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+2Ei(Sii)li + 2E2(Sii)l2 + E2(Si )li + E|(Si )L2 = 0

Reduction as before leads to the condition for e Siii = e Siv = 0:

in which two additional terms appear compared with the case of a "thin" corrector represented by (4.79). Since E1 and E2 are large, we will again assume the approximation that the contribution (Si) l1 to E Si is negligible and ignore this term. Assume, as before, E2 = 35 and that E1 = 20 giving (E2 — E1) = 15. Substitution of the values of Table 3.2 as before then gives for n' = 1.5:

So, separation of the elements of a doublet of a single glass also converts (4-79) into a form with a real solution for a strict RC telescope. However, we must remember the law, first pointed out by Ross in connection with field correctors, that a separated doublet of a single glass cannot correct both chromatic conditions Ci and C2, even if they are afocal. In this case, they are not afocal so chromatism is anyway present with one glass.

The above theory can readily be extended to a corrector consisting of three separated thin lenses. By analogy with the 3-aspheric plate theory of § 4.2.2, a solution of the three conditions E Sj = E Sjj = E Sjjj = 0 must also exist with thin lenses. Because of the greater parametric freedom of lenses (power and bending), E Sjv = 0 can also be corrected. This applies to both strict RC and classical Cassegrain telescopes. The third lens resolves, in principle, the conflict between Sj and Sjj correction which exists for classical Cassegrain telescopes. But, as with plates, high individual lens powers may emerge from the solution which may lead, in practical cases, to limitations of higher order aberrations. Aspheric surfaces may help here, as indicated by Paul [4.4] or Wynne [4.23].

4.3.2.3 Summary of the results of the above theory. As Wynne [4.5] has pointed out, thin lens theory becomes less accurate for elements close to the image. Nevertheless, the above theory reveals clearly by Eq. (4.79) that, for a strict RC, a "thin" corrector of a single glass cannot provide a solution of all four conditions for a real glass, irrespective of the number of elements. For such a strict RC telescope, a real monochromatic solution can be achieved at once for 4 conditions by one of the following devices, using a doublet corrector:

a) Maintain the strict RC form and introduce 2 different glasses into a thin doublet corrector with as large a difference of refractive index as reasonable. The chromatic aberrations can then also be corrected at the cost of secondary spectrum.

b) Establish a quasi-RC form of the 2-mirror system to deliver some coma to the thin doublet corrector consisting of one glass. Some chromatic aberration is inevitable but there is no secondary spectrum. This solution also applies to singlet lens correctors.

c) Maintain the strict RC form and the single glass doublet but introduce a finite separation between the lenses. With the single glass, the chromatic residuals will tend to be worse.

Of course, combinations of (a) and (b), or (a) and (c) are also possible. The former was the solution chosen by Violette [4.3].

The situation for a classical Cassegrain corrector is similar for the above relaxation possibilities. However, there is an important difference. A solution is theoretically possible for a thin, single glass doublet if the distance g from the image is chosen to give a feasible refractive index according to (4.72). But for modern values of m2, this may be an unacceptably large value of g. Furthermore, as has been mentioned above and shown by Wynne [4.23]

[4.42], the bending of the doublet elements to produce the coma correction normally introduces too much spherical aberration. These problems can be resolved by using three separated lenses.

4.3.2.4 Third order theory for Cassegrain telescopes without correction of (£ Si)Tel. In the above theory, we have considered normal Cassegrain telescopes delivering an image from the 2-mirror system corrected for spherical aberration, i.e. (£ Si)Tei = 0 or acceptably small. This also enables the mirror system to work without corrector. It will always be the case if a corrector is added to an existing telescope.

If a telescope is designed always to work with the corrector, this restriction is not necessary, giving the following general formulation for a thin, single-glass doublet corrector:

ESlii = (SlIl)Tel + (Slll)cor + 2E (Sll)cor + E2(Sl) cor = 0

Here (Sii)Tel will include a stop-shift term in (Si)Tel, and (Siii)Tel includes stop-shift terms in (Sii)Tel and (Si)Tel. The required corrector aberrations are

(Siii)cor = — [(Siii)Tel + 2E(SH)cor + E2(Si) cor] = f" (Kli + KL2))

(Si)Tel, (Sii)Tel and (Siii)Tel are given from the defined values of Z and £ from Table 3.5. Then, as before, the requirement for £ Siii = £ Siv = 0 gives

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