(W )Fi = -5(5Wi*) gives from Eqs. (3.243), (3.244) and (3.256) the compensation condition d =( n^)^, (3-257)

if n' — 1.5. The sign of this compensation effect is important. The Schmidt primary has undercorrection of spherical aberration. To correct this, the Schmidt corrector plate introduces compensating overcorrection which is stronger in the blue, weaker in the red. The filter plate adds further overcor-rection, thereby pushing the corrected wavelength of the total system further into the red. Since d > 0 in all real cases, addition of filters will therefore always worsen the spherical aberration correction for wavelengths on the UV-side (shorter) of the central correction wavelength A0. In deciding on the value of A0 for a new Schmidt telescope, the mean filter thickness should be taken into account. If red filters are thicker than blue ones, this will improve the total correction.

3.6.2.4 Field aberrations of a Schmidt telescope. As stated in § 3.6.2.1, the Schmidt telescope is corrected monochromatically for S j, S jj and S jjj but still has, from its symmetrical nature, a field curvature equal to the focal length of the primary (Fig. 3.27). As in the Couder telescope, this can be corrected by a field-flattening lens. This device was first suggested by the Scottish astronomer Piazzi Smyth in 1874 [3.35]. In Eq. (3.20), the factor y for the aperture beam height appears in S j, S j j and S j j j: at the image therefore, a powered surface has no effect. However, it does affect the Petzval sum S jv . In the case of the Schmidt telescope with a field curvature in the same sense as the mirror and equal to 1/f', the error can be corrected by a convex lens placed as near to the image as is technically acceptable. The general form of the refraction (or reflection) expression given in Eq. (2.33) can be written for a thin lens in the form n' n ' 1 1

For a plano-convex lens in air, the power K is

The Petzval sum of such a field-flattening lens is, from the definitions preceding (3.20),

For compensation of the field curvature of a Schmidt telescope of focal length f[, we require

Such field-flattening lenses (singlets) are not free from other optical aberrations, in particular transverse chromatic aberration (C2 in Eq. (3.223)) and distortion. In practice, because the lens has a certain finite distance from the image surface, there is also some coma. If the whole system is designed to include the field flattener, compensations are possible such that the only effective disadvantage is a slight increase in chromatic aberrations. A detailed treatment is given by Linfoot [3.29].

For larger Schmidt telescopes with f' > ca. 0.5 m, it is possible to bend the photographic plates to the required spherical form. Alternatively, film can be used.

Figure 3.27 shows that the plateholder inevitably obstructs the incident beam. This limits the free field according to the obstruction ratio Dph/D, where D is the telescope diameter and Dph that of the plateholder:

where (upr)max is the semi-field in radians and N the f/no. Eq. (3.262) shows that large fields are only possible with reasonable obstructions if N is small, i.e. with high relative apertures. In practice, plateholders take up a larger angle than the free field. The diameter Di of the primary required to avoid vignetting of the field is

In most practical cases, Di/D ~ 1.5. This relation assumes, of course, that the pupil is at the corrector plate. Although this is the normal case, it should be remembered that the correction of the terms Si, Sii and Sm makes the Schmidt telescope very insensitive to stop shift (see § 3.4). Nevertheless, there are good reasons for retaining the basic geometry. Any shift of the stop from the corrector will increase the necessary diameter of this most difficult optical element. Also field beams pass asymmetrically through the plate giving chromatic variations over the field.

There are two obvious physical sources of higher order field aberrations. Firstly, an oblique beam traverses the plate with an increase of optical path length of

SW' = n' (-^ - l) = n' (1 - cos UPr ) ^ n'% , (3.264)

compared with unity for the path length of equivalent axial rays. Secondly, there is a projection effect of y/ cos upr in the effective height of the plate for oblique beams because the corrector does not rotate to be perpendicular to the principal rays as it is for the axis. Bahner [3.5] gives the formula

f1' 16n'N3

for the angular aberration diameter produced by these two effects together, where Upr is the semi-field angle in radians, n' the refractive index of the corrector and N the f/no. With upr = 3°, N = 3, n' = 1.5, Eq. (3.265) gives a diameter of 0.87 arcsec. Much work was done in the pre-computer era on the analytical theory of higher order aberrations of the Schmidt telescope, for example by Baker [3.36] and Linfoot [3.29]. Although this work retains great interest for specialists, the universality of computers and sophisticated optical design programs has reduced its practical importance: higher order aberration effects are revealed exactly by ray tracing and spot-diagram representations [3.7] [3.37] (see also § 3.2.5.3). Third order theory, by contrast, will always retain its validity and importance because it reveals the fundamental possibilities and limitations of a given system.

Figure 3.29 shows spot-diagrams of the theoretical image quality of the ESO 1 m, f/3.0 Schmidt telescope with its original singlet corrector plate, plotted over the field for 24 cm x 24 cm plates bent to the optimum field curvature (±3.20° field).

3.6.2.5 Tolerances. Tolerances and alignment procedures will be dealt with in detail in RTO II, Chap. 2. Suffice it to say here that the Schmidt telescope is a favourable design from the point of view of the tolerances on position of the corrector plate except for lateral decentering. It will be shown in RTO II, Chap. 2 that the tolerances in this respect are a function only of the primary mirror and the distance of any secondary element (mirror or plate) from it. Sag or bending of the corrector plate is completely uncritical, since the optical path in the glass remains unchanged. Tilt of the corrector shifts the axis of the corrected image away from the centre of the photographic plate, leading to field asymmetry, but this is also not critical. The tolerance on the axial position of the plate can be deduced directly from Eqs. (3.225) in which spr1 is set to 2/' + <5z instead of 2/'. It is obvious that the coma introduced is <5z/2/' of that of the uncorrected primary, i.e. <5 z/4.

3.6.2.6 The achromatic Schmidt telescope. The price paid for the great gain in field coverage of the Schmidt telescope is its physical length and the spherochromatism introduced by transferring the asphericity on the primary (reflecting) to the singlet corrector plate (refracting). The latter can be virtually entirely removed if the singlet corrector is replaced by a doublet. This is the same principle as an achromatic doublet objective with the important

Fig. 3.29. Spot-diagrams for the ESO 1 m, f/3.0 Schmidt telescope with the original singlet corrector plate. Optimum curved field of radius 3050 mm and ±3.20° field for 24 cm x 24 cm plates to

Fig. 3.29. Spot-diagrams for the ESO 1 m, f/3.0 Schmidt telescope with the original singlet corrector plate. Optimum curved field of radius 3050 mm and ±3.20° field for 24 cm x 24 cm plates difference that the doublet objective is concerned to correct the first order longitudinal chromatic aberration C i of (3.223), whereas the achromatic Schmidt corrector is concerned to correct (SSi)c, the third order chromatic effect on Si. We saw, from (3.244) that the spherochromatism of the singlet plate is given by

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