Then E S = 0.01 f' = +f'Z for the primary, as required by the first equation of (4.5). The balance of terms in this example is instructive. The second term in Z for the typical case chosen brings a reduction of only 11.4% for the first plate and only 6.1% for the third plate, relatively insignificant relaxations compared with the plates for a parabolic primary. The individual plate contributions are respectively about 30, 37 and 6 times higher than the contribution of the quasi-RC primary. Since a spherical primary would give a contribution 0.25f', the contributions of plates 1 and 2 are even higher than this! Furthermore, these contributions measure the required aspheric-ity of the plates over the axial beam width. But the axial free aperture of the plates must be multiplied by (1 + Er) to avoid vignetting in the field, where Er is the E-value for the real field and aperture. Since the aspherici-ties increase with y4 compensated by a y2 term, the field surplus on the plate diameter rapidly increases the total asphericity for small g-values. We see, therefore, that the elegance of a multi-element corrector in correcting several conditions also exacts its price: the individual elements are very strong and this leads to higher order effects which limit the performance, as we shall see with practical examples.
Multiplying (4.28) by E1 and E2 respectively, gives the coma and astigmatism contributions of the first plate as r 1 f'(f' — 91 )(92 + 93) — (f' — 91)(f' — 92)(f' — 93)Z"
2 f'(f' — 91 )2(92 + 93) — (f' — 91)2(f' — 92)(f' — 93)Z
with similar expressions for the other two plates. With the values given above, the contributions of the third plate to coma and astigmatism are (Sji)3 = f'(—1.260) and (Siii)3 = f'(—23.94). The latter is thus 24 times larger than the astigmatism of the primary mirror we are correcting, again a measure of the sensitivity of the system and the fine tolerances required.
The chromatic variations of the aberrations Sj, Sjj, Sjjj will be proportional to the contributions of the individual plates but, in third order theory, will balance out as for the monochromatic aberrations themselves. From this theory, then, a set of plates of one glass will simply have the chromatic variation of the mirror contributions on the right-hand side of Eqs. (4.5). As stated above, this will only be about 10% for crown glass and the full spectral range of 1000 nm to 300 nm. Unfortunately, in practice, the situation is much less favourable because of higher order aberrations, both monochromatic, and above all, chromatic. Third order theory assumes the paraxial heights of the rays at the different elements are respected in all cases. This is not the case with strong plates: the largest plate disperses the rays to different heights on the other plates, above all the principal ray affecting E which is very sensitive for the coma and astigmatism contributions. Such 3-plate correctors are therefore essentially limited by the higher order chromatic aberrations, above all chromatic differences of coma and, to a lesser extent, spherochro-matism and astigmatism. These effects are mitigated by the balancing y2 term in the plate form, but are still determinant. The only way to eliminate these chromatic effects would be to achromatise each plate individually as an achromatic plate using 2 glasses. In practice, this is barely possible since the individual plates of the achromatic combinations would be many times stronger than their combination, also producing higher order effects apart from the technical difficulty and cost.
We shall return to this issue in comparing lens and plate correctors.
Wynne [4.5] stated in 1972 that the manufacture of such 3-plate correctors had not apparently been envisaged for the correction of a parabolic primary. I think this is still true today. The manufacture was seriously envisaged for RC and quasi-RC telescopes but, I believe, has never been carried out.
Schulte [4.11] investigated such a 3-plate corrector on the basis of Meinel's work [4.6] for the Kitt Peak 3.8 m, f/2.8 RC primary. A detailed investigation was carried out by Kohler [4.12] [4.13] and colleagues for the ESO 3.6 m, f/3.0 quasi-RC primary. This system has been re-calculated by Wynne [4.5] [4.14] for comparison with lens correctors. The original system (Wilson [4.15]) had 25% vignetting. Spot diagrams for an optimized form without vignetting and with extended spectral range were given for this 3-plate corrector plus field lens by Cao and Wilson [4.16]. These are reproduced in Fig.4.14 in a comparison with a 3-lens corrector. Since the latter gives superior chromatic performance, the 3-plate system was abandoned at the time. The relative merits are further discussed later in comparison with the lens corrector.
4.2.2 Cassegrain or Gregory focus correctors using aspheric plates
This possibility was referred to briefly in § 3.4 in connection with the addition of full-size aspheric plates in the object space of 2-mirror telescopes. We are concerned now with a real aspheric plate (or plates) near the Cassegrain (or Gregory) image which is then treated as a virtual plate in object space, as in the PF case above. For a single plate, we have from Eq. (3.220) in our simplified notation above for the fulfilment of the three aberration conditions
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