3.2.4 Analytical (third order) theory for 1-mirror and 2-mirror telescopes General definitions. The results of Table 3.3 are derived directly from the values of the paraxial ray traces. Before commenting on the results, it is better to develop the general analytical theory of such telescope systems, giving the third order aberrations in terms of basic system parameters without the need to trace paraxial rays.

Table 3.3 normalized the system data with y1 = +1, upr1 = +1, f = ±1, rf = ±1, H = n'u'rf = —1

Bahner [3.5] derives the analytical formulae using this normalization, a practice which has frequently been followed and produces a simplification in the derivation. However, by omitting the parameters y1 and f', defined as 1, this simplified formulation is no longer dimensionally correct. It also leads to confusion between the parameters y2 and L, which have the same numerical value with the normalization from Eq. (2.58). Furthermore, if the paraxial relations of Chap. 2 are applied appropriately, the general (i.e. non-normalized) procedure becomes quite reasonable. This is now given, the form of the relations being similar to that of Bahner, but in generalised form, and with sign changes arising from our use of the strict Cartesian sign convention of modern optical design. The relations are thus perfectly general if the sign convention of Eqs. (2.36) to (2.38) and of Tables 2.1 and 2.2 is rigorously followed.

Equations (3.20) can be re-written in the form

E Sjj = Ev (A )S0 + Ev (HE)Sf = Ev S0n + Ev Sfj E Sjjj = Ev (i )S0 j + Ev (HE)Sf j = Ev S0jj + Ev Sfjj E Sjy = — E,H 2Pc

E Sy = Ev (A )(S0jj — H2Pc) + Ev (HE)Sfjj in which S0, S0j, ... give the contributions due to a spherical surface, Sf, Sf j, ... those due to the aspheric form. The quantities A, y and u) for S0 are derived from the paraxial aperture ray with Eqs. (2.36) to (2.38), while the multipliers A and HE require A and ypr from the paraxial principal ray. In Tables 3.2 and 3.3, the entrance pupil was defined for the normal case as being at the primary mirror, i.e. spr1 = 0 in Eq. (2.36). However, this limitation is not acceptable for a general formulation, for which the entrance pupil is at spr1 from the first surface (primary). If the entrance pupil is to the left of the primary (i.e. in front of it), spr1 is negative.

The other quantities required for the evaluation of (3.24) are the aspheric parameter t , the Petzval curvature Pc and the Lagrange Invariant H.

Because practical use of telescopes is almost always confined to objects effectively at infinity, we retain this limitation here. However, the informed reader may prefer the general recursion formulae given in §, below, for calculating the aberrations of any system of n mirrors under general conditions. 1-mirror telescopes: single concave primary. The term "1-mirror telescopes" implies, of course, a single mirror with optical power and covers all folded forms with plane mirrors.

The paraxial ray trace equations (2.36) to (2.38) give with the definitions above Eq. (3.19):

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