impossible in practice. This result means that the correction of astigmatism requires a lens which, with a real glass with n' ~ 1.5 and in this position, would over-compensate the field curvature. Of the other conditions, (Sii)cor = 1 f' can be met by suitable bending from (4.52); but (Si)cor = 0 cannot be formally met if K = 0, which is in conflict with the field curvature condition. However, the axial beam width is sufficiently small, with small g-values, that the residual (Si)cor may be acceptable.
The same formulation applies to any lens corrector which is "thin", whatever the number of lenses involved. In particular, it applies to a "thin" doublet, substantially afocal except for the balance of power to correct the field curvature, if a single glass is used. This is clear if we write for the doublet from (4.65) and (4.66)
which lead to the same equation (4.72). However, the doublet will permit the formal correction of (Si)cor = 0 and (Sii)cor = 1 f' because of its extra degrees of freedom. But the limitation of (4.72) remains. It can be met by shifting the corrector about 2 2 times further from the image to give f '/g ~ 14. This is, in principle, possible for a doublet (though it may cause problems in practice), but would lead to problems of spherical aberration with a singlet lens. There are also chromatic aberrations which we shall deal with later.
There are two other ways we may relax the situation: by introducing glasses with different refractive indices n' for the basic wavelength (different dispersions will help us with the chromatic conditions anyway, but at the cost of secondary spectrum); and by departing from the classical Cassegrain form with one or both of the mirrors.
Consider first 2 different glasses in a doublet corrector. Eqs. (4.73) become
as the condition for Siv = 0. This relaxes the requirement of (4.72), as will be shown in detail in the more important application to RC telescopes, below. The different dispersions in a "thin" system also permit the correction of both chromatic aberrations C1 and C2, at the cost of secondary spectrum.
The other possibility, to allow a variation of the aspheric constants on the mirrors, leads to a quasi-classical Cassegrain. Suppose this mirror system still corrects (J^ Si)rei = 0, but introduces changes of coma ASii and astigmatism ASiii which will be functions of £. (The corrector must be capable of producing (S/)cor = 0, which is possible with a doublet; with a singlet near the image the value would not be zero but would be small). Then (4.68) becomes with a "thin" corrector:
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