Anastigmatic Schiefspiegler: For astigmatism, we have from (3.20)
and substitution from (3.343) and (3.345) in (3.348) gives
ri ri r2
For the anastigmatic Schiefspiegler y^ Sin = (Siii)i + (Sm)2=o, giving the condition
upri ri - 2di for anastigmatism of the "axial" image.
Rutten and van Venrooij [3.12(j)] give designs and spot-diagrams for three sorts of Schiefspiegler, the first being an anastigmatic system of the Kutter type with an aperture of 110 mm working at f/26. The radii ri and r2 are identical with -3240 mm, di = -965 mm and the angles of deviation are given as 5.234° and 13.666° for the primary and secondary respectively. The ratio is 2.611 and should be same as the ratio (upr2/upri) from (3.351). With the above numerical values, (3.351) gives 2.473. The coma-free condition (3.347) gives 3.834 for the ratio. The Schiefspiegler of Rutten and van Venrooij thus lies close to the geometry given by (3.351), slightly shifted towards the coma-free condition. This is a compromise recommended by Kutter. The "axial" spot-diagram of Rutten and van Venrooij shows a residual aberration, mainly coma, comparable in size with the Airy disk, a good correction.
Equations (3.347) and (3.351) prove one of Kutter's propositions concerning the basic Schiefspiegler: that it is impossible to set the secondary at an angle upr2 which corrects both coma and astigmatism for the "axial" image. This proposition is only strictly true for the normal telescopic case with the object at infinity. For a finite object distance, the magnification of the primary provides a further degree of freedom so that geometries are possible giving correction of both coma and astigmatism. Such a system was designed by Wilson and Opitz [3.106] as a very high aperture 2-mirror condensor for use in chemical analysis equipment, in which absence of central obstruction in the pupil and achromatic performance were important. Unfortunately, this solution is not applicable to telescopes since the primary is used at a magnification |mi| > 1.
The coma-free Schiefspiegler defined by (3.347) can be made anastigmatic by adding cylindrical power to the secondary mirror, giving a toroidal surface. Since such non-rotationally symmetrical surfaces are very difficult to manufacture, this solution is rarely of interest for amateurs. Kutter proposed a more practical solution by choosing a compromise between the coma-free and anastigmatic solutions of Eqs. (3.347) and (3.351) and adding a weak, tilted planoconvex lens in the exit beam about half way from the secondary to the image. This is to correct both the residual coma and astigmatism. The chromatic aberration is small and its effect can be reduced by adding a slight wedge to the lens. Rutten and van Venrooij [3.12(j)] give design data and spot-diagrams for such a Kutter catadioptric Schiefspiegler for an aperture of 200 mm working at f/20.
A further version developed by Kutter and also analysed by Rutten and van Venrooij is the Tri-Schiefspiegler using a weakly concave tertiary which folds the beam back at right-angles to the original incident beam. Their design for aperture 200 mm at f/14.7 does not seem fully optimized. In principle, with 3 powered mirrors having free tilt values, a correction for the "axial" image for both coma and astigmatism must be possible. The secondary and tertiary are spherical, while the primary has an elliptical aspheric form to correct the spherical aberration.
Further designs for amateurs are given by Leonard in Mackintosh [3.48] under the names Yolo and Solano reflectors. The Yolo is a Tri-Schiefspiegler consisting of 3 concave spherical mirrors arranged for freedom from coma. The astigmatism is additive for concave mirrors and Leonard corrects this by making the primary or secondary, or both, toroidal. If necessary, spherical aberration is corrected by aspherising the primary in addition. The Solano reflector is also a Tri-Schiefspiegler, but with a convex tertiary. By using long radii, toroidal surfaces are avoided in this design. Further references to such systems are given by Leonard in Mackintosh [3.48] or in Rutten and van Venrooij [3.12(j)]. Of particular interest are articles by Buchroeder [3.107] and Kutter [3.108].
Finally, it should be mentioned that a perfect Schiefspiegler can be made as an excentric (off-axis) section of a centered, 2-mirror telescope, as shown in Fig. 3.95. If the centered telescope form is an RC form, the Schiefspiegler will have the same field correction, but improved by the reduction to the small excentric aperture. Although this form is theoretically simple and gives optimum imagery, it is of little interest, in practice, for amateur telescopes because the off-axis aspherics are asymmetrical and extremely difficult to make. If the mirror powers are not too high and the off-axis angles are kept to a minimum, as Leonard proposes in his Schiefspiegler, the off-axis sections can be approximated by toroidal surfaces.
The pure form of Fig. 3.95 is sometimes used in spectrograph design at the professional level to overcome central obstruction problems.
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