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Fig. 4.2. Transformation of a real corrector plate to a virtual plate in object space

Fig. 4.2. Transformation of a real corrector plate to a virtual plate in object space aspheric constant a is defined by Eq. (3.221) as

SS} = 8(n' - n)ay4 , which is reduced by the factor (g/f ')4, where g is the distance of the plate from the image. The wavefront aberration contributions of the real and virtual plates are identical over the respective axial beam widths.

In the PF case, the paraxial transformation for the virtual plate is simply spi = f' (, (4.2)

in which f' = | f' |, the focal length of the primary, and g is defined as a positive quantity with g < |f'11. If, for our PF case, we place the fictitious plane secondary in contact with the primary, we can substitute in Eqs. (3.220) above m-2 = -1, L = f, di =0, £ = 0, uPr i =1 giving, with our usual normalization1 of y1 = f', the conditions E Si = -f'Z + SS} = 0

E S// = - 2 f/ + f SS} = 0 E S/// = f/ + ( f")2 SS} = 0

It should be noted that this normalization with y1 = f ', thereby eliminating the factors (y1/f') in various powers in Eq. (3.220), is only valid in the general case if the conditions are corrected to zero, as in Eq. (4.3). This must be borne in mind for cases where a telescope is deliberately not corrected for spherical aberration (E Si) because of desired compensation of the aberration in a subsequent instrument.

As explained in § 3.4, if Si = Sii = 0, the stop position in the system has no effect on the three Seidel aberrations and may be considered to be at its normal position, the primary. Then, with our normalization f' = y 1 = Upr 1 = 1 , it follows at once that

from (4.2) and simplify the notation with

(SSI)V = Sv for aspheric plate number v in a multiple plate corrector. Then (4.3) becomes for a 3-plate corrector:

The quantity Z defines the asphericity of the primary according to Table 3.5

z=^m3 (1+bs1), in which m2 = —1 in our PF case. If Z = 0 in (4.5), we have the conditions for the correction of a parabolic primary, essentially the same as those given by Wynne [4.5] with a slightly different normalization.

4.2.1.1 One corrector plate. The first two equations of (4.5) immediately enable us to prove the first fundamental property given by Paul [4.4]. For a parabolic primary (i.e. a conventional Newton telescope) and a single corrector plate:

The only solution requires E = ro or, from (4.4), 9 = 0. The plate is in the image plane and would require infinite asphericity. So a single plate at a finite distance 9 can correct the coma of the parabolic primary but introduces, from (4.4), spherical aberration e SI = S. = 2 ( f—g) <«>

If the primary is allowed to assume a non-parabolic form giving a real solution in the case of a single plate corrector, we have from (4.5)

ES1 -giving z=2 