From the signs given in Tables 2.2 and 2.3, the supplement term is negative for the Cassegrain and positive for the Gregory, giving values near the parabola, but hyperbolic for the Cassegrain and elliptical for the Gregory (Cases 4 and 8 of Table 3.2). The difference from the parabolic form only becomes appreciable for small values of m2.
In the limit, afocal case, Eq. (3.115) is singular. If we substitute L = m2(f1 — d1) from Eq. (2.75), the supplement is again zero in the afocal case, confirming that the classical and aplanatic forms are identical, i.e. parabolic.
We have seen that the aplanatic supplement for the primary is small for normal values of m2. In manufacture, this is insignificant unless the primary is tested against a full-aperture flat in autocollimation, giving a null-test (RTO II, Chap. 1) in the classical case. The supplement for the secondary amounts to about 14% in the cases of Table 3.2 with m2 = ±4. It increases rapidly for smaller values of m2. It is important to note that, while the apla-natic forms of the secondaries are still hyperbolae or ellipses to a third order, these conic sections are no longer such that the foci are at the primary and secondary images, as is the case for the classical forms. This is a consequence of the fact that the primary image is not free from spherical aberration in the aplanatic case.
The amount of spherical aberration at the prime focus is given from Table 3.5 and Eq. (3.30):
a negative value for the Cassegrain (/ positive) and a positive value for the Gregory (/ negative). The differences between the values for surfaces 1 and 1* in Cases 4 and 8 of Table 3.3 for S7 correspond to Eq. (3.117). They are small (-0.581 in the RC case) compared with the total contribution of the primary as a spherical mirror (+16) but larger than that of a spherical primary of the same final focal length / (compare Case 1 of Table 3.3). The aberration thus makes the prime focus unusable without a corrector. However, it is fortunate that the primary form in the (commonly used) RC case is hyperbolic, since this is favourable in the design of PF field correctors, correcting above all the field coma (see Chap. 4).
We now consider the astigmatism of the aplanatic 2-mirror telescope. The general relation for £ Sm is given by Eq. (3.61) and in Table 3.5. Comparison with Eqs. (3.22) shows that the term in spr1 contains the spherical aberration while that in spr1 contains the coma, both of which are zero. So, even if spr1 = 0, there is no change in astigmatism. If, therefore, for IR observation the stop is placed at the secondary, there is no effect on , in an aplanatic telescope, whereas in the classical telescope £ Sm is changed because £ Sjj = 0. The astigmatism of the aplanatic telescope is yi f' (J + di) + d?,
Substituting from (3.108) gives c A _ V /' /V , dM 2
Comparing with the classical telescope, Eq. (3.96), we find a similar result except that d1 is replaced by d1/2 in the bracket. In the RC case, / is positive, d1 negative, so there is a modest increase of astigmatism. In the aplanatic Gregory, there is a modest reduction. Cases 4 and 8 of Table 3.3 reveal these effects in the normalized case.
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