Similarly, a single-axis solution is possible by analogy with the first, 2-axis solution above, using an afocal feeder telescope (Fig. 3.88). As in the system of Fig. 3.85, it is possible to image the pupil more or less on to M4. But the final f/no is always a tied function of the diameter of M2: the smaller M2 becomes, the larger the final f/no. This is the same law as that governing the 2-axis solutions of Figs. 3.80 and 3.83. The steeper the spherical primary, the more important correct pupil transfer to M4 will be if this mirror is correcting the bulk of the spherical aberration.

In spite of the analogy of Fig. 3.88 to the first, 2-axis solution of Fig. 3.80, and of Fig. 3.85 to the second, 2-axis solution of Fig. 3.83, it is important to realise that the 2-axis solutions are fundamentally more relaxed and natural because of the Schmidt characteristics of M3. These characteristics are impossible in the single-axis solutions, although the pupil transfer to M4 can still be achieved. In the single-axis cases, therefore, three aspheric mirrors will be necessary following the generalised Schwarzschild theorem, whereas, as we saw above, the two-axis solutions can give good field correction with only 2 aspherics. This important difference will always mean that the aspheric forms in single-axis solutions are more complex and more extreme.

The two basic 2-axis solutions given above will lead, with normal (i.e. not excessively small) obstruction by M2, to final f/ratios of the order of f/6-f/7, as we have seen. Their geometry precludes fast f/nos. Suppose, however, M4 in Fig. 3.83 is withdrawn somewhat from the primary beam, then it could be made steeper and the image removed sideways with a Newton flat. But this final image position is no longer on the alt-axis and has an inconvenient position. Furthermore, a steeper concave M4 worsens the field curvature. If, therefore, a further mirror is to be introduced, it seems better to give it a more constructive role than that of a Newton flat. Figure 3.89 shows a 2-axis system with five powered mirrors where M4M5 form a Cassegrain telescope. If M4 is made very steep (~f/1.0), then high relative apertures (equivalent to Schmidts) are possible with modest obstruction; furthermore the field curvature can be corrected. A lightly modified version of this system has been set up [3.94], giving at f/2.805 spot-diagrams with d80 < 0.2 arcsec over 30 arcmin field diameter and d80 < 0.7 arcsec over 60 arcmin diameter. The field curvature has a small overcorrection residual from the convex M5, but it would probably be possible to eliminate this. The primary and secondary mirrors are spherical, as in Fig. 3.83, and the pupil is transferred to a plane between M4 and M5.

Fig. 3.90. A 2-axis solution with 4 powered mirrors proposed by Sasian (1990). Either Mi or M2 is spherical, M3 is toroidal

A very interesting 2-axis solution (Fig. 3.90) with 4 powered mirrors and correct pupil transfer to the final mirror has been proposed by Sasian [3.100]. He recognizes the importance of active correction by an element at the pupil, in his case the last mirror of the system as in the 2-axis concept of Fig. 3.80. Sasian's system resembles the Korsch geometry of Fig. 3.72 (b), but the plane mirror Mp in that system is replaced by a powered, concave mirror M3. Because it is inclined at 45° to form the second axis (Sasian terms this a "bilateral" configuration), M3 has to have a toroidal form. It images the pupil P2 on to the final mirror M4. Because of the pupil transfer, Sasian obtains interesting solutions with only 2 aspherics: the toroidal M3 is not aspheric

RjM4

Fig. 3.90. A 2-axis solution with 4 powered mirrors proposed by Sasian (1990). Either Mi or M2 is spherical, M3 is toroidal and either Mi or M2 is spherical. The d80-values of the spot-diagrams remain well within 1 arcsec for fields of 6 arcmin diameter, the case with the spherical secondary being somewhat better. This solution is to be compared with the 2-axis system of Fig. 3.83, which gives markedly better imagery with both primary and secondary spherical because the toroidal mirror for transferring the pupil is replaced by a flat. But Sasian's system has one mirror less, albeit at the cost of manufacturing an accurate toroidal surface. He saw his system as of particular interest with a spherical primary, because of the simplification of manufacture.

The pupil transfer to the fourth mirror is fundamental to successful field correction if a steep primary of spherical form is used. An interesting 2-axis variant of the single-axis system of Fig. 3.85, also permitting (in principle -it is not clear whether it is the case here) such pupil transfer to M4, is given by Baranne and Lemaitre as one of the corrector proposals for the TEMOS telescope [3.96] [3.97]. This is shown in Fig. 3.91. In this concept, the form of M2 can be varied actively according to the requirements of the various corrector systems. For the corrector shown, M1 is spherical and M2,M3,M4 aspheric. The spot-diagrams are within 0.4 arcsec over a field of 20 arcmin diameter. M4 is placed within the central obstruction and this geometry can not permit a Schmidt-type pupil transfer by M3, as given by the geometry of Fig. 3.83. The latter permits a better image quality with only two aspher-ics, the secondary being spherical. Since M4 is symmetrically placed on the opposite side of the axis from M3, it follows that the final focal length will be doubled in comparison with Fig. 3.91.

If imagery of the pupil on M4 is abandoned, many other variants of 4-mirror telescopes are possible. If the primary is not spherical, but parabolic or some similar form, the pupil transfer is not important. A system with favourable geometry is the double Cassegrain (Fig. 3.92) proposed by Korsch [3.88]. This has 4 independent mirrors, in distinction to the (normally impracticable) double-pass Cassegrain referred to above, which has 4 reflections from 2 mirrors. Since pupil transfer to M4 cannot be achieved, the field performance is vastly better with a primary of parabolic form. With a spherical primary of f/1.7, superb performance (50 nrad) for the rms blur diameter is possible over only 1 arcmin field diameter, whereas a larger field with this quality is possible with an f/0.6 primary of parabolic form.

Fig. 3.92. Double-Cassegrain 4-mirror telescope with intermediate image after M2, proposed by Korsch (1986)

Other designs using fast spherical primaries, above all in an Arecibo (fixed primary) concept, have been proposed by Schafer [3.101] and Shectman

In summary, it appears clear that designs with 4 powered mirrors, either as 2-axis or single-axis solutions, have great promise in the further development of the astronomical telescope. They permit excellent field correction for both conventional and fast f/nos and are, unlike designs involving aspheric plates in the pupil, no more restricted in aperture than classical Cassegrain or

Fig. 3.92. Double-Cassegrain 4-mirror telescope with intermediate image after M2, proposed by Korsch (1986)

RC telescopes. Furthermore, the image position and baffling characteristics are just as favourable, the application of active optics even more so. Of particular interest are designs with a spherical primary, a spherical secondary as well being a further bonus for the basic 2-axis designs given above. The more general application of such designs requires improvement in the reflectivity of the mirrors: protected silver seems to offer the greatest promise for large optics, multi-coating for smaller mirrors.

Single-axis solutions of the types shown in Figs. 3.85, 3.87, 3.88, 3.91 and 3.92 will all have increasing problems of higher order aberrations as the corrector system formed by M3 and M4 is made smaller or more compact (steeper curvatures on these mirrors).

3.7 Off-axis (Schiefspiegler) and decentered telescopes 3.7.1 Two- and three-mirror Schiefspiegler

Up till now we have been concerned solely with centered optical systems in which the elements with optical power all have a common axis or, in the case of spherical surfaces, are normal to the common axis. Of course, centered optical systems do not exclude deflections of the beam by plane mirrors which have no optical power and hence no optical axis. The restriction to a unique axis of symmetry leads to the Hamilton Characteristic Function dealt with in § 3.2.1 and the aberration types of Table 3.1.

If powered optical elements are tilted in one plane, which we will define as the tangential plane normally given by the plane of the paper on which the system section is represented, then symmetry to a line in space, the common optical axis, is abandoned. There is, however, still symmetry about the tangential plane. If the elements are tilted in two dimensions, then this symmetry is also lost. All those aberration terms which were eliminated by symmetry about the optical axis in Hamilton theory are present in the general case, so the theory becomes much more complex.

The first use of a tilted element in telescope design was, in fact, the Herschel type "front-view" single mirror reflector of Fig. 1.1. This was afflicted by the field aberrations of the paraboloid as the stop of the system. For the Herschel mirrors working at f/10 or longer, the limiting field coma was small and offset by the gain in light gathering power through avoidance of a second reflection at a speculum mirror with reflectivity of about 60% at best.

According to Riekher [3.39(g)], the first Cassegrain-type telescope with tilted components was the Brachy telescope of Forster and Fritsch in 1876. Both mirrors were spherical, requiring low relative apertures to give adequate correction of spherical aberration and a correspondingly long construction.

The first systematic investigation of the possibilities of a telescope with two tilted mirrors was published in 1953 by Kutter [3.103], followed by a second book in 1964 [3.104]. He introduced the German term "Schiefspiegler"

(oblique mirror) which has since become the generic name for this type of telescope. Kutter addressed himself particularly to amateurs, his specific purpose being the removal of the central obstruction in the Cassegrain telescope. It will be shown in § 3.10, dealing with physical optical aspects, that the central obstruction significantly reduces the contrast of imagery of surface detail in amateur size telescopes. Above all for observation of the moon or major planets this is a disadvantage for amateurs.

A detailed description of the Kutter Schiefspiegler and derivatives is given by Rutten and van Venrooij [3.12(j)]. Kutter also started with spherical mirrors like Forster and Fritsch. He set up the three solutions shown in Fig. 3.93. In solution I, both coma and astigmatism are present in the "central" ("axial") image point; in solution II, astigmatism is corrected but not coma; in solution III, coma is corrected but not astigmatism. As with Forster and Fritsch, the use of spherical mirrors imposed low relative aperture mirrors. This, in turn, imposed a low secondary magnification m2 of about 1.7; but even then the final f/no was f/20-f/30, values normally only adapted to visual use with small telescopes. Kutter proposed a secondary with the same radius as the primary, giving simplified manufacture and testing as well as a flat field.

Although the general theory for the aberrations of a finite field is highly complex, this is not so for the imagery of the "axial" image point shown at I, II, III in Fig. 3.93. For this point, the field is effectively zero and both reflections can be considered as if the stop were at the mirror concerned. A simple and instructive example is the classical Czerny-Turner arrangement

for monochromators or spectrographs [3.105], the theory of which is also discussed by Schroeder [3.22(c)]. The basis of the Czerny-Turner system is a pair of concave spherical mirrors arranged in so-called Z-form and tilted in such a way that the coma is zero for the "axial" image point I' (Fig. 3.94). Normally a grating is placed in the parallel beam which, because of the asymmetry of the incident and diffracted angles of the beams, produces a compression or expansion of the beam incident on M2. In Fig. 3.94 a beam compression is shown. We can apply the Eqs. (3.334) and (3.336) to derive the coma Sjj as though we were dealing with two primaries with a parallel incident beam. From (3.20)

rl r2

for coma compensation. The Czerny-Turner condition is therefore um = (riV / (yiy , (3.340)

Upri \rij / \yij in which (y2/yi) is the beam compression ratio and upri and upr2 are the tilt angles in Z-form.

An important point to note with all such tilted systems is that the above aberration calculations for the "axial" image point are performed as though the stop were at each surface of the system: stop shift terms only have meaning for the field of the Schiefspiegler. It follows that the coma and astigmatism of the "axial" image point are independent of the form of the mirrors, whether spherical or aspheric: they are solely functions of the mirror radii, the apertures and the geometry of the arrangement. The Czerny-Turner arrangement is simpler than the Schiefspiegler because there is a collimated beam between the mirrors, so that both can be treated as "primaries". The "Schiefspiegler" then provides an elegant application of the general recursion aberration formulae (3.334) and (3.336). We shall consider two cases, the coma-free Schiefspiegler and the anastigmatic Schiefspiegler.

Coma-free Schiefspiegler: Using, from (3.20), (Sii)v = -yvAvAv^ (-N with the parameters derived from (3.336), we have

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