## Optical Profile of Aspherical Reflective Gratings

Similarly to all-reflective Schmidts, the shape of an aspherical reflective grating that minimizes the field residual aberrations is obtained by Lemaitre's condition k = 3/2 (Sect. 4.3.1). Thus, the local radial curvatures of the grating are extremals at its center and at its clear aperture edge, and of opposite sign for these radii (see Fig. 5.10).

Because of the beam anamorphosis of the diffracted beams, and since the incident beams are usually of circular cross section - diameter 2rm -, one defines the basic camera f-ratio as Q = f/d = R/4rm = R/4ym in the (y, z) plane containing the grating central line, where ym = rm is the semi-length of this line. Since the grating lines are parallel to the spectrograph slit, Q is called nebular f-ratio.

Summarizing the results in Sect. 4.4 for all reflective grating spectrographs, such as for all-reflective telescopes, the best images are obtained with design parameters k = 3/2, i.e. M = 3/64 Q2 with Q = R/4ym = R/4rm. (5.75)

With recent developments of large size spectrographs designed with f-ratios as fast as f/1.5, it is useful to summarize hereafter both high- and third-order representations of a grating optical shape. The following representations are for an incident principal beam of circular cross section.

• Axisymmetric gratings (ft = 0): For a normal diffraction mounting [see (4.23)], ft = 0, the grating shape is axisymmetric and the contour of its clear aperture is an ellipse. If the spectrograph only uses a spherical camera mirror - curved field -, then the best under-correction factor is s = cos2^. From (4.26) and denoting p = r/rm a dimensionless radius, the grating shape is cos2^m ^ Anrn cos2^m , 2 4\ /c-7^

Zopt =——— V ~—t ^ oS»3/1 ,-~A3p2 -p4) rm, (5.76a)

0 < p < 1/cos a in dispersion direction x, where An are given by Table 4.1.

The location of the circular null power zone ro is outside the clear aperture radius rm and defined in the third-order by po = ro/rm = %/k = \J3/2. For higherorder correction profiles, the determination of po can be accurately derived from dh/dr2 |o = -d2z/dr2 |rm.

• Bisymmetric elliptical gratings (ft = 0): For off-normal diffraction mountings, ft = 0, the grating shape must be with biaxial symmetry. Without a field-lens flat-tener and for a spherical camera mirror, then s = cos2qm. From (4.28), the grating shape is

F cos po+cosa 24"6 R

{0 < y < ym in nebular direction y, 0 < x < xm in dispersion direction x, where An coefficients are given in Table 4.1. Assuming a collimated incident beam of circular cross-section and aperture 2rm, the iso-level lines of the surface are ellipses represented by cos2ft x2 + y2 = constant. (5.76c)

In the third-order approximation, the elliptical null powered zone located outside the clear aperture is the ellipse cos2ft x2 + y2 = r2 = f rm. (5.76d)

For a collimated beam of circular cross-section, 2rm in diameter, the contour line of the grating clear aperture - as well as in a Littrow mounting -, is the ellipse cos a x2 + y2 = rm, i.e. xm = rmf cos a, ym = rm. (5.76e)

These relations are useful for the design of basic spectrographs with a curved focal surface. The undercorrection ratio s = cos2qm provides the best blur images along the central spectrum but would be slightly different for a balance with sideways spectra or multi-object spectroscopy. A slight modification of s allows maintaining the k = 3/2 condition either for the latter aberration balance or for the design of flat fielded systems with a positive corrector lens often used as the detector cryostat window.

• Aspherization replication process: In all the following Sections, the aspher-ization of a reflective grating is achieved via two replication stages by use of an intermediate active optics submaster as follows. Starting from a classically plane diffraction grating - "master grating" -, the first stage is performed on the plane surface of the submaster when in a null stress state. Then, the deformable submaster is aspherized in an opposite shape to that of the final grating replica on a rigid substrate which is obtained in the second stage.

5.4.3 Axisymmetric Gratings with k = 3/2 and Circular Built-in Submasters

Aspherized reflective gratings with a geometrical profile k = 3/2 are easily generated from the replication of built-in deformable submasters. In the axisymmetric grating case - corresponding to the normal diffraction angle A = 0 for the central wavelength - the best design is a circular active submaster with a built-in radius at the null powered radius a = ro. The aspherized grating is used with an elliptic clear aperture included into the built-in circle; this condition, rm/cos a < ro, is always satisfied for usual incidence angles.

Let ZSub be the shape of the active submaster when in a stressed stage and assume that it was figured in a flat shape when unstressed (ZSphe = 0). The aspherization of a grating replica Zopt is realized if the components of the active optics co-addition law write

ZElas = ZSub, ZSphe = 0 and ZSub + ZOpt = 0, (5.77)

where ZElas is the elastic flexure.

• Constant thickness submasters for slow focal ratio cameras: The design of active optics submasters for making gratings of low f-ratio spectrographs can be easily derived in the third-order optic theory, i.e. limited to the Cv 1 and Sphe 3 terms. In this case the elasticity theory provides a constant thickness solution which is a built-in plate bent by a uniform load. Hence, the submaster for grating replications is a vase form made of a constant thickness plate built-in to a thick outer ring.

From the identification of r4 terms of ZElas and ZSub in (5.5) and (5.76a) respectively, we obtain, with a uniform load q, q

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