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than those given from third order theory above. This could only be true in very extreme cases because such higher order form deformations are also converging functions following the general aberration function: the higher order wavefront effects are far lower than the third order effect of any powered optical element. The only clear exception to this general rule is the case of an element without optical power (plane mirror or plate) which is aspherized only with higher order terms.

3.7.2.2 Angular decenter of the secondary (rotation about its pole).

The contribution (Sji)s,s,2 leading to the second term in (3.352) of the Schief-spiegler analysis of decenter above gave directly the effect on coma of rotating the secondary through the angle upr This expression can be written for the general rotation angle upr2 of the secondary, positive in the sense shown in Fig. 3.96, as

With

With our paraxial substitutions given above for m2 = — r, y2 /y1 = Ra and (y1/r1)2 = m2/16N2, this reduces immediately to in \ 3 f RA(m2 - 1) ,, (Comat)2,rot = - 16 f -n-Upr2 (3.375)

By definition in the derivation of this formula, the stop is considered as being at the secondary mirror, so the result of (3.375) is independent of bs2, the form of the secondary. If upr2 is expressed in radians, the angular coma in arcsec is given by (206265/f')(Comat)2rot. Expressing upr2 in arcsec, we have, more directly

(5U'p)0oraa] ^ = - ^ ^ ^ 1 (upr2)arCsec arcsec (3.376)

For the geometry of Table 3.2, N = 10 as in Table 3.20 and a rotation upr2 = 60 arcsec, the angular coma according to (3.376) is only 0.380 arcsec, identical for the Cassegrain and Gregory forms apart from the sign (negative for the Cassegrain). Since 60 arcsec is quite a coarse rotation, causing a pointing change of

2upr2L/f' = 27 arcsec or almost two orders of magnitude more than the Comat effect, it is clear that this source of decentering coma is far less sensitive than the lateral (translation) decentering coma given in Table 3.20, values corresponding to S =1 mm for a 1 m telescope at f/10.

3.7.2.3 Coma-free (neutral) points. For any 2-mirror telescope system, there will be a point somewhere on its axis for which the translation coma (Coma()j and the rotation coma (Comat)2rot will cancel out. This is called the coma-free point or neutral point for coma. Another important neutral point, independent of the shape of the secondary, is the neutral point for pointing, which is always at the centre of curvature of the secondary.

If the distance behind the secondary (towards the light) to the coma-free point is denoted by —zcfp , then the translation S of the pole of the secondary by a rotation through upr2 is

Inserting this in Eq. (3.362) and combining the resulting relation with the rotation coma of Eq. (3.375), we have for the total coma in the general case

(Comat)i