This should be compared with Eq. (4.9), the equivalent expression for the single (Gascoigne) plate correcting a hyperboloid for spherical aberration and coma. Eq. (4.21) for the 2-plate corrector gives

with g1 = f'/10, g2 = f'/20. The Gascoigne plate with g = f'/10 gives from (4.9)

The 2-plate corrector correcting Si, Sii and Siii therefore requires a much stronger hyperbola for the spherical aberration correction than the Gascoigne plate and this places it normally outside the useful range of RC or quasi-RC telescopes. In other words, its use in the PF requires a mirror combination unfavourable for field correction at the Cassegrain focus.

There remains one interesting possibility: a 2-plate corrector with a parabolic or hyperbolic primary correcting E Si and E Sii only, without ESiii. From (4.5) for the general case with a hyperbolic primary, we have the conditions

giving

Clearly a positive value of Z (i.e. bs1 < -1 with m2 = -1) reduces the power of the plates compared with the case of a parabolic primary with Z = 0. From the third equation of (4.5) the astigmatism is given by

Substituting for E from (4.4) gives or

E SJJJ = 1 / '2 — / 'Z(/' — gi)(/' — g2) (4.25)

Finally, substituting for Z from Table 3.5 with m2 = —1 gives the form equivalent to that given by Gascoigne [4.9]:

E Sjjj = 1 / '2 + / (1 + bsi)(/' — (i)(/' — (2) (4.26)

Gascoigne gives an approximate form for the second term on the reasonable assumption from practical application that g1,g2 ^ /'. Simplifying in this way, the second term of (4.26) reduces to

/' (1 + b )(/' — (i)(/' — (2) — /'3 (1 + bsi) — (1 + bs1)- —--

With this approximation and taking into account a factor for different normalization, the second term of Gascoigne should have a factor 8 in the denominator which is missing in [4.9]. This is important for the general conclusions of the potential of correctors consisting of two aspheric plates. Taking (1 = /'/10, (2 = /'/20 and bs1 = —1.03 as a typical RC value, the approximate form of (4.26) gives

i.e. the compensating second term is only 1/10 of the first term, which gives the astigmatism for such a system with a parabolic primary. This modest compensation is a consequence of the small departure from the parabola of RC primaries and confirms the conclusion from (4.21) that a much larger departure from the paraboloid (bs1 = —1.351) is required to correct all three aberrations with a 2-plate corrector with the above values of (1, (2. Without the factor 8 in the denominator of his second term, Gascoigne's formula gives 80% compensation by the second term which would indicate the 2-plate corrector can give excellent correction of all three aberrations with modern RC primaries. Unfortunately, this is not the case.

With the above values of (1, (2, the first term of (4.26) gives Sjjj = 15/' for a 2-plate corrector and a parabolic primary correcting Sj = J^Sjj = 0. Using (4.12) we can compare this with the astigmatism of a single plate corrector of similar mean size (( = 0.075/') correcting a hyperboloid, giving Sjjj = 7.17/', less than half that of the 2-plate corrector with parabolic primary. However, from (4.9), the corresponding eccentricity of the primary for ( = 0.075/' in the singlet case is bs1 = —1.162, much more favourable than a normal RC case. If bs1 = —1.03 for a typical RC, the singlet Gascoigne plate requires from (4.10) ( = 0.0148/' and gives Sjjj = 34.33/' from (4.12), over twice that of the parabola, 2-plate case using much larger plates.

The general conclusion is that the PF corrector with 2 aspheric plates is only of practical interest for a special PF telescope with hyperbolic primary of high eccentricity (bs 1 ~ -1.35). This was the suggestion of Paul [4.4]. It is not a practical solution for parabolic or normal RC primaries, even if larger plates are used, because of the residual astigmatism.

4.2.1.3 Three corrector plates. We can apply the 3 equations of (4.5)

directly to correct the three conditions S1 = Sii = Siii = 0 for any value of Z. Meinel [4.6] and Wynne [4.5] give the solution for the parabolic primary and subsequent work, which we shall consider in more detail, has extended the application to RC and quasi-RC primaries. We can derive the general solution from (4.5) which is

Si = f' [{2(E2 + E3) + 1} — E2E3C] , (E1 — E3XE2 — E1)

with equivalent expressions for S2 and S3. Substituting for E from (4.4) reduces (4.27) to c g2 T!f'(g2 + g3) — (f' — g2)(f' — g3)Z'

If the second term of the numerator is set to zero for a parabolic primary, (4.28) is the same as the result given by Wynne [4.5] apart from small differences of normalization. If the suffixes 1,2,3 are taken in the direction of the light, then 1 refers to the first (largest) plate with the largest g. The denominator in (4.28) is therefore negative. For S2 it is 'positive and for S3 negative. Since the numerator is positive in all cases, these are the signs of the aspherics on the three plates: 1 and 3 as for a Schmidt plate, 2 the opposite like a Gascoigne plate for a hyperbolic primary. Figure 4.4 shows this situation schematically for 1-, 2- and 3-plate correctors. For 2-plate correctors the sign of the front plate will depend on whether the conditions Sii = Siii = 0 are corrected for a strongly hyperbolic primary (positive front plate); or Si = Sii = 0 for a parabolic or RC primary (negative front plate).

Theoretically, a solution of (4.5) exists for any plate spacing and any asphericity of the primary; but small values of g and small differences in the 3 values will lead to very high plate asphericities. Meinel suggests a maximum g of the order of 0.15/'. Let us take, as an example, g1 = 0.15/', g2 = 0.10/', g3 = 0.05/' and bs1 = —1.04 for a typical quasi-RC giving Z = +0.01 from (4.8). Eq. (4.28) then gives

Fig. 4.4. Schematic appearance of aspherics on PF plate correctors: (a) Gas-coigne plate (singlet) corrector for RC hyperbolic primaries; (b) 2-plate corrector for strongly hyperbolic primaries correcting y) Si = E Sii = E Siii = 0; (c) 2-plate corrector for parabolic or RC primaries correcting E Si = E Sii = 0; (d) 3-plate corrector for parabolic or RC primaries correcting E Si = E Sii = E Siii = 0

Fig. 4.4. Schematic appearance of aspherics on PF plate correctors: (a) Gas-coigne plate (singlet) corrector for RC hyperbolic primaries; (b) 2-plate corrector for strongly hyperbolic primaries correcting y) Si = E Sii = E Siii = 0; (c) 2-plate corrector for parabolic or RC primaries correcting E Si = E Sii = 0; (d) 3-plate corrector for parabolic or RC primaries correcting E Si = E Sii = E Siii = 0

Transposing suffixes 1 and 3 gives for the third plate

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