a of the semi-infinite ends and conversely.

• Approached values of the maximal stresses: The maximum stress at the surface of the mirror can be accurately derived from finite element codes. However, a basic estimate of the maximum radial and tangential stresses, orr and ott, can be easily obtained at the center of the shell from Love-Kirchhoff hypotheses of the plane plate theory (cf. Sect. 1.13.6). In this simplified case, whatever the boundaries, we have and the coefficient a2 of the flexure, in (6.57), is a2 =

which then leads to

This maximum stress at the center of a plane plate provides approximate stress values for cases 1 and 2 in the above tables. Compared to the tensile rupture stress c103sec = 90MPa for Zerodur in Table 5.2 (cf. Sect. 5.2.5), the maximum stress obtained from Table 6.3 for cases 1 and 2 lead to a safety factor of at least ~9. The similar two cases in Table 6.4 lead to a safety factor of at least ~18.

For the third case of a built-in and movable edge, used to form a closed shell, the maximal surface stresses are at the edge; From the plane plate theory, one easily shows that these stresses are derived from orr(rN)/arr(0) = 2/(1 + v) which gives 1.61 for Schott Zerodur. For the edge one also substitutes thickness tN to t0 in (6.64). Hence, from case 3 in Table 6.3, we find a safety factor of at least ~ 8. For the same case in Table 6.4, this factor is at least ~13.

Although the bending stresses always dominate in the deformation of these shells, their exact values can be determined from finite element analysis.

6.6.3 Concave Paraboloid Mirrors with a Central Hole

Concave mirrors with a central hole require the introduction of some conditions for the active optics aspherization. We will see that the lack of stiffness caused by the removed material at the hole can be compensated by a local extra-thickness around the hole. In addition, boundary conditions at the hole edge must be found to allow the aspherization by a uniform load.

• Conditions for a holed mirror aspherizable by active optics: In most astronomical primary mirrors, the size of the hole is smaller than 1/6 or 1/5 of its clear aperture diameter. The high-order flexure terms, such as the r6 and r8 terms which provide the optical shape, have extremely small values in the region near the hole. Hence, the study of a holed geometry can be accurately done within the third-order theory of optics. For a mirror at f/2, the edge region is little larger than f/12 or f/10. From these features, use of the plane plate theory of constant thickness plate is appropriate.

Let a (instead of rN in the latter sections) be the outer clear aperture radius of the mirror. Considering a built-in edge at r = a, we want to obtain the same flexure as that of a plain mirror (see Case 3 in Sect. 1.13.11). Introducing a radius r = b somewhat larger than that of the hole radius r = c, we state that the flexure in the region corresponding to the optical clear aperture must be where the load q is negative for a partial vacuum (in situ aspherization) and positive for an air pressure (stress figuring).

Now if the region c < r < b around the hole is with the increased rigidity

and submitted to the same load q, the general form of a flexure satisfying Poisson's equation is

Z2 = -q- \ r4 + Cia2r2 + C2a2r2lnr + C3a4lnr + C41, c < r < b, (6.67) 64Dh where the five constants Ci and Dh are unknowns determined by the continuity and boundary conditions.

The radial shearing forces and radial bending moments are respectively

After substitutions of z1 and z2 and their associated derivatives, the results for the shearing forces are

At the junction of the two zones, the continuity condition for the load implies that Q2r(b) = Q1r(b). Therefore, this can be only achieved if

C2 = 0 , and if a circular force f = nq~ = 2 qc (6.70)

2nc 2

per unit length is exerted along the hole edge. The superposition of force f applied at r = c with the load q applied in the region c < r < a can be considered as providing the equivalence of a uniform load q applied all over the surface of a plain mirror (Fig. 6.10).

The unknown constant C4, which holds for the condition z1(b) = z2(b), is not involved in the determination of the three remaining unknowns C1, C3, and Dh/D1. These are solutions of the following conditions,

Fig. 6.10 Geometry and equivalent load configurations for a holed mirror aspherizable by active optics

dZ2 dr

After solving this system, and since th/t1 = (Dh/D1)1/3, we obtain t± = f + 2 [(1 + v)a2 - (3 + v)c2]c2 11/3 (6_2)

for a built-in holed mirror. We note here that there exists a solution th = t1 for v = 1/4 and c/a =0.620 which is completely useless in practice.

Referring to Case 2 in Sect. 1.13.11, where the mirror edge is simply supported, the associated flexure z1 is straightforwardly obtained by the substitution

1+v in (6.65). Hence, the same substitution in the above equation leads to the thickness ratio t± = {1 + 2_(3 + v)(a2 - c2)c2_11/3 (673)

t1 I + (1 - v) [(3 + v)a2 - ((1 + v)b2)](b2 - c2 )j (. )

for a simply supported holed mirror. For instance, for a = 1, b = 0.2, c = 0.1, and v = 1/4, (6.72) gives th/t1 = 1.238 whilst (6.73) gives th/t1 = 1.231, thus a quite similar result.

The small difference between these two results for the usual sizes of a central hole, say, c/a < 0.2 and b/a < 0.25, allows neglecting some of the above b2 and c2 terms compared to the a2 terms. Hence, (6.72) and (6.73) join together into the simplified form [17]

th ti

Whatever the supporting conditions at the mirror edge, this general thickness-ratio relation provides the complete geometry in the region of the hole for active optics aspherization. This relation directly applies for implementing a central hole in any of the three cases previously treated of a meniscus shell, a vase shell, and a closed shell. Therefore, a concluding result for the active optics aspherization of holed mirrors is as follows.

^ When implementing a small central hole of radius c in a meniscus, a vase or a closed shell, except for a local increase of the thickness in a narrow region c < r < b and use of a small circular force f at the hole edge for the continuity of the load q, the mirror geometry for r > b remains unchanged.

• Parabolization of a holed concave mirror by in situ stressing: The mirror geometry for the active optics aspherization of a paraboloid with a central hole is first determined for a plain mirror. In a second stage, an increased constant thickness in the region c < r < b surrounding the hole of radius c is determined from (6.74) provided the two conditions attached to it are satisfied. The geometry in the region r > b remains unchanged after the geometric modification for the hole implementation. This allows the determination of holed meniscus, vase or closed shell geometries without change of their associated boundaries at the edge r = a = rN or at the base of the outer cylinder N +1.

For instance, we consider hereafter the case of an f/1.75 holed vase shell mirror parabolized by in situ stressing. A prototype mirror of 186-mm clear aperture diameter semi-built-in at a = rN = 95 mm was made of Schott Zerodur vitroceram (cf. E and v in Table 1.10). 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 = 11, allows the iterative determination of the normal thickness distribution {tn} of a plain vase shell. All successive ring-shell elements are 9.5 mm width. Relations (6.74) provide the thickness-ratio th/t1 in the normal direction for the design of the final holed vase shell (Table 6.5).

In the final design, the outer cylinder of axial length tz,N+1 is modified into a more compact L-shaped ring providing an equivalent rigidity. From the third condition in (6.71), we have shown that no bending moment is necessary at the hole edge. After spherical figuring and polishing at the radius of curvature RSphe while the mirror was unstressed, the paraboloid is obtained by permanent loads q and f applied in situ. The uniform load q is obtained by partial vacuum inside the mirror whilst, from (6.70), the small ring force f = qc/2 at the hole edge provides the load continuity (Figs. 6.11 and 6.12).

A helium-neon point source located at the mirror focus allowed optical testing by double pass of a collimated beam reflected on a flat mirror. Interferometric analysis of this prototype mirror showed that the double-pass reflected wavefront errors from a perfect sphere were smaller than X/5 ptv.

• Parabolization of a holed concave mirror by stress figuring: From the theorem stated in Sect. 6.6.1, if the mirror is aspherized by stress relaxation after spherical figuring under stress, then the geometry is the same as that of in situ stressing of a spherical surface. Only the uniform load q and circle force f change to opposite signs but remain with same absolute values (Fig. 6.13).

During the figuring, this requires use of an outer reaction ring to absorb the pressure reaction of the load. Because this ring must be located near the edge of the optical surface this alternative presents some practical difficulties. These can be partly avoided by use of a closed shell made of two sealed vase shells in which air or liquid pressure applies the load q during the stress figuring.

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