in which the chromatic coefficients are, by analogy with Eq. (3.20):
Longitudinal chromatic variation of focus: (longitudinal colour)
Chromatic variation of wavefront tilt: (lateral colour)
Chromatic variation of spherical aberration: (spherochromatism, Gauss-error)
Chromatic variation of coma (colour coma): Chromatic variation of astigmatism: (colour astigmatism)
The second group of terms in (3.222) are identical wavefront functions to the monochromatic aberrations Si , Sii, Sm. Note that we have omitted the chromatic variation of Siv , although in the general use of refracting optics, for example in photographic objectives, this term may not be negligible. It is omitted here because, as stated above, the total optical power of refracting elements in modern telescopes is normally small. The Petzval sum is correspondingly small and its chromatic variation a negligible factor. By contrast, we shall see in Chap. 4 that the chromatic variations of spherical aberration and, above all, of coma and astigmatism, are often the limiting factors of corrector systems.
The first group of terms in (3.222) are the chromatic variations of the first order (Gaussian) terms in Table 3.1. While these monochromatic terms can be focused out by movement of the secondary mirror in a Cassegrain telescope or by a change of pointing, their chromatic variations given in (3.222) cannot be eliminated. They are therefore serious factors limiting the performance if a wide spectral band is employed. For narrow spectral bands they can be "focused" or "pointed" out. This is not possible for the chromatic aberrations of the second group.
The second term, the lateral colour, produces - for a centered optical system with symmetry axis - a radial spectrum of a point image whose length depends on the coefficient C2 and the field radius rf. Atmospheric dispersion is a similar aberration phenomenon produced by the atmosphere except that the spectrum is radially symmetrical to the zenith point in the sky. This will be dealt with in Chap. 4.
(SSii)c = Ev AAyA (^) + Ev HE(SSI)c (SSm)c = Ev A2yA (Ut) + Ev(HE)2(5S*i)c
The chromatic aberrations (SSj)c, (SSjj)c, (SSjjj)c in (3.223) each consist of two terms, as for the monochromatic aberrations of (3.20). The first terms result from the optical power of lenses, as do the first order aberrations C1 and C2. The second terms (SS|)c, (SSn)c and (SS*iu)c are due to aspheric surfaces. For aspheric plates, these terms are the only chromatic effects. From (3.221) we have
SSI = 8(n' - n)ay4 = 8A(n)ay4 , the symbol A in all these formulae meaning the change (n' — n) causing refraction. The chromatic variation is then
in which Sn = (n'2 — n[) — (n2 — n1) for the wavelengths X2 and Ai in the two media before and after refraction. In the case of an aspheric glass plate in air, for which the index is taken to be 1 with negligible dispersion, the quantity A(Sn) is simply the dispersion (n' — ni) of the glass. An analogue interpretation applies to the quantities A () and A (^ ■ "J1) in the terms for spherical surfaces with optical power.
Since the stop-shift effect due to dE on C2 was not given in Eqs. (3.22), it has been added to the equation for C2 in (3.223). The equivalent stop-shift terms for (SSjj)c and (SSjjj)c have been omitted, as they are the same as those given in (3.22).
3.6 Wide-field telescopes
3.6.1 The symmetrical stop position: the Bouwers telescope
The problem of making a satisfactory primary had so dominated the development of the reflecting telescope that it was axiomatic for about 270 years that it would also form the stop (pupil) of the system, since this used its surface with maximum efficiency. Recognition that field aberrations (coma, astigmatism and distortion) followed the formulation of third order aberration theory by Seidel in 1856 [3.4], and many of his successors involved above all in the development of photographic objectives, led to deep understanding of the significance of pupil position relative to the constructional elements of the system. It therefore seems almost amazing that nobody before B. Schmidt in 1931 [3.31] considered the possibilities of stop shift into the object space in front of the primary. Schwarzschild and Chretien, both concerned to correct field aberrations to produce wider fields and in possession of the necessary aberration theory, failed to realize this simple possibility.
Figure 3.25 shows the fundamental form of a wide-field telescope without correction of spherical aberration. It consists of a stop at the centre of curvature of a spherical mirror. The principal rays are normals to the sphere. The system has no axis except that of the stop. Its field with constant image
quality is 180°, only limited by vignetting of the stop. If the stop were rotated about its centre to remain normal to a principal ray of varying field angle, the theoretical field is 360°, though the complete spherical mirror would, in practice, block the entry of any light. The system has only two aberrations, field curvature and a spherical aberration, which is identical for all field directions apart from stop vignetting. This reduces the beam width in the t-section giving an asymmetrical effect on the spherical aberration looking like astigmatism; but the system has no astigmatism in the real sense. It is also free from coma. It has the field curvature of the concave primary equal to its focal length, giving an image surface on a sphere also centered on the stop. These properties are obvious from the geometrical symmetry of the arrangement. Schmidt's great achievement was to recognize this: the addition of a device to correct the spherical aberration was simply a corollary, although technically very difficult to achieve.
The equations of Table 3.4 must lead to the same conclusions. Setting bsi =0 for the spherical primary, we have
Now spr1 = 2f[, both being negative. This gives at once:
Now spr1 = 2f[, both being negative. This gives at once:
This result is also evident from the basic equations (3.20) since A = 0 for principal rays normal to the mirror at all field angles.
It is often useful to deal with all 1-mirror and 2-mirror solutions with one single set of formulae. The 1-mirror case can be treated with the 2-mirror formulae of Table 3.5 by treating it as the limit case where the secondary becomes a folding flat with m2 = — 1 (Fig. 2.15). Then £ = 0 and, with bs1 = 0, Z = — 1. The equations become:
f spr1 f f
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