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system with finite conjugates on both sides (Fig. 2.5), the normal definition is the transverse magnification as defined in Eq. (2.23). A compound telescope (Gregory or Cassegrain) is afocal on the object side but usually gives images at a finite distance both from the primary and from the secondary. In the Gregory case, the corresponding image heights n1 and r/2 are negative and positive respectively, giving a negative transverse magnification. In the Cassegrain case, both heights are negative, giving a positive transverse magnification. This is the definition used in Schroeder's book [2.6]. However, both forms - and in modern times particularly the Cassegrain because of its predominant use - can be used in the afocal mode invented by Mersenne (Fig. 1.3). We shall see in Chap. 3, from their remarkable aberrational properties as both aplanatic and anastigmatic telescopes, that the Mersenne telescopes may be seen as the fundamental forms of the reflecting telescope. We recall, too, that the original refracting telescope of Fig. 2.8, with ocular, was also afocal. Above all, since modern Cassegrain telescopes have relatively small secondaries with high telephoto effects, the aplanatic Ritchey-Chretien form only departs minimally from the Mersenne afocal form: the primaries of such telescopes have a hyperbolic form of eccentricity only slightly higher than the Mersenne parabola. For generality, therefore, it seems desirable to define the magnification of the secondary in a way that is also valid in the afocal case. The obvious parameter to use would be the ratio of the field angles, as in Eq. (2.41) for the afocal refractor. Unfortunately, in the focal case, the angular magnification of the secondary defined as upr2/upr1 deviates from the transverse magnification n2/n1 because of the discrepancy between the exit pupil (and maybe the entrance pupil) and the principal planes. However, the signs emerging from this definition are instructive: with our Cartesian ray trace system of Table 2.2, the angular magnification of the secondary is negative in the Cassegrain telescope. But since the angular magnification of the primary is also negative (normally -1), the total angular magnification is positive. This is why, if we were to look at the moon through an afocal Mersenne Cassegrain telescope, using the secondary as an ocular as Mersenne proposed, we would see the moon upright, albeit with minute angular field because of the unfavourable exit pupil position (Fig. 1.3).

We see, therefore, that the sign of the magnification of an optical arrangement in general or a Cassegrain secondary in particular, can depend on the measuring parameter chosen. Nevertheless, the discrepancy of the values given by the definitions in the focal case between transverse image heights and field angles is not acceptable. Fortunately, there is an excellent alternative which removes this discrepancy and is also applicable to the afocal case. This definition of the magnification of the secondary m2 is simply the ratio of the final to the primary focal length, a ratio which is anyway fundamental to telescope designers. This also gives a direct measure of the telephoto effect. With this definition, we have for the focal case from Eq. (2.24) and the Lagrange Invariant of Eq. (2.27)

in which the normal transverse magnification ratio n2/n1 is multiplied by n2/n1. Now in Table 2.2 for the Cassegrain case n2 is positive and n1 is negative while, as mentioned above, r/2 and n1 are both negative. It follows also that f1 is negative, with f positive. The definition of Eq. (2.55) therefore gives the same value for m2 as the transverse magnification, but with reversed, that is negative sign in the Cassegrain case. Bahner [2.11] also defined m2 as f /f1, but he used a non-Cartesian sign convention of special form, not suitable for general application, giving f1 as positive. But other non-Cartesian ray tracing systems, such as those of Conrady [2.12] or Hopkins [2.13], give f1 as negative, in agreement with the strict Cartesian system. Since the definition of Eq. (2.55) can be applied perfectly generally to both focal and afocal cases, this has been adopted for this book. The resulting negative value for m2 in the Cassegrain case then inevitably produces discrepancies in the equations for the aberration theory compared with other treatments such as that of Schroeder. It is not a question of right or wrong, but a question of preference. The choice according to Eq. (2.55) has been deliberately made with the aim of following an absolutely consistent Cartesian scheme throughout this book, also giving preference to the focal lengths as fundamental parameters rather than the transverse image heights. Correspondingly, of course, m2 is positive for the Gregory telescope.

One could define the telephoto effect relative to the primary simply as

but the true telephoto effect T in the normal photographic sense of the equivalent focal length relative to the constructional length L is given by

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