Fig. 49.—Spherical Aberration of Convex Lens.
For in a convex lens with spherical surfaces the rays striking near the edge, of whatever color, are pitched inwards too much compared with rays striking the more moderate curvatures near the axis, as shown in Fig. 49. The ray a′ b′ thus comes to a focus shorter than the ray a b.
This constitutes the fault of spherical aberration, which the old astronomers, following the suggestions of Descartes, tried ineffectually to cure by forming lenses with non-spherical surfaces.
Fig. 50.—Spherical Aberration of Concave Lens.
Fig. 50 suggests the remedy, for the outer ray a″ is pitched out toward b″ as if it came from a focal point c″, while the ray nearer the center a″′ is much less bent toward b″′ as if it came from c″′. The spherical aberrations of a concave lens therefore, being opposite to those of a convex lens, the two must, at least to a certain extent, compensate each other as when combined in an achromatic objective.
So in fact they do, and, if the curves that go to make up the total curvatures of the two are properly chosen, the total spherical aberration can be made negligibly small, at least on and near the axis. Taking into account this condition, therefore, at once gives us a clue to the distribution of the total curvatures and hence to the radii of the two lenses. Spherical aberration, however, involves not only the curvatures but the indices of refraction, so that exact correction depends in part on the choice of glasses wherewith to obtain achromatization.
In amount spherical aberration varies with the square of the aperture and inversely with the cube of the focal length i.e. with a²/f³. It is reckoned as + when, as in Fig. 49, the rim rays come to the shorter focus, as-, when they come to the longer focus.
In any event, since the spherical aberration of a lens may be varied in above the ratio of 4:1, for the same total power, merely by changing the ratio of the radii, it is evident that the two lenses being fairly correct in total curvature might be given considerable variations in curvature and still mutually annul the axial spherical aberration.