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This can be compared with the form of Eq. (3.75). The form (3.81) is singular in the case of the parabola.

It is a matter of individual preference, but I personally prefer the basic form of Eq. (3.11), as derived by Schwarzschild [3.1], and the derived form using bs of Eq. (3.75) which leads naturally to the general aspheric form of (3.77). The only weakness of the conventional (Schwarzschild) definition of 6s is revealed by Eq. (3.11): the actual amount of the aspheric third order contribution Az to the sagitta z is given by c3 1

where Ns = D = 2'D is the aperture number if the surface is used as a telescope mirror with an infinite object. Az is therefore proportional to the inverse cube of the aperture number Ns. This means, for a given aperture height y, that the aspheric contribution Az for an f/1 parabola with 6s = -1 is the same as for an f/10 hyperbola with 6s = -1000. It is important to bear the aperture number of the surface in mind when interpreting the practical significance of a given value of 6s.

3.2.5.2 Higher order aberrations, the Abbe sine condition and Eikonals. In § 3.2.1, the "orders" of the monochromatic aberrations arising out of Hamilton's Characteristic Function were discussed and the terms of the first three orders (first, third and fifth) were given in Table 3.1. The number of terms in an order increases quite rapidly as the order increases. Ray tracing embraces, by definition, all the orders, within the accuracy given by the computer. With modern computers, therefore, the effect of all the higher orders is given at once from the difference between ray tracing results and third order calculations, the latter being converted to lateral aberrations for comparison by the formulae of § 3.3.

After the formulation by Seidel [3.4] for third order aberrations in 1856, there was considerable interest in developing the theory of higher orders, particularly to allow the explicit calculation of fifth order effects. Another area of intensive research was concerned with formulations for the total optical path through an optical system. In 1873, Abbe [3.9] discovered what is perhaps the most important relation of this type covering all the orders in aperture of coma (but only the third order in the field): the Abbe sine condition. Abbe's work was concerned with imagery in microscopes, but the condition had been discovered in another form by Clausius [3.10] in connection with energy transfer from thermodynamic considerations. The sine condition for freedom from coma is expressed in the normal (focal) case by sin U' sin U

u' u in which U and U' are the aperture ray angles to the axis of finite rays in the object and image space respectively and u and u' the paraxial equivalents. In the afocal case, the equivalent condition is

in which the aperture ray angles are replaced by the aperture ray heights. For the normal telescope case of an object at infinity and a finite image distance, the sine condition is simply

The Abbe sine condition refers to the so-called linear coma, i.e. the sum of all coma terms in Table 3.1 which depend linearly on the field parameter:

ComaLm =i k3i<rp3 cos 0 +i k5i<rp5 cos 0 +i k7iap7 cos 0 + ...

It is not trivial to prove that ComaLin, so defined and when set to zero for its correction, is the equivalent of the Abbe sine condition of Eq. (3.83). The proof is given by Welford [3.6] and includes the derivation of the Staeble-Lihotzky condition. This condition expresses the requirement for zero linear coma in the presence of uncorrected spherical aberration. Only if the latter is zero does the Staeble-Lihotzky condition reduce to the Abbe form of Eq. (3.83), which therefore applies only to aplanatic systems corrected for both spherical aberration and linear coma.

Equation (3.83) shows that the concept of principal planes, introduced in § 2.2.1 (Fig. 2.1) in the treatment of Gaussian optics for the paraxial domain, is no longer valid if coma is to be corrected with finite apertures. Eq. (3.83) requires that the planes be replaced by spheres, concentric to the object and image, as shown in Fig. 3.2. The spheres are, of course, simply wavefronts defined as passing through the principal points P and P'. The sine condition confirms what one would intuitively expect from the wavefront concept of image formation.

Following Abbe, up to the effective implementation of computers, much elegant mathematical work was carried out in the hope of developing higher order aberration theories and techniques of calculating optical path lengths by extensions of the theorems of Fermat and Malus (see § 2.2.2). This led to the "Eikonal" of Bruns in 1895 and the modification (Winkeleikonal) used by Schwarzschild [3.1]. Accounts of this work and complete literature references are given by Czapski-Eppenstein [3.11(a)] and by Herzberger [3.7]. Modern computers, with their efficiency in ray tracing, have reduced the significance of this theory. Nevertheless, for a deeper understanding of the function of optical systems, many aspects retain great value, but it cannot rival third order theory in its significance and simplicity.

Modern developments in telescope optics involving active optics, as described in Vol. II of this work (RTO II, § 3.5) and relating particularly to the ESO New Technology Telescope (NTT) and Very Large Telescope (VLT), also make extensive use of aberration theory. However, in the basic layout of the active support systems for thin primary mirrors, higher order aberration theory is a useful rather than essential tool, since Zernike polynomials (§ 3.9) or natural vibration modes (RTO II, § 3.5.4) provide the information required. But classical higher order theory has assumed considerable significance in the general theory of decentering and in practical techniques for the alignment of the elements of modern telescopes (RTO II, § 2.2.1).

Of the earlier work on higher order aberration theory given by Herzberger [3.7], the most notable in the modern context is that of Buchdahl [3.154]. Buchdahl developed systematically the concept of quasi-stable parameters for a given order, equivalent to the use of paraxial parameters in third order theory. Probably the definitive treatment of this subject was given by Focke in 1965 [3.155], although its assimilation requires much hard work. Fock-e's aim is a modernised treatment unifying the theories of Schwarzschild, T. Smith [3.156] and Herzberger. T. Smith's classic paper investigated the dependence of aberration coefficients of any order on the object and stop positions. Focke gives explicit formulae, based on the Schwarzschild angle Eikonal and the Smith stop-shift formulae, for refraction at a surface which can be spherical or aspheric to the fifth order. The formulae cover all third order and fifth order terms, separated into spherical and aspheric parts. He also derives the Herzberger formulae, with the link to the Schwarzschild coefficients, and shows that they are in exact agreement with those of Buchdahl and have a simple form for spherical surfaces. This simplicity is lost for aspheric surfaces and Focke does not develop the Herzberger treatment for this case.

More recently de Meijere and Velzel [3.157] have analysed the dependence of third and fifth order aberration coefficients on the definition of the pupil coordinates. They show that the normal definition leads to 9 independent fifth order terms, in agreement with Table 3.1, whereas two other definitions lead to 12 independent fifth order terms. This is of considerable historical interest, because Petzval [3.11] [3.151] stated clearly already in 1843 that there were 12 such terms (see Footnote in § 3.2.2).

3.2.5.3 System evaluation according to geometrical optics from ray tracing. Traditional representations of system quality according to geometrical optics were based on limited information from hand-traced rays, usually as transverse aberrations (intercept differences in the image plane). Such representations can still be very revealing and are still used [3.8]. However, fast computers have led to the universal use of spot-diagrams as a measure of geometrical image quality. They were introduced by Herzberger [3.7] [3.37]. To generate a spot-diagram, the entrance pupil is (usually) divided up into a rectangular grid with square mesh and a finite ray is traced through the system for each mesh intersection point [3.12]. Sometimes, other pupil divisions are used, such as concentric circles, but this gives less complete sampling of the pupil. The number of rays required will depend on the number of aberration orders present in the system and usually lies in the range 50 - 300. The usual spot-diagram (SD) representation is a matrix of points, the columns representing the different field heights with the SD for the axis at the bottom, while the rows represent different wavelengths (Fig. 3.3). Each matrix represents an image plane at a chosen axial focus point. Full understanding of the image potential often requires several such SD-matrices for different focus points.

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