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Fig. 5.11 In-situ aspherization of a vase-form mirror by uniform partial vacuum. For obtaining k = 3/2 with a mirror in Zerodur (v = 1/4), from Table 5.5, possible design parameters are for instance b/a = 1.15 and t2/ti = 2.811

slightly decreasing from center to edge. Reflective Schmidt telescopes are usually with focal ratios lower than f/2.5 or f/3, so the Sphe 5 term remains small.

5.3.3 Bisymmetric Circular Primaries with k = 3/2 - MDM

• Non-centered systems: In the third-order theory, the optical profile is represented by (5.20c), that is a bi-axial symmetry figure where the astigmatism correction mode is included. In this case, vase-form multimode deformable mirrors - or vase-form MDMs - derived from the above vase-form but equipped with radial arms provide convenient solutions by active optics.

These MDMs, mainly discussed in Chap. 7, allow the co-addition of Cv 1, Sphe 3, Astm 3, and Astm 5 modes by means of a uniform loading and axial forces applied to the radial arms (cf. Table 7.2 and Fig. 7.9-Down).

5.3.4 Bisymmetric Circular Primaries with k = 0 - Tulip Form

• Non-centered systems: Although k = 3/2 provides the best balance of field aberrations, let us consider k = 0 for which the primary mirror is perfectly flat in its paraxial zone. Also in this case, active optics methods provide interesting solutions with tulip-form mirrors. For 100% encircled energy, the field residual images will be about three times larger than for k = 3/2 (see residual image variation with k in Fig. 4.9 - solid lines), but this could be a small handicap for the design of moderate field reflective Schmidt telescopes of fast f-ratio as first priority. With k = 0, the two first terms in (5.20c) vanish, so the primary mirror shape is

Zopt = 29^ [-(1 - 2t)p4 + 2rp4cos29 +...] rm, (5.23)

where the second term is Astm 5.

Tulip-form mirrors (see Sect. 3.3.2) can provide the compensation of Sphe 3 represented by the first term of (5.23). Avoiding the above configuration which requires

Fig. 5.12 Tulip-form thickness distribution of a circular primary mirror in the special case k = 0. (Left) Thickness T40 and central force F40 in reaction to the mirror edge providing the Sphe 3 correction. (Right) T42 = T40 and perimeter force Ft2cos20 generated - via a ring - by two orthogonal force pairs which provide the next Astm 5 correction [18]

Fig. 5.12 Tulip-form thickness distribution of a circular primary mirror in the special case k = 0. (Left) Thickness T40 and central force F40 in reaction to the mirror edge providing the Sphe 3 correction. (Right) T42 = T40 and perimeter force Ft2cos20 generated - via a ring - by two orthogonal force pairs which provide the next Astm 5 correction [18]

a uniform external loading q, one can select a configuration using a central external force F40 in reaction to the edge p = 1 of the mirror. We have seen that with q = 0 the two roots found for A40 in Sect. 3.3.2 (VTD-2) are oci = 8/(3 + v) and a2 = 2. This leads to the thickness distribution (3.30a),

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