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Fig. 5.9 Rupture tensile stresses of cylindric bent glass samples in function of the stressing time T. The samples were ultra-flat photographic plates from Kodak (Loom)

where p =14 and Ois = 96.6MPa provides the rupture of the Kodak plates in a one second delay. This curve is denoted [6] in Fig. 5.9. This law gives ault(103 s)= 59 MPa for Kodak slightly quenched plates, which is in good agreement with the value of 50 MPa given by Schott for BK7 optical glass (not quenched). For fused silica, our measures give ault(103 s)=75MPa.

Use of the experiment values a103s and law (5.19b) with p=14, allowed us to determine the rupture tensile stresses - or ultimate tensile strengths - of some brittle materials for various loading delays (Table 5.2).

Depending on whether the material is stressed just for the figuring - each day or for a month - or if in a permanent in situ aspherized state, the time-dependent values a in Table 5.2 allow us to adopt a design value aTmax of the tensile maximal stress for the active optics process, thus minimizing the breaking risk.

From the results in Table 5.2 we obtain the following conclusions.

^ Fused silica and polycrystalline sapphire are the most convenient isotropic refractive materials to achieve high deformations in active optics aspherization.

Except when the ultimate tensile stress ault is expressly given by glass manufacturers as corresponding to a loading time of 103 s, manufacturers generally indicate a recommended stress a which includes a safety factor for an indefinitely long load duration; thus ault for 103 s is not known by the user. For instance, with a BK7 glass working under permanent stress, Schott recommend to not exceed a tensile stress of 10 MPa. However, it has been proved as possible to aspherize several 20-cm BK7 corrector plates under a tensile stress of 20 MPa.

Some clarifications concerning the maximum tensile stress and also the time-dependent law itself ought to be given by the glass manufacturers.

5.3 Reflective Correctors

### 5.3.1 Optical Figure of the Primary Mirror

The basic optical parameters defining the shape of the reflective corrector - or primary mirror - are obtained from Tables 4.1, 4.2, and Eq. (4.9). In addition, Lemaitre's condition of minimizing the residual aberrations in the field of view (Sect. 4.3.1) entails a position of the null power zone set by k = 3/2, i.e. ro/rm = %/k ~ 1.225. Thus, the mirror local powers are extremals at the radial zones r = 0 and r = rm, corresponding to local curvatures of opposite sign (Fig. 5.10).

Summarizing the results of Sect. 4.3 for centered or non-centered reflective telescopes, the best images are obtained with design parameters k = 3/2, i.e. M = 3/64Q2 with Q = f/d = R/4rm. (5.20a)

It may be useful to derive simplified representations of the primary mirror shape. In the third-order theory, if the two first coefficients of Zw in (4.6) are approximated by their main part, namely A2 ~ M and A4 ~ -1/4, and denoting P = r/rm the dimensionless radius, we obtain the following simplified optical surface representations.

• Axisymmetric circular mirrors: For a centered system used off-axis (s = cos (i+^m)),

Zopt ^ Col(9i+3>m) (3P2 - P4) rm with 0 < p < 1. (5.20b)

Zopt ^ Col(9i+3>m) (3P2 - P4) rm with 0 < p < 1. (5.20b)

Fig. 5.10 Aspheric section of a reflective Schmidt primary mirror. Lemaitre's condition k = 3/2 provides the best balance of the field residual aberrations. An axisymmetric corrector mirror, Zopt ^ 2kp2 — p4 = 3p2 — p4 with p = r/rm G [ 0,1 ], shows a positive power all over its aperture. The null powered zone is at p = Vk ~ 1.225, thus located outside the clear aperture. This condition realizes the balance of the second derivative extremals in setting opposite maximum curvatures at the vertex p = 0 and clear aperture edge p = 1 (see Chap. 4)

Fig. 5.10 Aspheric section of a reflective Schmidt primary mirror. Lemaitre's condition k = 3/2 provides the best balance of the field residual aberrations. An axisymmetric corrector mirror, Zopt ^ 2kp2 — p4 = 3p2 — p4 with p = r/rm G [ 0,1 ], shows a positive power all over its aperture. The null powered zone is at p = Vk ~ 1.225, thus located outside the clear aperture. This condition realizes the balance of the second derivative extremals in setting opposite maximum curvatures at the vertex p = 0 and clear aperture edge p = 1 (see Chap. 4)

• Bisymmetric circular mirrors: For a non-centered system [ s = 1, deviation angle 2i at the primary, t = 1 sin2 i (see Table 4.2)],

-(1-2t)p4 + 2Tp4cos29] rm, 0 < p < 1, (5.20c)

where the origin of 9 is in the symmetry plane (z, x) of the telescope (x = r cos 9).

• Bisymmetric elliptic mirror: For a non-centered system (s = 1) and a primary illuminated by circular beams of diameter 2rm, an equivalent representation from (4.21c) is

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