Fig. 4.33. Points in the field for the calculation of spot-diagrams, from Wang and Su [4.65]

Fig. 4.34. Spot-diagrams for lensm corrector type I for a field diameter of 45 arc-min with a 7.5 m, f/2 paraboloid. Rotation angles of the lensms are 0o, 0o, i.e. zero dispersion. Circle diameter = 1 arcsec. Reproduced from Wang and Su [4.65]

0.76 and 0.85 arcsec, the maximum being 1.11 arcsec. The field covered is somewhat larger than that of Wynne and Worswick (Fig. 4.29) who are using one more "lens" in the total system and a less steep paraboloid. But these authors used no aspheric surface.

The variations in quality of the lensm correctors of Wang and Su with different rotations (dispersions) are very small.

Figure 4.35 shows another example with lensm corrector type II for the same field of 45 arcmin diameter and zero dispersion. The mean diameters of the spot-diagrams for 434, 350 and 1014 nm respectively are 0.35, 0.87 and 0.96 arcsec with a maximum of 1.35 arcsec. Adding an aspheric to the

Fig. 4.35. Spot-diagrams for lensm corrector type II for a field diameter of 45 arcmin with a 7.5 m, f/2 paraboloid. Rotation angles of the lensms are 0o, 0o, i.e. zero dispersion. Circle diameter = 1 arcsec. Reproduced from Wang and Su [4.65]

Prism 1 Prism 2

Prism 1 Prism 2

Fig. 4.36. Optical design of the LADC - schematic (after Avila, Rupprecht and Beckers[4.78])

first (larger) lensm (type III lensm corrector) improves these values to 0.33, 0.65 and 0.63 arcsec with a maximum of 0.97 arcsec. The same system at maximum dispersion gives 0.33, 0.66 and 0.65 arcsec with a maximum of 1.07 arcsec, a very minor increase.

The extra lens element of types II and III seems to bring little gain over type I. Since ADC in this system is achieved with the minimum number of normal corrector elements, it seems a most interesting solution. It should be noted that the front meniscus lens has a diameter only 9% of that of the primary and g/\f[ | = 0.08. This relatively low value accounts for the effectiveness of the aspheric surface, as discussed above.

Finally, it should be said that ADC does not necessarily have to be introduced as part of the telescope optics. If slit or fibre instruments are used, it will then have to be done before the image plane, but such an ADC can still be seen as part of the auxiliary instrument and can be directly adapted to its field requirements. This solution was adopted, for example, in the ESO NTT.

This symbiosis of telescope and instrument is illustrated in another form by the more recent development for the ESO VLT unit telescopes when combined with the two FORS (Focal Reducer and Spectrograph) instruments [4.77] at the Cassegrain focus. For direct imagery, but above all for spectroscopy of extended sources and for multi-object spectroscopy where the slits cannot be aligned along the direction of the atmospheric dispersion, it is not possible to include the ADC in the instrument. The size of such telescopes of 8 m aperture makes the classical solutions discussed above, which were developed for telescopes of half this size or less, both difficult and expensive. Above all, the necessary size of the dispersive prisms was prohibitive for the elegant double-prism pair (zero deviation) system of Wynne. Apart from the high cost of the individual prisms with significant prism angles y, the absorption of the two glasses with appreciable thicknesses would be a major problem. Furthermore, because cementing of the prism pairs in sizes of the order of 0.5 m would be too dangerous, oiled contact surfaces would be essential. But this would also be technically problematic.

A solution better adapted to such large telescopes was proposed at ESO by J. Beckers and designed in detail by G. Avila and G. Rupprecht [4.78]. This system has been called the "Linear Atmospheric Dispersion Corrector" (LADC). It consists of two single prisms, but with significant separation as shown in Fig. 4.36. The whole unit is about 900 mm in diameter and has a length of 1570 mm. It is part of the Mi cell/M3 tower construction. The figure arbitrarily defines the red principal ray emerging from the telescope exit pupil as being on the axis of the telescope. Since the two prism angles y are identical and reversed, there is a deflection e of the axis, depending linearly on the separation, but no pupil tilt. The latter property was a fundamental requirement of the VLT system. The axis shift is automatically compensated by the guide probe. No rotation is required in an alt-az mounted telescope since the dispersion is always in the altitude direction. The blue field ray is shown with vastly exaggerated field angle . Bearing in mind that the deviation of a small angled prism is (n — 1)7 if the incidence angle remains small, then the emerging rays from prism 2 are parallel to the incident rays on prism 1. The separation d is adjusted so that the emerging coloured rays focus on the telescope focal plane. In practice, prism 2 remains fixed at a suitable distance from the image plane, while prism 1 is moved along the axis. The larger the dispersion field angle, the larger d must be. Prism 1 produces the effective dispersion correction, part of which is removed by prism 2. The positive balance arises from the rule stated for the Wynne system above, that the compensating dispersion of a prism is proportional to its distance (lever arm) from the focal plane. The real role of prism 2 is therefore simply to remove image and pupil tilt. In the zenith, d = 0 and the two prisms form a parallel plate.

Since there is only one glass type, fused silica can be used giving optimum transmission, an important property bearing in mind that the system cannot be routinely removed from the telescope, even if it is not required for observations close to the zenith. A disadvantage, inherent in the system compared with the 2-glass doublet system of Wynne and Worswick, is that the matching of the atmospheric dispersion is inevitably less precise. However, in the most critical case of the high spatial resolution imaging mode with a field diameter of 3.4 arcmin, the authors show that, for the spectral range 350—850 nm and a zenith distance Z = 50°, the atmospheric dispersion is reduced from 1.00 arcsec to 0.12 arcsec at the field centre and from 1.14 arcsec to 0.18 arcsec at the field edge. The broad-band correction is therefore between 88% and 84% for a field diameter of 3.4 arcmin and the residues have, in practice, proved undetectable. Scaling factor variations of the field would be very serious for multi-object spectroscopy with FORS using slits of about 0.5 arcsec, but the variation produced by the LADC is less than 0.01 arcsec and therefore negligible. If the FORS instrument has to be moved to another unit telescope, the LADC unit must also be removed and remounted with it, since the presence of the LADC facility is considered essential for high quality FORS operation.

4.5 Focal reducers and extenders

4.5.1 Simple reducers and extenders in front of the image

In §, systems were discussed in which refracting elements were used in combinations with Mangin mirrors and lens objectives, e.g. Fig. 3.67. In this spirit, between 1828 and 1833, Barlow (see ref. [3.39] in Chap. 3) carried out experiments with liquid lenses and, as a by-product, introduced the negative Barlow lens as a device for increasing the focal length. This was a small-field telephoto system. In principle, a positive lens as a focal reducer is completely analogue; but as a "wide-angle" system for a given detector size it has problems of field aberrations that do not occur with the Barlow lens. In both cases, the system must function with a nearly telecentric pupil because of the position of the telescope exit pupil behind the secondary, far away from the Barlow or reducer (Shapley) lens.

A focal reducer or extender must operate by definition with an axially corrected telescope: otherwise it is a special form of corrector with positive or negative power.

Since a focal reducer (FR) or extender (FE) has significant power by definition, it must be achromatised in the classic sense with two different glasses. The simplest form is therefore a thin doublet achromat. This can be treated by the same thin lens theory as that given above in § for quasi-afocal doublet correctors applied to classical Cassegrain and RC telescopes. However, the situation with FR and FE is much less favourable since the total power is finite and prescribed by the magnification. In the case of RC telescopes, the situation of Eqs. (4.78) applies, where stop-shift terms are absent and a perfect FR or FE would have Si = Sn = 0 with compensation of the RC Siii and Siv residues. Then (4.82) would apply, in principle, for the correction of £ Sm and £ Siv. For available values of n| and n'2, the condition can only be fulfilled with a small negative total power of the FR or FE. This is in fundamental conflict with the prescribed finite power determined by its magnification. The direction of the FE is more favourable than that of the FR since it compensates the RC residues. But significant power (magnification) is bound to overcompensate. The required power Kfe is dependent on the distance gpE from the original telescope focus and is given from (2.8), for a "thin" FE, by mFE = ^-7^- > (4.95)

1 + KFEgFE

where gpp and mpp, the magnification, are positive quantities. Since the FE has mpp > 1, it follows from (4.95) that Kpp must be negative. In the limit case withgpp =0, mpp = 1 and the FE becomes a field lens. If Kppgpp = —1, then gpp = —fpp and 'mpp = ro, the limit case of an afocal Galilean telescope. From (4.52) we have the simple situation for a "thin" FE that

(Siii)pp « (JZ K)pp and (Siv)pp « ( K1 + K? ) , if there are no stop-

\ 1 2/pp shift terms affecting the astigmatism. The overcompensation by (Siv)pp can be mitigated for a given (^Z K)pp by choosing n!x and n'2 to be as high as possible, but there is nothing to be done about the astigmatism except to make gpp as large as possible to reduce (^ZK)pp. The larger the mpp, the worse the situation becomes. The requirement that (Si)pp = (Sn)pp = 0 is the same as that for a normal achromatic objective and can normally be met, provided gpp is not too small and mpp not too large. But the quasi-telecentric stop produces supplementary coma.

The situation for an FR with an RC telescope is the same except that the residual RC aberrations increase those of the FR instead of compensating them to some extent. Furthermore, the reduction in linear field means the angular field for a given detector size is increased, whereas it is reduced with the FE. For both these reasons, FE are more favourable for mpp = 1 /mpR than FR.

With a classical Cassegrain, the coma of the telescope produces a stop-shift term in astigmatism which introduces a further complication. As with the quasi-afocal doublets, the correction of the telescope coma may require a bending which prohibits correction of spherical aberration. For most practical cases, residual coma from the quasi-telecentric effects in the FE or FR is more serious.

Barlow lenses are very popular with amateur telescopes of small size, usually for visual use. An excellent description of the possibilities is given by Rutten and van Venrooij [4.66]. They give a design for a cemented achromat for a 200 mm, f/10 Schmidt-Cassegrain telescope (in theory, a similar situation to the RC case above) without precise indication of the glasses used. Normally, common glasses are used, but (following from the theory above) high index lanthanum glasses give better performance provided a considerable dispersion difference (Abbe number) is available. The design of Rutten and van Venrooij has the positive (flint) lens towards the incident light in agreement with Hartshorn [4.67]: the reverse order gives worse field aberrations. The authors give spot-diagrams for the combination with the Schmidt-Cassegrain, the performance being acceptable (relative to the Airy disk) over a field diameter of 40 mm for a Barlow magnification of 2 and a final focal length of 4000 mm (about ' ). The field limitation is by coma and astigmatism.

Rutten and van Venrooij also give an equivalent design for a focal reducer with magnification 0.55. Again the positive lens (crown) leads. The linear field diameter given is 20 mm (about 1°) with acceptable performance compared with a 25^m circle set by grain size for photography. Because of its "thin lens" nature, its field performance is limited mainly by astigmatism.

Of course, the design of such systems with a modern optical design program is a trivial operation. Separation of the elements gives additional design freedom (e.g. coma) at the cost of chromatic errors and increased air-glass surfaces. Such systems are more in the domain of powered correctors, dealt with above.

Simple Barlow (diverging) doublets are capable of correcting linear fields comparable with their own diameters [4.67] for moderate magnifications (« 2) on small telescopes, the diameters being of the order of a tenth of the primary (or objective) diameter. For larger telescopes, the linear field corrected for a given doublet size will be the same, but the angular field will decrease linearly with the telescope size with fixed f/ratio. If it is wished to correct a larger angular field, the doublet must be made larger and placed further from the focus, giving increased problems of chromatic aberrations (including secondary spectrum) and, possibly, spherical aberration. The performance deteriorates if the Barlow magnification increases because the total power, and hence that of the individual lenses, must increase. Similarly, if the relative aperture of the telescope beam incident on the Barlow doublet is increased, the problems of correction of spherical aberration and coma grow rapidly.

4.5.2 Wide-field focal reducers (FR) as a substitute for a prime focus

This was the subject of a study carried out at Carl Zeiss for the 3.5 m, f/3 to f/8 MPIA telescope. A resume of the results was given by Wilson [4.15]. The following properties of an ideal focal reducer were listed:

a) It should optically replace the PF, providing a similar field with similar quality over the whole spectral range.

b) It should use the normal telescope secondary (RC or quasi-RC in the case considered).

c) It should yield a convenient position of the final focus.

d) The final focus should be so arranged that the image receiving apparatus (including IR) can be used without causing unacceptable obstruction.

e) It should not cause construction problems such as a long overhang.

f) It should have as small a length and weight as possible to reduce handling problems.

g) It should have as few optical elements as possible.

h) It should contain only UV-transmitting elements.

This is a formidable list and it may be doubted whether a full solution exists. Above all, the linear field size implies at once that this general problem is quite different from the simple small-field doublet systems discussed above. In the above 3.5 m telescope, the required 1° diameter field of the f/3 PF, with corrector, gives a linear field at the f/8 Cassegrain of nearly 490 mm diameter. The magnification is 3/8 = 1/2.67, appreciably stronger than the small telescope examples given above. Wide-field focal reducers (FR) without an intermediate image. Such a system is the wide-field, large-scale Gaussian equivalent of the simple doublet solutions given in § 4.5.1. The biggest problem for the wide-field extension is the position of the exit pupil of a Cassegrain telescope far in front of the FR so that the FR is effectively working telecentrically. This situation is already a limitation with the small-field doublets discussed above, but becomes far more serious with large linear fields.

Space limitations will normally rule out mirror solutions for an FR without an intermediate image. With lenses, the diameter of the front lenses must be larger than that of the virtual image, i.e. over half a meter for the above 1° field requirement. Figure 4.37 shows schematically a basic design restricted to 5 lenses, the largest having a diameter of 654 mm, covering a field of 0.9° diameter. With such diameters, the choice of glasses is very limited. Spot-diagrams are shown in Fig. 4.38. The monochromatic correction is favoured by increasing the length and is within 0.5 arcsec. The lateral chromatic aberration C2 and the higher order chromatic aberrations are extremely serious even at modest fields. The system would only be useable over narrow spectral ranges and, even then, over a modest total spectral range because of chromatic differences of coma and astigmatism. Such linear fields cannot be covered by a practical lens system for a significant spectral range. The basic requirement a) above cannot be met. Achromatisation of the last singlet lens in Fig. 4.37, which balances the astigmatism and field curvature, would produce an improvement, but the finite thickness and powers of the individual doublets are fatal for the higher order chromatic performance.

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