Since, in our formulation using a dummy flat secondary in § 4.2.1, both f' and g are defined as positive quantities with g < f', this condition cannot be fulfilled. In other words, such a thin, afocal doublet cannot fulfil all three conditions with a parabolic primary.

If we accept that S/ = 0, and use the second and the third equations of (4.47) to achieve E S// = E S/// = 0, we can derive at once

(S/)cor = E S/ with the parabolic primary as

or, substituting (4.48)

In his derivation, Ross [4.21] assumed the astigmatism of the primary was negligible and simply required the corrector to be free from astigmatism. (It should be noted here that the field curvature of the primary is even weaker and can be considered as a small compensation of its astigmatism. In the PF case, it is not necessary to introduce it as a separate condition in Eqs. (4.47), since we shall mainly be dealing with afocal correctors for reasons of achromatism). This approximation gives in our notation

the form also given by Gascoigne [4.9] with a different normalization factor. This result given by Ross was of fundamental importance in corrector development since it confirmed in explicit form the implicit conclusions of Sampson. It enabled Ross to conclude that the correction of coma and astigmatism with a thin, afocal lens system inevitably leaves finite spherical aberration whose amount depends (roughly linearly) only on the parameter g, i.e. the distance from the focal plane relative to the focal length. The powers and shapes of the lenses are immaterial. By implication, it must also have been clear to Ross that the result implied independence from asphericity on the corrector as well, in contradiction with his first paper [4.20]. However, this important corollary was stated neither by Ross [4.21] nor by Paul [4.4] and was first clearly enunciated in a classic paper by Wynne [4.23] in 1949 in a general analytical treatment of field correctors for parabolic mirrors. We shall formally prove this below.

Since Ross was interested in correcting existing parabolic primaries, the possibility of correcting the spherical aberration at the primary, as proposed by Sampson, was not acceptable. Instead, he fixed limits on the amount of Sj. Converting (4.51) to the disk of least confusion in arcsec (see Eq. (3.184) and Eq. (3.190)), Ross postulated a mean value of g//' = 0.04, giving an image spread of 1.7 arcsec for an f/5 paraboloid and 5.8 arcsec for f/3.3 (Palomar). The value of g//' was a delicate compromise: if made too small, the powers and/or bendings of the lenses will be so strong that higher order field aberrations will dominate.

Following Hopkins [4.22], the "central" contributions of a thin lens (stop at the lens) can be expressed as

2 XL

where n' is the refractive index, Kl the power, Xl a dimensionless "shape factor" and Yl a dimensionless "magnification factor" 2. The paraxial ray height y is proportional to g. Yl is largely determined by the converging ray bundle from the primary mirror; but Kl and Xl are free parameters in the equations for (Sj)l and (Sji)l. There are therefore, as Ross pointed out, an infinity of solutions to be chosen between powers and bendings.

Ross noted that the spherical aberration could be reduced by introducing a small separation - but this inevitably introduces chromatic aberration from (4.46). The finite thicknesses of the lenses also disturbed the achromatism. The transverse chromatic condition C2 then requires glasses with slightly different dispersions (7% with the thicknesses Ross chose). He preferred to

2 It should be noted that Eq. (4.52) is simply a more general form, using different parameters, of Eq. (3.271).

correct the longitudinal condition C1 using identical glasses, giving a small departure from afocality and a slight transverse chromatism. The secondary spectrum is, with identical glasses, zero irrespective of the total power of the system.

Ross proved that the Seidel distortion (Sy)cor of the corrector is also uniquely fixed by the parameter g by the relation thereby increasing rapidly with reduction in g and in conflict with the spherical aberration condition of Eq. (4.50). Distortion has been neglected in our general treatment of telescope systems because it is usually small enough to be of no consequence when calibrated. Field correctors, however, can introduce appreciable distortion: even when calibrated, significant distortion may lead to objectionable photometric effects.

In his pioneer work, Ross investigated many doublet arrangements in detail. He concluded that the best solutions put the negative lens at the front (nearer the mirror), particularly for small values of g/f', say < 0.05. For larger g-values, viable solutions were also found with the positive lens at the front, but spherical aberration and distortion values were less good than in the reverse order. In all cases, the total power of the corrector was negative. His original system for the 60-inch, f/5 Mt. Wilson telescope primary had an 8-inch aperture and g = 0.05f'.

An elegant general theorem concerning the impossibility of correcting S/ as well as S// and S/// for a parabolic primary and a corrector consisting of any number of thin lenses in contact was given by Wynne [4.23], who showed that correction of all three conditions is only possible in the limit case where the total power of the system, including the primary, is zero. Such an afocal total system is the strict equivalent of an afocal Mersenne beam compressor using two confocal paraboloids, which is also anastigmatic. Such a beam compressor was used by Paul [4.4] (see Fig. 3.73) as the basis for his 3-mirror telescope proposal. A similar possibility would exist, in principle, with the above afocal system with lenses. But, in practice, as Wynne points out, the secondary spectrum would preclude its use. The Paul telescope would be preferable in every way, including obstruction aspects.

In general, one could say that, although Ross's work effectively introduced practical PF correctors, the spherical aberration of his doublet solutions for paraboloids would not be acceptable today. He mentions [4.21] that "a system consisting of three lenses will be discussed in another paper". Apparently this paper never appeared, but Wynne [4.5] [4.24] has given details of such a design realised for the 200-inch, f/3.3 Mt. Palomar primary - see Fig. 4.7 taken from Wynne's paper [4.5]. Ross used his compact doublet, with negative lens at the front in combination with a thin, strongly curved meniscus placed before the doublet at about twice the g-value. This gave good correction on axis, but according to Wynne [4.24] about 3 arcsec of monochromatic comatic image

Fig. 4.7. The 3-lens Ross corrector for the Mt. Palomar 5 m, f/3.3 parabolic primary (schematic, after Wynne [4.5] [4.24])

spread at 10 arcmin off-axis, apart from chromatic difference of coma. Wynne gives a modified design, with thicker meniscus and inverted powers on the doublet, giving much improved field performance. The Ross 3-lens designs were more satisfactory for less steep paraboloids such as the 120-inch, f/5 Lick primary.

Apart from his pioneer work on aspheric plate correctors, Paul [4.4] carried out in 1935 a detailed analysis of lens correctors, based only on the earlier paper of Ross [4.20]. Paul considered first the case of a single lens, pointing out that it was only of theoretical interest because of the inevitable chromatic aberration. The equations for the three conditions ^Si, ^ Sii, ^ Siii are identical to those given for an afocal doublet in Eqs. (4.47), except that the definition of E given in (4.48) has to be modified to take account of f' = |f'|, which changes the length metric in our normalized system. However, this has no effect on the validity of Eqs. (4.47). The proof that it is impossible to correct all three conditions with a parabolic primary therefore remains valid. Paul gives solutions either for the correction of ^ Si = Sii = 0 with finite £Siii; or, following Ross, with Sii = Siii = 0 with finite Si. In principle, the two available parameters Kl and Xl from (4.52) are sufficient to satisfy the two conditions; but Paul realised that with the necessary small distance from the focus, a practical solution led to extreme bendings Xl. He therefore introduced a third parameter, an aspheric surface, to relax the requirements, enabling an equiconvex lens to be used. Taking g = 0.201f'|, he derived a residual ^ Si, for the system with ^ Sii = Siii = 0, equivalent to a disk of least confusion of 1.7 arcsec for the 60-inch, f/5 paraboloid of Mt. Wilson.

Although Paul does not state this explicitly through an equation equivalent to (4.49) for the inevitable residual spherical aberration resulting from the paraboloid-lens combination, it is clear that he understood this limitation because he states that the solution of all three conditions requires Z = 0 in (4.47) and that the primary must have hyperbolic form. The subsequent aspheric on the lens was solely to relax the shape factor Xl, giving a better practical solution for ^ Sii = Siii = 0 for a parabolic primary, not an attempt to correct Si which Paul knew was inevitable.

Paul then treated the case of an afocal doublet of a single glass, following Sampson [4.2] and Ross [4.20]. Again, Eqs. (4.47) express the requirement for the correction of the three conditions and Paul explicitly suggests correcting

E Si by a hyperbolic form of the primary. With g//' = 0.05, he deduces the necessary eccentricity bs1 = -1.22 for the primary from a formula equivalent to (4.50) for (Si)cor, dependent only on g//'. Paul noted that this departure from the parabola is much less than that proposed by Sampson [4.2] (bs1 = - 1. 944 - see above) and that such a solution would be applicable to the first RC telescope, then just completed by Ritchey. This telescope has been referred to in §§ and 3.7.2 above. The constructional data are given in Table 3 of Bahner's book [4.25]. With a primary of f/4.0 and an RC focus of f/6.85, the secondary magnification m2 = -1.71. This value is far lower than those typical of modern RC telescopes and led, from Eq. (3.114) or (3.115) to a much more eccentric primary. This favoured at that time Paul's proposal for such correctors for RC primaries, a basic idea which was to be very fruitful 30 years later.

We shall now consider several specific cases, starting with the proof of the important theorem of Wynne [4.23] that an aspheric surface brings no advantage for a thin, afocal corrector for a parabolic primary, irrespective of the number of lenses. Adding an aspheric extends the Eqs. (4.47) to

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