The astigmatic lines have the length 25r/ast m.

The angular aberration (diameter) of the round image at best astigmatic focus is

(Su'n)ast m = -Sl11 rad = -Sl11 (206 265) arcsec , (3.208)

p' ' n'yi n'yi again on the assumption, as for the coma case, that the field angle upri contained (as u^rl) in the expressions for Sm is given in radians.

Similarly, the image diameter produced by the field curvature effect IprIm is

n yl or in angular measure

(SvL)fc m = - (2Sl11 + SlV) (206 265) arcsec (3.210)

3.3.4 Distortion (SV)

The distortion coefficient Sv was not calculated in Table 3.3 or considered in the general formulations because it has little significance for the one- or two-mirror telescopes we have dealt with. Distortion can become more important if elements are introduced near the image, as we shall see in Chap. 4. In any event, the coefficient Sv can be readily calculated from the formulae in Eqs. (3.20) and (3.21).

The wavefront form of distortion is very useful as the starting formula for calculating its value but is not, in itself, a very practical measure. Distortion is not an error of quality of an image point: it is simply in the wrong place compared with the linear scale of the system as defined by its focal length f' according to Eq. (2.102). So the wavefront aberration corresponds simply to a wavefront tilt whose value depends on the cube of the field size (see Eq. (3.21).

Depending on its sign, distortion will be either "barrel" or "pincushion", as shown by the distortion of a square field in Fig. 3.22.

For astrometric work, fixed distortion in a system can normally be calibrated out. If large distortion is present, a more serious consequence is photometric. The pincushion distortion of prime focus correctors, for example, leads to a light intensity fall-off with increasing field radius, an effect which can be a nuisance with non-linear detectors like photographic plates.

Fig. 3.22. Distortion: (a) barrel, (b) pincushion

3.3.5 Examples of conversions

It is instructive to consider examples of the above conversion formulae for some of the cases of Table 3.3.

The angular aberrations, which are normally the most practical form for astronomical telescopes because of the direct comparison with the seeing, are independent of the size (linear scale) of the telescope and depend only on the f/no, field angle and aberration coefficients. Since upr1 is expressed in radians, the dependence on upr1 in the case of astigmatism brings, for practical field angles, a large reduction factor compared with the linear factor upr1 in the case of coma.

The angular aberrations are given in Table 3.10 for a 1-mirror telescope with spherical primary (Case 1 of Table 3.3), a classical Cassegrain (Case 3) and an RC telescope (Case 4). In each case, the telescope has a final image beam of f/10 and the semi-field angle Upri is 30 arcmin or (1800/206 265) rad. For the Cassegrain cases 3 and 4, m2 = — 4 as in Table 3.3.

The signs are the opposite of those of the coefficients Sj, Sjj, Sjjj if n = +1, i.e. for the two 2-mirror cases. This simply reflects the fact that a positive wavefront aberration corresponds to a negative ray height change Sr/' in the image plane. The sign is most important in the case of the asymmetric aberration coma, where a positive value in Table 3.10 with n = +1, and a negative coefficient Sjj in Table 3.3 imply that the point of the coma patch is directed towards the field centre as in Fig. 3.4 (a).

3.3.6 Conversions for Gaussian aberrations

Although the two first order (Gaussian) aberrations are not usually treated in books on geometrical optics as "aberrations", because they can be removed by small positional adjustments, they are, nevertheless, in practical telescopes

Table 3.10. Angular spherical aberration, coma and astigmatism for three telescope cases of Table 3.3, with an f/10 image beam and a semi-field angle upri of 30 arcmin

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