One further special case is worth noting, that of annulling the spherical aberration for rays passing through the lens in both directions. By proper choice of glass and curvatures this can be accomplished to a close approximation and the resulting form is shown in Fig. 56. The front of the crown is notably flat and the rear of the flint conspicuously curved, the shape in fact being intermediate between Figs. 52b and 52c. The type is useful in reading telescopes and the like, and for some spectroscopic applications.
Fig. 56.—Corrected in Both Directions.
There are two well known forms of aberration not yet considered; astigmatism and curvature of field. The former is due to the fact that when the path of the rays is away from the axis, as from an extended object, those coming from a line radial to the axis, and those from a line tangent to a circle about the axis, do not come to the same focus. The net result is that the axial and tangential elements are brought to focus in two coaxial surfaces touching at the axis and departing more and more widely from each other as they depart from it. Both surfaces have considerable curvature, that for tangential lines being the sharper.
It is possible by suitable choice of glasses and their curvatures to bring both image surfaces together into an approximate plane for a moderate angular space about the axis without seriously damaging the corrections for chromatic and spherical aberration. To do this generally requires at least three lenses, and photographic objectives thus designed (anastigmats) may give a substantially flat field over a total angle of 50° to 60° with corrections perfect from the ordinary photographic standpoint.
If one demands the rigorous precision of corrections called for in astronomical work, the possible angle is very much reduced. Few astrographic lenses cover more than a 10° or 15° field, and the wider the relative aperture the harder it is to get an anastigmatically flat field free of material errors. Astigmatism is rarely noticeable in ordinary telescopes, but is sometimes conspicuous in eyepieces.
Curvature of field results from the tendency of oblique rays in objectives, otherwise well corrected, to come to shorter focus than axial rays, from their more considerable refraction resulting from greatly increased angles of incidence. This applies to both the astigmatic image surfaces, which are concave toward the objective in all ordinary cases.
Fortunately both these faults are negligible near the axis. They are both proportional to tan²{u}/f where u is the obliquity to the axis and f the focal length; turn up with serious effect in wide angled lenses such as are used in photography, but may generally be forgotten in telescopes of the ordinary F ratios, like F/12 to F/16. So also one may commonly forget a group of residual aberrations of higher orders, but below about F/8 look out for trouble. Objectives of wider aperture require a very careful choice of special glasses or the sub-division of the curvatures by the use of three or more lenses instead of two. Fig. 57 shows a cemented triplet of Steinheil’s design, with a crown lens between two flints. Such triplets are made up to about 4 inches diameter and of apertures ranging from F/4 to F/5.
Fig. 57.—Steinheil Triple Objective. Fig. 58.—Tolles Quadruple Objective.