## O

365 436 546 656 1014

Fig. 3.3. Normal representation of spot-diagrams

For the given focus, each matrix represents the field plane for the various wavelengths on the chosen (arbitrary) scale, on which the SD give the geometrical point-spread-functions (PSF) in a vastly magnified form. Their scale is normally indicated by a circle whose diameter is given in arcsec for telescope systems, to permit direct comparison with the seeing. Catoptric systems, with only reflecting elements, will have a single-column spot-diagram matrix for a given focus. In such catoptric systems, we shall show a "through-focus" matrix in which the rows represent different focus shifts from the nominal focus.

In this book, the primary purpose is to present the theory and properties of various telescope forms in a coherent way. Examples of achievable quality will be given in typical cases with spot-diagrams. For more complete comparisons with SD, the reader is referred to Rutten and van Venrooij [3.12] and Laux [3.8].

3.2.6 Analytical expressions for a 1-mirror telescope and various forms of 2-mirror telescopes (Classical, Ritchey-Chretien, Dall-Kirkham, Spherical Primary)

3.2.6.1 Introduction. The results derived in § 3.2.4 will now be used to derive the basic properties of 1-mirror telescopes and various common forms (Cassegrain and Gregory) of 2-mirror telescopes. The theory is given in wave-front aberration, whose conversion to other forms is given in § 3.3.

It should be noted that the classical forms of reflecting telescope only corrected one aberration: spherical aberration, giving image correction on the axis. The necessary theory was given by Descartes in 1634. This theory was sufficient for the long focal lengths and small visual fields for nearly three centuries! The problems lay not in lack of theory, but in manufacture, test x arcsec x arcsec methods, mirror support, the mechanics of mountings and operation. Only in 1905, impelled by the field demands of the slow, but increasingly dominant photographic emulsions and the dramatic advances of the reflecting telescope due to Ritchey (see Chap. 5), came the revolutionary work of Schwarzschild [3.1] [3.13]. This introduced third order theory into reflecting telescope design, in a remarkably complete form, and laid the way for all modern designs taking field correction into account.

Schwarzschild proved, using his definition of the asphericity of Eq. (3.11), that two aspheric mirrors in any geometry with sufficient space between them could correct two Seidel aberrations. I call this the "Schwarzschild theorem". The two lowest terms from the point of view of field dependence (<r) in Table 3.1 are S/(<r0) and S//^1). Schwarzschild's work thereby opened the way to aplanatic telescopes, a term introduced by Abbe [3.9] for optical systems corrected for S/ and S//.

We shall see in §§ 3.4 and 3.6.5 that the generalization of the above Schwarzschild theorem [3.13] is of central importance in assessing the possibilities of telescopes containing more elements than two aspheric mirrors and is, indeed, the basis of all modern telescope solutions.

3.2.6.2 1-mirror telescopes. Since plane mirrors introduce no aberrations, the single centered concave mirror (i.e. one mirror with optical power, or "powered" mirror) covers all the normal prime focus (PF) cases: direct PF, Newton focus and folding-flat Cassegrain (Fig. 2.15). The aberrations are given in Table 3.4.

Since any telescope must be free from (S/)1, the first equation gives the requirement bs1 = -1, the parabolic form. A spherical primary gives

The second term in the bracket

for the coma (S/i)1 is also zero, which shows that the coma is independent of the stop position spr1 because (S/)1 is zero, in accordance with the stop-shift formulae of Eqs. (3.22). The coma is then given by

For a spherical mirror, or any non-parabolic form, the result of Eq. (3.87) only applies if the stop is at the primary; otherwise, the general form of Table 3.4 must be used.

For the astigmatism, the third formula reduces, for a parabolic form, to

This is not independent of the stop position spri because the coma is not zero: only the last term of the stop-shift formula in (3.22) vanishes because of zero (S/)i. As with the coma, if the stop is at the primary, the astigmatism is independent of its form.

If the normalization of Table3.3 is used with yi = + 1, /i = —1, upri =+1 and the stop is placed at the primary with spri = 0, then (S/)i = 0, (S//)i = — 0.5 and S/// = + 1.0 as shown, in agreement with Eqs. (3.87) and (3.88).

Historically, there is another form of 1-mirror telescope which is not axi-ally centered: the Herschel form (Figs. 1.1 and 2.10). The central part of the observed image then corresponds to the field angle upri of the mirror tilt. This leads to the coma and astigmatism of Eqs. (3.87) and (3.88). Because of the dependence on the first and second power of upri respectively, the astigmatism is negligible for small tilts and the coma is dominant.

3.2.6.3 2-mirror telescopes in normal Cassegrain or Gregory geometry The results of § 3.2.4.3 enable us now to derive analytical expressions for the third order aberrations of various 2-mirror solutions. In this section, we shall confine ourselves to those solutions based on normal Cassegrain or Gregory geometry: other 2-mirror systems will be treated in the following section. The theory given here will enable us to understand the (normalized) third order values of Table 3.3, derived from Eqs. (3.20) and paraxial ray traces. The conversion to lateral or angular aberration is given in § 3.3. At this stage, we are only concerned with the relative values of the third order coefficients arising from different telescope solutions.

It should be remembered that the formulae we have derived above are completely general for 2-mirror telescope forms. It follows that the derivations below apply to both Cassegrain and Gregory forms. Which form applies will depend on the signs of the parameters m2 and / (negative and positive respectively for Cassegrain; positive and negative respectively for Gregory -see Tables 2.2 and 2.3), and on the values of di and L which are negative and positive respectively in both cases.

a) Classical telescope: classical Cassegrain and Gregory. These forms are defined by a primary mirror of parabolic form (bsi = —1) giving a prime focus corrected for (S/)i. From Eqs. (3.30) and (3.31) it follows that

Similarly, the secondary mirror must also contribute zero spherical aberration (SI)2. From Eqs. (3.41) and (3.43), we have

## Telescopes Mastery

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