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Using the same dimensionless quantities as in (5.82), where q is substituted by F/na2, a numerical integration of (5.83) provides the dimensionless rigidity D(p). Since theoretically infinite at the origin, the integration of the rigidity is carried out from a small radius, e.g. p = a/(10R), by starting with the corresponding D-value derived from the previously obtained VTD for A40 coefficient alone (5.88). After successive increments of the radius, next iterations are made by adjusting the starting D-value up to D = 0 for the radius p = a/R.

This elasticity design was followed up by construction of a submaster in quenched Fe87 Cr13 stainless steel. The rear side and the outer collar and ring were machined in a single piece - forming a holosteric substrate - by a computer-controlled lathe. This submaster allowed obtaining the first aspherized grating by an active optics replication process. The substrate of the reflective grating replica was Zerodur from Schott (Fig. 5.22).

Fig. 5.22 He-Ne interferograms of a plane replica grating, 102x102mm, 1,200i/mm, when stressed on the tulip-form submaster. The next replication provided the aspherized grating on a rigid Zerodur substrate for an f/2xf/1.8 spectrograph, 14° curved field, XX [300-500nm], a = 28°7, ft = 0° at 400 nm, R =360mm, collimator beam diameter 2rm = 90 mm. Submaster: alloy Fe87 Cr13, active diameter 2a =170 mm, A40 = 2.812 10~9mm~3, A60 = 3.2 10~14mm~5, thickness t(a/2) =8 mm (Coll. LOOM - HORIBA-Jobin-Yvon [17])

A laboratory spectrograph in the ultraviolet spectral range XX [300-500nm] was built for optical evaluation of the grating spherical aberration correction (Sphe 3 and Sphe 5). A circular cross-section collimated beam, 90 mm in diameter, illuminated the 1,200 i/mm grating onto an elliptic area 90x100 mm in size. At the wavelength 400 nm (ft = 0), the output beam from the camera mirror was with an f/2xf/1.8 anamorphosis. Both Fe arc and low-pressure Cd lamp provided spectral line widths sharper than 5 ¡m over the 15° curved field of view.

The spectra obtained with this grating showed that the resulting spectral resolution were in accordance with the theoretical blur images for k = 0 in Sect. 4.4.5 (see Fig. 4.14). This also demonstrates that the active optics replication technique perfectly applies to the aspherization of diffraction gratings.

5.4.5 Bisymmetric Gratings with k = 3/2 and Elliptic Built-in Submasters

Considering now the case of a diffraction angle ft = 0 for the center of the output spectrograph field, the best design is still provided by the geometry k = 3/2. The shape for these gratings, now defined by (5.76b), is made of iso-level lines that are homothetical ellipses. From (5.76c), their semi-axis ratio is ay/ax = cos ft. Assuming hereafter a circular incident beam of aperture 2rm, the following properties are useful to the design of the deformable submaster.

^ The flexure is generated from a built-in elliptical contour determined from ft in (5.76d),

^ The final grating clear aperture is determined from a by the elliptical contour (5.76e).

Let us denote ZSub the submaster flexure. If the grating aspherization process is a double replication, then the active optics co-addition law writes

where ZOpt is the figure of the final aspherized grating replica. If the grating is directly used on the submaster, then the sign of its flexure - and then the uniform load q considered hereafter - must be opposite.

• Constant thickness submaster for slow focal ratio cameras: For camera focal ratios slower than f/3 or f/2.5, a constant thickness submaster with an elliptic built-in edge C defined by cos2ft x2 + y2 = 3 rm = r2 (5.92)

can generate an elliptic symmetry grating. The flexure shape of the inner built-in plate of the submaster is given by (5.27) in Sect. 5.3.5. The elliptic machining of such a submaster only consists of the inner elliptical cylinder of the contour ring, a groove for the O-ring seal and holes distributed over an ellipse for tightening the air pressure closing plate; all other surfaces can be axisymmetrical. The O-ring length is selected from the determination of a Legendre elliptic integral of second kind.

Representing the built-in contour C by (5.25), i.e. x2/a^ + y2/a^ = 1, leads to ax = ro/cos ft, ay = r0. (5.93)

The flexural sag from the center to the elliptic contour C is, from (5.28),

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