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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