as given in the definitions for (3.20). The remaining parameter is the aspheric parameter tv given in the definitions for (3.20) as i

For our purely catoptric system with reflexions only, this becomes

Finally, we have for the field curvature

for a purely catoptric system.

This completes the formulae enabling the calculation of third order aberrations for any system consisting only of mirrors in a quite general way, with or without normalization of aperture and field, without the need of numerical ray tracing. They provide a powerful tool for the analysis of any of the purely catoptric systems treated above and may be preferred by the informed reader, because of their generality, to the specific formulations for 2-mirror systems given earlier in this chapter.

An interesting analysis of the theory of 3-mirror telescopes is given by Robb [3.82]. Robb is concerned specifically with 3-mirror solutions like Korsch [3.73], and notes the complexity of the Korsch equations (3.317) et seq. He uses first order combinations of the constructional parameters to simplify the formulation of third order aberrations for the three mirror case. The same procedure is possible with the recursion formulae above if the parameters are inserted in Eqs. (3.20). But the simple recursion properties are then lost and the complexity increases rapidly for n surfaces where n > 2.

3.6.5.3 Other 3-mirror and 4-mirror solutions. Robb [3.82] considers a number of 3-mirror designs and gives an optimized form for the geometry shown in Fig. 3.77, for an angular field of 2.30°. The geometry is similar to that of the Willstrop telescope of Fig. 3.74, except that the strict afocal feeder is replaced by a slightly convergent beam and an optimization performed by the technique referred to at the end of § 3.6.5.2 above. This optimization was performed balancing third, fifth and seventh orders, the primary having a relative aperture f/1.94 and the final value being f/5.0. The mirror forms are hyperbolic to the third order, as in the Korsch system of Table 3.18 and all such flat-field, 3-mirror solutions.

Robb also draws attention to 3-mirror designs by Shack and Meinel [3.83] and Rumsey [3.84] [3.85]. The latter design is similar to the Loveday telescope, treated above, in that the primary and tertiary mirrors are combined; but with the important difference from Loveday that the central part for the tertiary is figured differently according to its specific requirements. The

Rumsey solution is therefore nearer to a Paul-Baker telescope, but with the position of the tertiary linked to that of the primary.

What, then, is the optimum geometry of a 3-mirror telescope? This will depend upon several parameters, particularly the angular field, the acceptable obstruction ratio and the baffling solution. An attractive form has been optimized by Laux [3.8(b)], combining aspects of the Paul-Baker system (Fig. 3.73) with those of the Korsch single-axis system (Fig. 3.72 (a)). It is shown in Fig. 3.78. The final image is behind the secondary, as in the Korsch system, giving a decisive advantage for baffling. However, M3 is virtually in the plane of the primary, following Baker or Rumsey. This is probably the most convenient constructional position and is more compact than the Will-strop or Robb geometries. However, a price must be paid with a relatively high obstruction ratio if a fast system (f/4) and large field of about 2.5° diameter are aimed for. The image quality over a flat field is excellent, within 0.2 arcsec for a field diameter of 2°; but the linear obstruction ratio is about 0.5. This system has been proposed for a 2.5m wide-field survey telescope, the LITE project [3.86], using a large CCD-array detector.

Fig. 3.78. 3-mirror system given by Laux (1993) for a fast, flat-field 2.5m wide-field survey telescope with f/2.18 primary and f/4.0 final image, with a field diameter of 2.0° to 2.5C

Korsch [3.87] has recently proposed a 4-reflection solution using 3-mirrors, the secondary in double pass, in connection with possible forms for a Next Generation Large Space Telescope. Korsch proposes two possible solutions with the same 4-reflection geometry shown in Fig. 3.79. The initial Cassegrain telescope forms an intermediate image between the secondary and tertiary mirrors, the tertiary being placed in the same plane as the primary as in the Rumsey form, but forms a separate mirror in the primary central hole. Korsch has pointed out that a 4-reflection solution is more favourable for the final image position and baffling than a normal 3-mirror solution [3.88]. In the best solution using an f/1.25 primary all three mirrors are aspheric [3.87]. The primary is then ellipsoidal and the system gives a maximum residual aberration of 10 nrad (effectively diffraction limited for 16 m aperture at A = 150 nm) over a field of 0.5° diameter. If the primary is made spherical, this quality is limited to a field of 1 arcmin diameter.

A further theoretical possibility on these lines would be a straight double pass Cassegrain, an extension of the Loveday telescope to 4 reflections with two mirrors. To produce a real image, the first reflection at the secondary must produce a slightly convergent beam. As the telephoto effect of the single-pass Cassegrain is greatly increased, the emergent beam will have a very slow f/no, since slight convergence implies a high telephoto effect. Furthermore, the third and fourth reflections can give no practical contribution to aberration correction, as is also the case in the Loveday telescope with the third reflection at the primary. So a double-pass Cassegrain normally has no practical interest.

The statement by Korsch [3.88] that the geometry of a telescope with 4 reflections is fundamentally more favourable (above all because of the final image position) than with 3 reflections is a profound truism. An even number of reflections places the image at the end of the system from the point of view of the incident light. Apart from the telephoto advantage, this was a major advantage of the Cassegrain over the Newton or prime focus forms. One-reflection or 3-reflection solutions will only normally be interesting for direct

Fig. 3.79. 3-mirror, 4-reflection telescope proposed by Korsch (1991) for a future large space telescope imagery, where the detector is compact and not too heavy; for spectroscopy or general purpose instruments they will rarely be suitable.

In §3.2.6.1, we referred to the generalised Schwarzschild theorem [3.13]. This states that, for any geometry with reasonable separations between the elements, n Seidel monochromatic conditions can be fulfilled by n powered mirrors or plates, in the general case with aspheric forms. We have seen (for example, the Mersenne afocal 2-mirror telescope or the Schmidt telescope) that a favourable natural geometry can satisfy n conditions with fewer than n aspheric elements. The generalised Schwarzschild theorem tells us that a system of 4 powered mirrors with 4 aspherics can anyway satisfy the four conditions Sj = Su = Sm = S/y = 0. However, the Petzval condition S/y requires only the sum of the powers to be zero: if the geometry is predetermined to fulfil this, three aspherics are sufficient to correct the other three conditions, as in the systems of Paul-Baker (Fig. 3.42), Korsch (Fig. 3.72 (b)) or Laux (Fig. 3.78). It follows that, if the geometry is fixed to give a zero or acceptable field curvature, one of the mirrors in a telescope with 4 powered mirrors can be spherical, the other three correcting the first three Seidel conditions. Such design principles are well known and an increasing number of designs with 4 powered mirrors (or 4 reflections) have been published over the last 20 years or so. However, hardly any have been realised, above all in larger diameters. The reason has been partly the obstruction problem, which limits the reasonable geometries available for 4 mirrors on a single axis, but, above all, the reflectivity problem. The most backward aspect of reflecting telescope technology today is the inadequate reflectivity of the simple unprotected aluminium coat, still the standard solution for large mirrors (see RTO II, Chap. 6). The notable advance of evaporating aluminium was perfected by Strong in 1933 [3.89], which means there has been no advance for large optics in over 60 years! In view of the immense technological advances in all other areas, this is a surprising and unbalanced situation. However, there is clear evidence that technical solutions giving enhanced reflectivity R are known [3.90]. Investment in their practical application is still required, but it seems clear that various forms of protected (or enhanced) silver coats seem very promising. It seems a reasonable expectation that durable coats, maintained by modern cleaning routines (RTO II, Chap. 6), will become available within the next decade with R > 0.95 instead of R ~ 0.80 for simple aluminium, even for very large mirrors. The "Optikzentrum" in Bochum, Germany, initiated such a research and development project in 1993 [3.91], but went bankrupt and closed down soon afterwards. If R = 0.95 can be achieved, the loss from 4 reflections will only be 19%, much less than in a currently normal Cassegrain telescope with R = 0.80. In view of their excellent optical design potential, telescopes with 4 powered mirrors could then be extremely attractive, provided the geometry is otherwise favourable.

The geometry of 4-mirror telescopes is best approached using the elegant properties of some of the 3-mirror solutions due to Paul, Baker, Willstrop and Korsch, discussed in § 3.6.5.1 above. The most fundamental are those of Paul (Fig. 3.73) and Willstrop (Fig. 3.74), exploiting the properties of the Mersenne afocal telescope and the Schmidt principle. If we consider extending the Willstrop concept to 4 mirrors, with an additional powered mirror, then, from the generalised Schwarzschild theorem, we can make one mirror spherical. Logically, this will be the primary, the largest and most expensive mirror in the system. Although the aspheric (parabolic) primary of the 10m Keck telescope is a notable success in the manufacture and active control of aspheric segments, it is generally agreed that a spherical primary of large size and steep modern form for a compact construction has significant technical and, above all, cost advantages, whether it be segmented or monolithic [3.13] [3.88] [3.92] [3.93] [3.94] [3.95] [3.96] [3.97]. Now, the Willstrop telescope used a classical Mersenne feeder telescope with confocal paraboloids, anastigmatic according to the equations of Table 3.6. These equations show that Su and Siii remain zero, even if Si = 0, provided that (initially) the entrance pupil is at the primary, i.e. spr1 = 0. This is simply a formal proof of the fact that the aspheric form of an optical element has no effect on Su and Sm if it is placed at the pupil. The Mersenne afocal feeder, equipped with a spherical primary and a secondary of parabolic form, then delivers a beam with field curvature and enormous spherical aberration from a steep primary, but no third order coma or astigmatism. The large spherical aberration Si must be corrected by the remaining two mirrors of the system, without spoiling the field correction. This is clearly most effectively achieved if the strongly aspheric mirror producing this correction of Si is at the transferred pupil. If we follow the Paul-Willstrop concept, the mirror M3 will be a Schmidt-type sphere centered on the exit pupil (behind M2) of the feeder telescope. This reimages the pupil back on itself, so that the transferred pupil is not accessible for a fourth mirror M4. This is an important conclusion: it is impossible for a single-axis, 4-mirror system to respect the Schmidt geometry of the Paul-Willstrop concept. Of course, other solutions with the three aspherics M2,M3 and M4 following the Schwarzschild theorem are possible (see below), also with a pupil transfer to M4, but not with Paul-Willstrop geometry. However, this geometry can be fully maintained, together with perfect pupil transfer, if a 2-axis solution is used [3.13] [3.94] [3.95] with a small flat mirror at the intermediate image I3 as shown in Fig. 3.80. The radius of curvature of M3, concentric to the exit pupil P2, is chosen such that the image I3, with P2I3 = I3M3 = I3P3, is placed at a convenient axial position for the altitude axis of an alt-az telescope and that M4 and P3 are conveniently placed outside the incident beam and with respect to the bearing of the alt-axis. The function of Mp is simply to image the pupil to an accessible position, so that the high (hyperbolic) asphericity of M4 has no effect on the field aberrations. M4 then images I3 with a magnification of about 2 to the normal "Nasmyth-type" focus I4 on the other side. The optical power of M4 will introduce modest field aberrations because its imagery is free from neither

coma nor astigmatism. But these relatively small aberrations can easily be corrected by lightly modifying the forms of M2,M3, M4 and the separations.

This extension of the Willstrop telescope, with a spherical primary and the other three mirrors M2,M3,M4 to a third order roughly parabolic, roughly spherical and hyperbolic respectively, gives such excellent imagery at about f/7 with an f/1.5 primary and the necessary higher order correction terms on account of the steep primary, that the field correction cannot be exploited for a large telescope with foreseeable detectors. It can be seen as a generic type of 4-mirror telescope with natural, relaxed geometry, giving a particularly well-conditioned solution matrix. In fact, the solution is so relaxed that the secondary M2 can also be made spherical, still giving excellent imagery with aspherics on M3 and M4 only. In the basic solution with an f/1.5 spherical primary (first, 2-axis solution), the Schwarzschild constants are bs3 = -0.951 and bs4 = -11.116. The data of this system are given in Table 3.19 and the spot-diagrams in Fig. 3.81. We see that the spot-diagrams have d80 < 0.1 arcsec for a field of ± 9 arcmin and d80 < 0.5 arcsec for ± 18 arcmin.

A further advantage of the 2-axis geometry is that the obstruction ratio is much more favourable than with a single axis. Table 3.19 is laid out for a linear obstruction ratio of about 0.25, both from M2 and Mp, giving some vignetting by Mp beyond ± 9 arcmin. But even for ± 18 arcmin, the ob-

Mirror |
Radius (mm) |
Separation (mm) |
Free diameter (mm) (paraxial) (±9 arcmin field) |

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