9 This sign definition for upr2, giving upr2 = upr1, is in apparent conflict with the definition in Fig. 3.96 which corresponds to upr2 = — upr1. The reason for this sign reversal is the following. Figure 3.96 gives the signs in our normal Cartesian ray tracing system, as applied to centered systems. But the Schiefspiegler is not a normal system, in so far as the application of our third order equations is concerned, with regard to the field parameter A1 = upr1 and A2 = upr2. In our Schiefspiegler theory from Eqs. (3.344) to (3.347) and also in our present case, the secondary is treated as a "second primary", but the light direction is reversed because of ni = —1. In the Z-form of the Schiefspiegler this brings a partial compensation of the coma, i.e. the signs of the coma introduced by the two primaries are opposite. This is particularly obvious in the case of the Czerny-Turner Z-form arrangement, where there is complete compensation of the two terms in Eq. (3.339), and a compensatory (subtractive) effect is obvious from the two halves of the system working as "telescopes" in opposition in one light direction. By contrast, in the so-called U-form of layout, the two components of the coma are additive, so this form should be avoided in practical systems.

If the sign reversal of the field angle is not introduced in the Schiefspiegler (Z-form) treatment, although it is, in fact, necessary and logical, the additive law of third order aberrations y) Sii = (Sn)1 + (Su)2 would have to be changed to a subtractive law with negative sign. But this would not be correct, since the third order treatment for the aperture parameters can be taken over without change from the normal centered case.

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