## Chn

Assuming that the submaster is flat when unstressed [see conditions (5.77)], the identification of the A40 coefficient of the flexure z40 = A40r4 with the coefficient in r4 of (5.76a) leads to cos2 çm A4 cos2 çm 1

The incident circular beam defines an elliptic pupil at the grating whose semi-axis lengths are rm and rm/cos a, so the submaster edge radius must be such as a > rm/cos a. From (3.30b), the t0 thickness parameter of the submaster is derived as a function of the central force F and of the material (E, v) by t0 =

cos a where R is the radius of curvature of the camera mirror.

Representations (5.88) and (5.89) completely determine the elasticity parameters of the submaster design for generating k = 0 geometry gratings. A classical way of obtaining such gratings is the double replication process by replicating first a plane grating on an unstressed submaster. The active optics co-addition law (5.77) applies and the final grating replica is of opposite shape to that of the stressed submaster.

The first developments of the active optics replication technique for obtaining aspheric gratings were carried out by Lemaitre and Flamand [17] with a metal submaster generating an r4 flexure (Fig. 5.21).

Fig. 5.21 Design of a tulip-form submaster generating z = A40?-4 aspherized gratings by central force and double replications. The active VTD is with t(a/2) = 8 mm and simply supported at the contour by a thin collar of diameter 2a = 170 mm. The VTD, collar and ring form an holosteric piece of quenched Fe87Cr13 alloy [17]

In fact, for the first aspherized grating built by the active optics replication method, the small Sphe 5 correction of the camera mirror was also taken into account in the design of the submaster thickness distribution. Although not really necessary for a 90 mm circular collimated beam whose central wavelength diffracted beam is merging from the camera mirror with an f/2xf/1.8 anamorphosis, this correction is easily obtained. Since Cv 1 = 0 and from the first terms in (5.76a), the active surface of the submaster is represented by Sphe 3 and Sphe 5 correction modes as

R3 R5

Zsub

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