## Aspherization of Concave Spheroid Mirrors

The active optics aspherization of a concave spheroid mirror requires some conditions relative to the sign and amplitude of the consecutive polynomial terms which represent the mirror figure. In certain cases, it is clear that those conditions entail that the active optics co-addition law (6.59) cannot provide a spherical surface for which an elastic flexure exists.

Table 6.5 Normal thickness distribution {f,,} for in situ parabolization of an f/1.75 holed vase shell. Mirror clear aperture 186 mm. Schott Zerodur. Load q = — 73.6kPa, a = 95 mm, ¿ = 25 mm, c = 15 mm, tf,/t\ = 1.3547. /?opt=650mm, Rsphe= 658.37 mm. /?piex= 51,126 mm, </?>=696.9mm

1. plain vase Shell. tx = 8 mm. cylinder outer edge radius ;'oe = 103 mm. [Units: mm]

6.057 6.071 6.104 6.152 6.216 6.295 6.387 6.488 6.594 6.696 36.000

2. holed vase Shell. tx = 8 mm. cylinder outer edge radius ;'oe = 103 mm. [Units: mm]

8.205 6.104 6.152 6.216 6.295 6.387 6.488 6.594 6.696 36.000

seal tape seal tape seal tape seal tape

partial vacuum spring partial vacuum spring

Fig. 6.11 Geometry from Table 6.5 and load configuration for the parabolization of an f/1.75 holed vase shell mirror by in situ stressing of a spherical surface

Fig. 6.11 Geometry from Table 6.5 and load configuration for the parabolization of an f/1.75 holed vase shell mirror by in situ stressing of a spherical surface

Fortunately, most - if not all - concave mirrors for telescope optics are represented by series expansions for which solution pairs (zSphe, zFlex) of the co-addition law can be found.

Primary mirrors of two-mirror telescopes which are not of paraboloid shape are mainly spheroids or hyperboloids belonging to the Ritchey-Chretien form (see Sect. 1.9.2). In general, the shape of these mirrors can be accurately approximated by a hyperboloid and their representation expansion limited to the three first terms.

Except RC telescopes especially designed for extended-field sky surveys -where R1 ~ R2 and equipped with additional two-lens correctors (see Fig. 6.18

in Sect. 6.6.6) -, most primary mirrors in RC telescopes have a hyperboloid figure which departs slightly from that of a paraboloid, i.e. their conic constants are generally included in the range k1 e [-1.20; -1.02]. Hence the thickness and load configurations, as developed in the previous section, do not differ widely from the case of a paraboloid. For this reason, we do not treat this case here.

Another special case is the wide-field three-mirror Rumsey telescope where all mirrors are hyperboloids with large negative conic constants, particularly for the tertiary mirror. Developments and results for this remarkable telescope form - a flat field anastigmat - are given in Sect. 6.6.7.

### 6.6.5 Aspherization of Cassegrain Mirrors

Most usual telescope forms are two-mirror systems where the primary and secondary mirrors are a paraboloid-hyperboloid pair (PH) or a Ritchey-Chretien with a hyperboloid-hyperboloid pair (RC). The afocal telescope form is a paraboloid-paraboloid pair (PP). In all possible forms with a convex secondary, this mirror is usually called a Cassegrain mirror. The PH, RC, and PP forms are with hyperboloid or paraboloid Cassegrain mirrors. Active optics aspherization of these mirrors by elastic relaxation after spherical figuring while stressed is always preferred.

However, some exceptions are non-conventional telescope forms such as, for instance, two-mirror systems with a spherical primary where the aspherization of the oblate ellipsoid Cassegrain mirror is more easily obtained by in situ stressing after spherical figuring without stress. This form was investigated to simplify the segmentation of large primary mirrors [2-4]. A 1.4-m prototype telescope with an in-situ aspherized vase shell Cassegrain mirror was built for imaging evaluations on the sky at Haute Provence observatory [14].

Returning to the case of paraboloid or hyperboloid Cassegrain mirrors, we develop hereafter some examples for these two cases.

• Paraboloid Cassegrain mirrors: In a two-mirror system, a paraboloid Cassegrain mirror is the second optical component of an afocal Mersenne form. This remarkable form is free from Sphe 3, Coma 3, and Astm 3, therefore anastigmatic (see Sects. 1.9.3 and 2.3). Although sometimes used in the past for a coude telescope arrangement, afocal forms are the basic systems of high-resolution telescope arrays.

Using Schott Zerodur vitroceram, a purely meniscus shell, an enlarged meniscus shell, and three varied vase shells have been computed with N = 10 equal width element rings (Fig. 6.14 and Table 6.6).

• Geometric scale factor: Similarly as for all the previous data given in Sects. 6.6.2 and 6.6.3 for concave mirrors, each parameter set defining the geometry of convex mirrors in Tables 6.6 can be scaled up or down provided the material elasticity constants E, v, and the load q are the same.

This general linear property also applies to the design parameters of paraboloid and hyperboloid convex mirrors in the examples below.

Fig. 6.14 Five varied shapes of Cassegrain mirrors for active optics aspherization by spherical figuring while stressed tz,N+l

Fig. 6.14 Five varied shapes of Cassegrain mirrors for active optics aspherization by spherical figuring while stressed

• Maximum stress: For a meniscus shell and vase shells where the outer cylinder is such that tx < tN, a basic estimate of the maximum stress at the center of the shell orr(0) and att(0) is given by relation (6.64) of the constant thickness plate theory. For the five convex paraboloids at f/2.5 in Tables 6.6, this relation leads to a maximum stress orr(0) = att(0) < 7.7 MPa. This value is well below the Zerodur tensile maximum stress for safety, aTmax = 22 MPa (cf. Table 1.10) and c103sec = 90 MPa (cf. Table 5.2) and thus provides a safety factor of ~ 11.7.

For a vase shell where the dimensions of the outer cylinder are tz,N+1 = tx = 2tN - the last cases in Tables 6.6 -, the maximum stress at the junction radius r = rN are dominating with o(rN) ~ 1.3 c(0). Although the bending stresses always dominate in the deformation of these shells, their exact values can be determined from finite element analysis.

• Execution of a paraboloid Cassegrain mirror: Two identical paraboloid Cassegrain mirrors at f/2.33 were parabolized by stress figuring for the 1.5 m telescopes of the Gi2t high-resolution interferometer near Nice. These two Mersenne afocal telescopes were designed for a beam compression k = R1/R2 = 20. The Cassegrain mirrors were made of Corning fused silica in low expansion ULE grade for which the Young modulus was found to be abnormally large, showing E = 85GPa, instead of E = 68.8GPa as given by Corning (cf. 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, allowed the iterative determination of the normal thickness distribution {tn} of the vase shell where all successive ring-shell elements have a rN/N radial width (Fig. 6.15 and Table 6.7)

From the result of iterations with the shell theory, the active optics superposition law was accurately satisfied. The resulting flexure writes, in [mm],

ZFlex = 0.722932 10-5 r2 - 0.16567 10-8 r4 - 0.4639 10-14 r6 - 0.162 10-19 r8

The dimensionless radius p = r/rN for which dzFlex/dr = 0 is po=1.038. The total flexure sag in the range p e [0; 1 ] is AzFlex=7.81 ¡m. The vase geometry of two sample Zerodur secondary mirrors were obtained by computer numerical control diamond turning. After spherical figuring under stress and elastic relaxation, the results from He-Ne interferometric tests showed that the paraboloid surfaces were obtained within X/12 ptv for each sample.

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