cor which have exactly the same form as the equivalent equations of (4.47) and lead to exactly the same spherical aberration residual of (4.49). The only advantage of an aspheric surface is, in the sense used by Paul, to relax the bending requirements for the lenses to achieve coma and astigmatism correction.
As stated above, the Eqs. (4.47) apply also to a single lens with power, as discussed by Paul. If an aspheric term Sp is added as in Eqs. (4.54), the conclusion is identical. Therefore, an aspheric surface cannot influence the residual spherical aberration of a "thin" corrector satisfying E S// = E S/// = 0 for a 'parabolic primary, irrespective of whether the corrector is afocal or not: the property comes from its "thin" nature, whereby all constructional parameters have the same value of E.
Let us consider now the possibility of a thin, afocal doublet combined with a parabolic primary to satisfy E S/ = E S// = 0 with E S/// = 0. Setting the first two equations of (4.47) to zero with Z = 0 gives for the astigmatism
bearing in mind that the central astigmatism (S///)cor = 0 because the doublet is afocal. From (4.48)
Again g may be chosen at will, as for the case of a corrector for E S// = ES/// = 0. With g = 0.05f', it follows from (4.57) that the astigmatism of the primary mirror, which is f', is increased by a factor 20 times in this case. Gascoigne [4.9] implies this solution is useless in practice because the astigmatism is excessive. It is instructive to compare it with the astigmatism of the Gascoigne plate as a PF corrector for hyperbolic primaries which was given in Eq. (4.12) as
For small values of g, the astigmatism of the thin doublet from (4.57) is about twice that of the Gascoigne plate with the same value of g. However, in the Gascoigne plate case, g is determined by the eccentricity of the primary, whereas the doublet corrector for a parabolic primary allows a free choice of g. With g-values ~ 0.05f', the astigmatism residual may be more favourable than that given by Gascoigne plates for typical RC primaries.
We will now consider more closely the proposal of Paul [4.4] to use a compact, afocal doublet with a hyperbolic -primary to correct E S/. We have from (4.47) and (4.50) with £ S// = £ S/// = 0
From (3.71), setting m2 = —1 for a Cassegrain with plane secondary, equivalent to the PF
Since f' and g are defined as positive, this proves Paul's statement that correction of all three conditions with a thin, afocal doublet requires a hyperbolic primary. Setting g = 0.05f' gives "si = —1.222, in agreement with Paul. Eq. (4.58) gives the approximate relation quoted by Gascoigne [4.9]
a good approximation because g must be small for modern RC telescopes. For the RC system of Table 3.2 with m2 = —4, Eq. (4.59) gives g ~ 0.0091f' for such a PF corrector. By comparison, the Gascoigne plate required from (4.9)
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