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Fig. 3.17 Configurations providing a Coma 3 mode deflexure from two point-force F = ±2aFo applied at 6 = 0 and 6 = n, usable in a star tracking system. (Left) VTD mirror. (Right) Hybrid mirror. Design parameters: telescope aperture 2aTei = 3.6m, acomatic mirror 2a = 0.16m, t0/a = 1/10, v = 0.305, O = f/d = 3.8, i = 0.005rad, F/aaTelE = 7.9510-9

The thin collars are readily obtained with a metal substrate such as quenched stainless steel Fe87Cr13. Designs of Coma3 mode mirrors deformed by a couple of point-force F and —F respectively applied at 9 = 0 and 9 = n, in the VTD and hybrid classes are displayed by Fig. 3.17.

In the CTD class, solutions for Coma 3 with meniscus or vase form distorted by clamped radial arms are given in Chap. 7. Radial arms are of preferable use for glass or vitro-ceram acomatic mirrors.

• Spectroscopy in convergent beams and acomatic gratings: Some astrophysical studies in slitless spectroscopy and low spectral resolution use a transmission grating located at a convenient distance before the telescope focal plane and detector (typical dispersion ~ 1,000 A/mm). Associated with a multi-passband filter, theses systems provide quick spectral selections of stellar-like objects, some of them being subsequently analyzed with slit spectrographs.

A plane transmission grating used in a convergent beam generates a coma aberration in the dispersed image. At a given dispersion order, this aberration can be corrected for the central wavelength of the spectra by using an aspherized grating. The design and realization of an acomatic grating for the Cassegrain focus at f/8 of CFHT has been carried out [19, 20, 33] with a 75 l/mm grating working in firstorder and providing a mean dispersion of 750 A/mm. After achieving the Coma 3 correction at A0 = 500 nm, the next residual aberration is Astm 3 which does not debase the spectral resolution; the remaining and dominating blur effect is the tilt of individual spectra with respect to the focal plane (Fig. 3.18).

Acomatic gratings are obtained by replication from an active optics submaster under stress. The submaster is designed in accordance with the CTD class (Fig. 3.12-Left) by using a vase form in quenched stainless steel Fe87 Cr13. A plane grating is deposited on the plane surface of the submaster when not stressed. A set of opposite ring-force at r = a and r = b, is generated, via two cylinders and thin collars, from an axially acting push-screw and pull-screw. These screws operate the point forces F = ±2aV0 in 9 = 0 and 9 = n azimuths. The acomatic grating replica is obtained on Schott UBK7 optical glass with a transparent resin of the same refractive index (Fig. 3.19).

ORDER 0 1

Fig. 3.18 Acomatic transmission grating working in first diffraction order. Focussing on residual tangential astigmatism at central wavelength 500 nm

ORDER 0 1

Fig. 3.18 Acomatic transmission grating working in first diffraction order. Focussing on residual tangential astigmatism at central wavelength 500 nm

Fig. 3.19 (Left) Vase form metal submaster in the VTD class generating acomatic gratings by replication under stress. (Right) He-Ne interferogram of the grating when collimated in zero-order with respect to a plane surface. (Down) Markarian 382 set-up spectra from a 75 </mm acomatic grating with the wide-field electronic camera at Cfht f/8 Cassegrain focus after Wlerick, Cayrel, Lelievre, and Servan [33] (Loom)

Fig. 3.19 (Left) Vase form metal submaster in the VTD class generating acomatic gratings by replication under stress. (Right) He-Ne interferogram of the grating when collimated in zero-order with respect to a plane surface. (Down) Markarian 382 set-up spectra from a 75 </mm acomatic grating with the wide-field electronic camera at Cfht f/8 Cassegrain focus after Wlerick, Cayrel, Lelievre, and Servan [33] (Loom)

### 3.5 Active Optics and Third-Order Astigmatism

The third-order astigmatism, Astm 3 mode, is defined by n = 2 and m = 2. This bisymmetric wavefront function, or mirror shape, is represented by the quadric surface

which is a hyperbolic paraboloid (saddle).

The solutions ai of this mode are derived from the representation of the load q by (3.14). In this relation, the first term in lnr is null whatever is A0, but the second term leads to a surface load q ^ A'0r-2cos29 which would be difficult to generate in practice, so that we consider hereafter A0 = 0. Then, the distributions for Mr, Vr, and q are all included by the third and remaining term of (3.11a), (3.16), and (3.14), respectively. After simplification, these terms are j

i=1 j q = 2(1 - v)A22 X a (2 + a) Ai r-2-ai cos29. (3.52c)

Since 2-D prismatic loads of the form q = [email protected] cos 9 are to be rejected as extremely difficult to achieve practically, the ai roots to retain are those corresponding to a null uniform load q = 0 ^ ai = 0 and a2 = -2.

Substituting ai in (3.9), the distributions of flexural rigidity are contained in the representation

With the notation of (3.22) and (3.23), let us write the rigidity and a dimensionless thickness as

D0 r2 D0

From (3.52a) and (3.52b), the bending moment and net shearing force are

Given the form (3.54) of the rigidity, two configurations can be derived such as described hereafter.

A solution in the CTD class is obtained when the coefficient A2 = 0 in (3.53): this solution is a plane or a moderately curved meniscus when deformed both by bending moment and net shearing force applied to its contour. Let V0 be the maximum amplitude of the net shearing force acting per unit length at the mirror perimeter r = a, so the ring force is

Vr{a} = Vocos29. From (3.55b), we deduce the amplitude of this force as

By setting A1 = D0, and since, with (3.22), D0Qaa' = Ai, the Ci and dimensionless thickness are

Similarly, the bending moment Mr{a} is derived from (3.55a). The parameters of this configuration are summarized as follows [19]:

^ A plate or a slightly curved meniscus generates an Astm 3 deformation mode z = A22r2 cos 29, if its thickness t = T22 t0 is a constant, and if a bending moment Mr and a ring-force Vr = V0cos29 are applied to its perimeter r = a:

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