As a typical example of practical values, the ESO New Technology Telescope has D = 3500 mm and an f/11 focus, giving f = 38500 mm. Then S = 5.36 arcsec/mm and S = 0.187 mm/arcsec.

The f/no, denoted by N for the final image of a compound telescope and N1 for the primary image, is given by

According to our sign convention, N is negative for a Gregory telescope and positive for a Cassegrain. Ni is a negative quantity. See Tables 2.1 and 2.2.

In § 2.2.5 above, the use of Gregory or Cassegrain systems in the afocal (Mersenne) form (Fig. 1.3) was mentioned. Such systems obey the same magnification laws as the conventional afocal refractor, given in § 2.2.4, whereby the compression ratio C is equal to the angular magnification m from Eq. (2.45). The principal use of such afocal forms is as a supplementary feeder system equivalent to afocal supplementary lens systems placed before an objective in photography. These increase the effective focal length. If the magnification of the afocal system is m = C from (2.45), then the emergent field angle upr = mupr and the objective forms an image of height mn instead of n . Now from (2.101) mn = muprf , so for the real object field upr the effective focal length has become mf . In the astronomical telescope afocal feed system, the purpose may also be to increase the effective aperture of the system being fed. A limitation of such afocal Gregory or Cassegrain feed systems is the unfavourable exit pupil position near the secondary combined with the field angle magnification.

For amateur use on a small scale, a conventional positive ocular may be used to observe the real image of the reflecting telescope. This will again follow the laws of the afocal refractor. Since the exit pupil of the Gregory or Cassegrain reflector is near the secondary, the ocular is working under similar conditions to those of a refractor of modest focal length.

2.2.7 "Wide-field" telescopes and multi-element forms

In this chapter, we have considered only the basic forms of telescope, all invented in the seventeenth century. Later forms with more elements and "wide field" forms also have design aspects requiring Gaussian optics. Above all, the theory of pupils is essential. From § 2.2.5, it is clear that there is already a major complication in the general Gaussian properties in advancing from one powered element to two separated powered elements. More complex forms are best treated by tracing a paraxial aperture ray and a paraxial principal ray through the system and deriving from these general formulae on the basis of aberration theory. (Alternatively, recursion formulae are given in § which enable the determination of the paraxial parameters required for any number of centered reflecting surfaces). This is the subject of Chap. 3.

3 Aberration theory of telescopes

3.1 Definition of the third order approximation

The limits of the theory of Gaussian optics were defined in Chap. 2 by the expansions of Eqs. (2.13) and (2.14):

All terms above the first were neglected. Snell's law of refraction reduces to a linear law in the Gaussian region. This gives, then, the theory of centered systems to the first order. If the next term is considered, we have third order theory. This is concerned with the lowest order terms which affect the quality of the image, whereas Gaussian optics is only concerned with its position and size. The aberrations affecting image quality are generally most important in the third order approximation, so the aberration theory of telescopes is mainly concerned with this approximation. The theory of higher order aberrations is extremely complex. It has not played any appreciable role in telescope development, since the exact total effect of all orders can be easily calculated by ray tracing using modern computers. But this in no way reduces the value of third order theory which remains essential for a correct understanding of the properties of different telescope forms.

In Eq. (2.13), the polynomial defines the form either of a refracting or reflecting surface, or of a wavefront defining the quality of an axial image point. We saw in § 2.2.2 (Fig. 2.3) that a perfect image point is associated with a perfectly spherical wavefront. Aberration theory is concerned with phase errors from this perfect spherical form which fall within the third order region. The rays are the normals to the wavefront, as shown in Fig. 3.1.

W = AA' is the wavefront (phase) aberration of the ray ABC passing, if W =0, through the image point Io on the principal ray E'OIo . The ray actually cuts the principal ray at B and the image plane at C. Its longitudinal aberration is BIo and its lateral aberration CIo . All three forms have their uses but W is, physically, the most meaningful. The lateral aberration is related to the angular aberration 5u' Interpreted, using the magnification i3

Fig. 3.1. Wavefront, longitudinal and lateral aberration

laws, in object space, this is particularly important in telescope systems. We shall give conversion formulae between W and 5u'p.

Let us consider now the precise forms that Eq. (2.13) assumes for the sphere with circular section and for other conic sections, all with axial symmetry. Let the z-axis define the direction of axial symmetry and the ordinate y the height in the principal section.

The equation of a circle referred to its pole is 1/2

Expanding this: 2 1 y4

or zSph

with c = 1/r. For a spherical surface, therefore, the constants of its section a\ and a2 in (2.13) are:

The first (Gaussian) term gives the equation of a parabola, but the nature of the conic section is only defined by the second term. If it is zero, the surface section really is a parabola with vertex curvature c. For an ellipse, the equivalent equation to (3.1) is z = a — a 1 —

y2 b2

where a and b are the semi-axes. The expansion gives

16 b2

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