Then, applying the recursion formulae (3.336) to the aperture ray as in the derivation of (3.352) from (3.342) and (3.343), we have, from (3.20)

(sii)V = โ ^ (1 โ ^ (m2 โ 1)^ + ,ypr2 , T2 (3.372)

This coma is the total coma since the primary has no contribution. Converted to (Comat) from (3.196), it leads, using the definition of t given above (3.19), to

TxJ r2 Vr2

This can be easily reduced using the paraxial relations given above (3.356) and the further relation y2 = (m-2 + 1) yi r2 = 2 f' , derived from (2.90) and (2.72). Inserting also ypr2 = Eq. (3.373) then gives

identical with our general result above of Eq. (3.362).

This derivation of the lateral decentering coma formula using a telecentric beam on the secondary and the recursion formulae (3.336) is certainly one of the most direct and simple.

Lateral decentering coma has been dealt with in detail because of its capital importance in practical telescope technology. Technical aspects, connected with telescope alignment and maintenance, will be discussed in RTO II, Chap. 2 and 3. The theory given above is based on third order theory as expressed by Eqs. (3.20). It may legitimately be asked how accurate this theory is in practice. The answer is, in most cases, remarkably accurate. Except in extreme cases of aperture and field, which are rare in most applications of telescopes, the aberration function given in Table 3.1 is a rapidly converging function. This is particularly the case for lateral decentering coma in respect of field effects, defined by the decenter which must be small in practice to give acceptable coma limits. It follows that only higher terms in aperture can normally be significant; but the convergence law will still apply.

Modern optical design programs permit decentering of elements both by lateral shifts and by rotations, enabling exact calculations of decentering aberrations. Coma will always be by far the most important term because of its third order linear dependence on field compared with the quadratic dependence of astigmatism. However, for modern telescopes using the edge of the field and particularly for anastigmatic telescopes, the effects of decentering astigmatism at the edge of the field may be significant. In such cases, freedom from decentering coma is no longer a sufficient condition. This is also true for modern, high quality, active telescopes like the ESO Very Large Telescope (VLT), for which the image analyser of the active optics system is near the edge of the field. Asymmetries in the astigmatic field produced by decentering astigmatism may then give wrong results for the astigmatism correction at the field center. In the VLT, therefore, decentering astigmatism is corrected resulting in virtual elimination of lateral decenter, even though this is not necessary for the correction of decentering coma. This whole matter is dealt with in detail in ยง 2.2.1 of RTO II.

It is sometimes asserted that higher order variations from strict conic sections defined by Eq. (3.11) will lead to more severe centering tolerances

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