## D2 A22

^ For a meniscus mirror with four arms (km = 4), the intensity and direction of these forces are represented by

E t3

Fig. 7.5 Degenerated configuration of a four-arm meniscus-form mirror generating an Astm 3 mode. Only two pairs of opposite forces Fck act on folded arm ends in the directions 6 = 0 and n/2. With b = a and y = 1, condition (7.22a) gives c/a = 1/2. V Fa^ = 0 andFc i = Fc 3 = -Fc,2 = -Fc,4. The built-in contour condition is realized by arches providing a best azimuth modulation of Mr (a, 6) and Vr(a, 6) for large deformations

Fig. 7.5 Degenerated configuration of a four-arm meniscus-form mirror generating an Astm 3 mode. Only two pairs of opposite forces Fck act on folded arm ends in the directions 6 = 0 and n/2. With b = a and y = 1, condition (7.22a) gives c/a = 1/2. V Fa^ = 0 andFc i = Fc 3 = -Fc,2 = -Fc,4. The built-in contour condition is realized by arches providing a best azimuth modulation of Mr (a, 6) and Vr(a, 6) for large deformations

7.4.3 Single Astm 3 Mode and Degenerated Vase Form

Comparing the geometries between vase form and meniscus form, from (7.22a), we obtain the following result.

^ A vase-form configuration provides much shorter arms than for a meniscus form, hence improves the stability of the flexure mounting by compact designs.

A degenerated configuration of a four-arm vase-form mirror was designed from condition (7.22a) for Fa,k = 0. Several vase forms were built in Hooke's linear stainless steel alloy for large flexural sags. With respect to higher-order modes of the astigmatism family, the reduction of the interferograms showed that the purity of this mode was characterized by low amplitude harmonics, |A42/A22| < 0.023 and |A62/A22|< 0.005 (Fig. 7.6).

Fig. 7.6 Degenerated configuration of a four-arm vase-form mirror providing an Astm 3 flexure mode. This solution satisfies condition (7.22a) for Fak = 0. Hence only four Fck forces act on the ends of folded arms clamped at r = b. Substrate: quenched Fe87Cr13 stainless steel, v = 0.305. Geometry: clear aperture 2a = 100mm, 2a/t1 = 20, Y = 1/27, b/a = 6/5, c/b = 0.7582, then c/a = 0.9098. (Left) Rear view of the mirror. (Right) He-Ne interferogram of the flexure [Loom]

Fig. 7.6 Degenerated configuration of a four-arm vase-form mirror providing an Astm 3 flexure mode. This solution satisfies condition (7.22a) for Fak = 0. Hence only four Fck forces act on the ends of folded arms clamped at r = b. Substrate: quenched Fe87Cr13 stainless steel, v = 0.305. Geometry: clear aperture 2a = 100mm, 2a/t1 = 20, Y = 1/27, b/a = 6/5, c/b = 0.7582, then c/a = 0.9098. (Left) Rear view of the mirror. (Right) He-Ne interferogram of the flexure [Loom]

• Note on single Astm 3 mode: The above CTD configurations can be compared to the VTD obtained by use of a cycloid-like distribution simply bent at their edge by an outer ring modulation (see Fig. 3.25).

### 7.5 Meniscus Form and Segments for Large Telescopes

The segmentation of telescope mirrors is an inevitable step in the development of large telescopes. Aspherical mirrors such as off-axis segments of an optical surface can be obtained with vase or meniscus-form MDMs. Using meniscus segments, the stress figuring method was applied for the construction of the primary mirror of the 10-m Keck Telescope (see Lubliner & Nelson [42], Nelson et al. [50] ). With this method, the aspherization of an off-axis segment, whose optical shape is denoted zopt, is the result of co-addition of an elastic relaxation zElas with a spherical shape zsph obtained by a tool of full lap aperture. Thus, the active optics co-addition law writes

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