## Parabolization of Concave Mirrors

Active optics parabolization of concave mirrors can be theoretically carried out by one of the two above processes. Unfortunately, because of practical difficulties to realize in a single piece a closed shell aspherized by inner air pressure and stress figuring, use of a vase or a meniscus shell aspherized by in situ stressing is preferred. However, from the above theorem, the following results are valid for both processes [6, 20].

The coefficient of the first term in flexure (6.57) is a2 = 1 /2RFlex. From the co-addition law (6.59), the first quadratic terms - i.e. the curvature terms - of the expansions must satisfy i = 1 ^ 1/Ropt = 1/Rsphe + 1/RFlex (6.60a)

whilst the sums zSphe + zFlex at any higher order exactly cancel. Hence, from (6.57) and the expansion of a sphere (see (1.38c) in Sect. 1.7.1), next coefficients of the flexure write

There are infinitely many a2i sets satisfying the active optics co-addition law. Since the sign of a2 is opposite to that of any higher-order coefficient a2i we can always find a flexure coefficient set with a balanced shape. From (6.57) and a third-order approximation, one shows that an acceptable range for obtaining a balanced flexure zFlex is when

2 a4 rN

where the left and right limit values correspond to dzFLex/dr = 0 for r/rN = 1/2 and 2, respectively. An optimal ratio is when the volume to remove is minimal; it can be shown that this is achieved for a null slope of the flexure located at r/rN = 1/= 0.7598... which corresponds to -a2/a4rN = 2/V3 = 1.1547... Because zFlex is derived from the displacements wn, un in (6.55) which are dependent on the shell boundary conditions, a strict condition for the above null slope radius-ratio would be excessive. For this reason, we assume as acceptable any flexure solution where the null slope radius r0 is such as

We consider hereafter the three shallow shell geometries defined in Sect. 6.4.1 and use Schott Zerodur substrates (cf. the elasticity constants in Table 1.10). For instance, we can solve the inverse problem for 40-cm or 2-m clear aperture mirrors made of N = 10 meniscus elements.

• Parabolization of 40-cm clear aperture mirrors: The calculations of 40-cm optical diameter mirrors with N = 10 elements use a 2-cm radius increment; the radius of the circle where the load reaction applies is rN=r10=20 cm. The uniform load is q =±80kPa ~ ±0.8 Atm where, as stated in Sect. 6.6.1, the negative sign is for a depressure (in-situ stressing) and the positive for a pressure (stress figuring). The iteration process allows us to determine the associated radii of curvatures and normal thicknesses {tn} (Table 6.3).

• Parabolization of 2-m clear aperture mirrors: Similar calculations of 2-m optical diameter mirrors with N = 10 elements use a 10-cm radius increment; the radius of the circle where the load reaction applies is rN=r10=1 m (Table 6.4).

• Concluding remarks: Whatever the above shell geometries and associated boundaries, we obtain the following results.

^ The thickness distributions {tn} always increase from center to edge. In all cases, the faster the f-ratio, the larger the relative thickness increase.

In the design shown by Fig. 6.6, an optical clear aperture of 2rN is not exactly achieved because the inner part of the cylinder N +1 locally increases the rigidity. In order to effectively obtain a clear aperture up to 2rN in diameter, slight modifications of the shape and position of the outer cylinder must be made. Since the radial thickness tx of the cylinder always satisfies tx/rN C 1, the inner and outer radii of the cylinder can be modified such as respectively rN - tx/2 ^ rN and rN + tx/2 ^ rN + tx (6.62)

without significant change of the results in Tables 6.3 and 6.4. From the axial thickness tz,N+1 used for the calculation, the geometry of the outer cylinder can be somewhat modified for the final mirror design ; equivalences can be obtained by a widened local base and a shortened axial thickness.

To prevent from local stress concentrations at the link of the meniscus with the outer cylinder, a toric inner junction of radius RJ must be made such as

An example of such a modified vase shell is shown in Fig. 6.8.

One shows that if a plane plate of constant thickness t is clamped at r = a into a solid of semi-infinite axial and radial thickness, then the radius RJ of the inner junction - at the rear side of the plate - must approximately satisfy RJ/t ~ 1 - v. This value provides the deformation sag corresponding to that given by the thin plate theory for a built-in radius a realized by a square-corner junction. This result - which also includes the flexure effect of the shearing forces - was derived from the results of deformation tests of samples having various two-dimensional shapes (Fig. 6.9).

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