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A schematic view of these seven designs of two-mirror anastigmats is displayed by Fig. 4.\.

The construction of Schmidt telescopes with a refractive corrector plate rapidly developed up to the largest instruments of Palomar and Tautenburg observatories having \.2 and \.4-m aperture plates, respectively (Ross [65]). Modifications for shortening the design have been proposed by also aspherizing the mirror (Wright [92], Vaisala [83]), then changing its optical properties from anastigmat to aplanat.

Various extensions of the basic Schmidt design have been proposed and built:

(i) Schmidt-Cassegrain form designs with a mirror pair having equal curvatures for a resulting flat field and internal focal surface (Baker [2]), or with a mirror pair having different curvatures for a resulting curved field and external focal surface (Burch [\3]) such as developed on the Mariner and Skylab probes (Momtgomery et al. [53], Courtes [\9]) or with monocentric Cassegrain mirrors (Linfoot [39], Sigler [77]) for obtaining an enlarged scale of sky.

(ii) Extended field of view has been achieved with an additional refractive Maksutov [45] lens and/or meniscus (Baker [3], Wynne [93, 94]).

Fig. 4.1 The class of two-mirror anastigmatic telescopes. The mirror curvatures ci, C2, are obtained with the positive sign before the square root. The negative sign provides virtual image systems. The Petzval curvature is Cp = — p / f', where f' is the effective focal length. In design (1), the secondary mirror degenerates into an infinitely small shell

Fig. 4.1 The class of two-mirror anastigmatic telescopes. The mirror curvatures ci, C2, are obtained with the positive sign before the square root. The negative sign provides virtual image systems. The Petzval curvature is Cp = — p / f', where f' is the effective focal length. In design (1), the secondary mirror degenerates into an infinitely small shell

(iii) Objective spectroscopy with a single prism (Schmidt and Wachmann [70]), and with a Fehrenbach normal dispersion prism (Fehrenbach [20, 21]), where the prism is located just in front of the corrector.

(iv) Catadioptric Schmidt cameras for spectrographs originally used curved films and are now widely equipped with flat field flatteners. Among the extremely large diversity of these designs, one may cite for instance special arrangements by Baranne [5] who invented the white pupil transfer [4] and some designs by Richardson [64]. Particular camera designs were elaborated with solid or semisolid forms (Hendrix [24], Schulte [75]) which provide, for the same geometry, faster f-ratio instruments in the proportion of the index of refraction N (these fast systems have been reviewed by Schroeder [74] and by Wilson [88]). Imaging focal reducers using a folded mirror in a refractive Schmidt have also been proposed with flat field designs (Su et al. [80]).

(v) Catoptric Schmidt telescope forms have been investigated and built (Lemaitre [29]) leading to a particular correcting mirror shape for obtaining the optimal angular resolution (Lemaitre [31]) over the field and whose monolithic corrector can be aspherized by active optics (Lemaitre [32]). In 2008, this all-reflective form led to the achievement of the giant Schmidt Lamost (Wang, Su et al. [86, 87], Su and Cui [82]) whose input pupil is 4 m in diameter and whose primary mirror is made of 24 active segments.

(vi) Quasi-catoptric spectrographs using aspherized reflective gratings - and a singlet lens field flattener as the only refractive element - have been designed and built for faint object spectrographs (Lemaitre and Flamand [30], Lemaitre [33, 34], Lemaitre and Kohler [36], Lemaitre et al. [37], Lemaitre and Richardson [38]).

A third-order aberration theory of refractive Schmidts has been carried out by Schmidt [67] and Stromgren [79] for minimizing the chromatism, by Caratheodory [14,15] for a more accurate determination of the on-axis stigmatism, and by Linfoot [40] for the Cassegrain form.

We develop in the next Section a high-order wavefront analysis of Schmidt systems. The basic design of Schmidt telescopes with refractive plates is displayed by Fig. 4.2 and the first built telescope by Fig. 4.3.

The length of a Schmidt telescope is twice that of the focal length. In order to avoid the field vignetting the size of the spherical mirror is the size of the corrector increased by two times the size of the field of view. Ratios between f/3.5 and f/2.7 and 4-5° fields of view were mainly adopted for 0.6-1.4 m refractive corrector plates. For large field sky surveys such as performed with Palomar, ESO and UK telescopes using the pure Schmidt form, the size of the residual aberrations and seeing degradation due to the atmosphere was matched with the resolution element of the detector. The sphero-chromatism of the corrector plate limits the optical performance of the system for broad wavelength ranges. Solutions for reducing this effect will be presented. New developments, such as all-reflective designs and spectro-graphic cameras using aspherical reflective gratings, need the treatment of a more elaborate theory of aberrations than the third-order one.

Active methods for obtaining refractive correctors, reflective correctors and as-pheric reflective gratings by in-situ stressing or elastic relaxation are described in Chap. 5.

4.1.2 Wavefront Analysis at the Center of Curvature of a Spherical Mirror

In order to determine the optical shape ZOpt to be provided by corrective optics when located at the center of curvature C of a spherical mirror, it is useful to determine, at this location, the shape Zw(r) of a wavefront emitted by a point source S after its reflection by the spherical mirror (Fig. 4.4).

If we consider now that the light propagates in the opposite direction, then the point source S becomes the stigmatic image of the Schmidt system. If the spherical mirror is used alone and if the object point is at infinity, the Gaussian focus G of the mirror is then located midway between segment CV where V is the mirror vertex.

Returning to the case where S is a source point, let R be the radius of curvature of the mirror, and denote

where M is a dimensionless parameter which is not necessarily small but may take any negative value between 0 and -1 for Schmidt cameras working at finite distance (Fig. 4.4). For an object at infinity, M is positive and relatively small; for reasons of field balancing of the residual aberrations, the M-value generally differs from zero.

In a cylindrical coordinate system, let Zw(r) be a wavefront issued from the source point S after reflection by the spherical mirror and holding at its center C. The wavefront surface can be expressed by the even terms of the polynomial series

Fig. 4.4 Wavefront from a source point holding after reflection at the center of curvature of a spherical mirror

Fig. 4.4 Wavefront from a source point holding after reflection at the center of curvature of a spherical mirror

where An coefficients are dimensionless. For large spectrographic cameras having an f-ratio as fast as f/1.5 or f/1, the higher-order terms of Eq. (1) have to be determined for an accurate optical correction. This determination is also required for solving the elasticity problem of the design of deformable optics that allows obtaining aspherical plates, mirrors, and gratings.

The shape of the wavefront Zw(r) can be obtained from the condition of a constant light path which is defined on the z-axis by 2 x SV + VC = (3 - M)R /2, so that for a point M at the mirror the sum of positive lengths SM = t1 and MZ = t2 is equal to this path. A set of relations can be deduced from the geometrical properties of Fig. 4.4, where u and v are respectively the angles of the segments ZM and CM with the z-axis. These relations are ti +12 = (3 - M)R/2 (4.2a)

where the unknowns are Z^, u, v,t1, and t2. Let us denote some dimensionless quantities with respect to the mirror radius of curvature R, as follows

Zw = Zw/R = XAnpn, L = t1/R, L2 = t2/R, p = r/R. (4.3)

The equation set (4.2) becomes

Each An coefficient can be obtained by remarking that

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