the light generally applied (e.g. white light), which is dispersed by refraction, and monochromatic (Gr. μóνος, one) aberrations produced without dispersion. Consequently the monochromatic class includes the aberrations at reflecting surfaces of any coloured light, and at refracting surfaces of monochromatic or light of single wave length.
(a) Monochromatic Aberration.
The elementary theory of optical systems leads to the theorem; Rays of light proceeding from any “object point,” unite in an “image point”; and therefore an “object space” is reproduced in an “image space.” The introduction of simple auxiliary terms, due to C. F. Gauss (Dioptrische Untersuchungen, Göttingen, 1841), named the focal lengths and focal planes, permits the determination of the image of any object for any system (see Lens). The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis (the symmetrical axis of the system) are infinitely small, i.e. with infinitesimal objects, images and lenses; in practice these conditions are not realized, and the images projected by uncorrected systems are, in general, ill defined and often completely blurred, if the aperture or field of view exceeds certain limits. The investigations of James Clerk Maxwell (Phil.Mag., 1856; Quart. Journ. Math., 1858, and Ernst Abbe[1]) showed that the properties of these reproductions, i.e. the relative position and magnitude of the images, are not special properties of optical systems, but necessary consequences of the supposition (in Abbe) of the reproduction of all points of a space in image points (Maxwell assumes a less general hypothesis), and are independent of the manner in which the reproduction is effected. These authors proved, however, that no optical system can justify these suppositions, since they are contradictory to the fundamental laws of reflexion and refraction. Consequently the Gaussian theory only supplies a convenient method of approximating to reality; and no constructor would attempt to realize this unattainable ideal. All that at present can be attempted is, to reproduce a single plane in another plane; but even this has not been altogether satisfactorily accomplished, aberrations always occur, and it is improbable that these will ever be entirely corrected.
This, and related general questions, have been treated—besides the above–mentioned authors—by M. Thiesen (Berlin. Akad. Sitzber., 1890, xxxv. 799; Berlin. Phys. Ges. Verh., 1892) and H. Bruns (Leipzig. Math. Phys. Ber., 1895, xxi. 325) by means of Sir W. R. Hamilton’s “characteristic function” (Irish Acad. Trans., “Theory of Systems of Rays,” 1828, et seq.). Reference may also be made to the treatise of Czapski-Eppenstein, pp. 155-161.
A review of the simplest cases of aberration will now be given. (1) Aberration of axial points (Spherical aberration in the restricted sense). If S (fig.5) be any optical system, rays proceeding from an axis point O under an angle u1 will unite in the axis point O′1; and those under an angle u2 in the axis point O′2. If there be refraction at a collective spherical surface, or through a thin positive lens, O′2 will lie in front of O′1 so long as the angle u2 is greater than u1 (“under correction”); and conversely with a dispersive surface or lenses (“over correction”). The caustic, in the first case, resembles the sign > (greater than); in the second < (less than). If the angle u1 be very small, O′1 is the Gaussian image; and O′1 O′2 is termed the “longitudinal aberration,” and O′1R the “lateral aberration” of the pencils with aperture u2. If the pencil with the angle u2 be that of the maximum aberration of all the pencils transmitted, then in a plane perpendicular to the axis at O′1 there is a circular “disk of confusion” of radius O′1R, and in a parallel plane at O′2 another one of radius O′2R2; between these two is situated the “disk of least confusion.”
The largest opening of the pencils, which take part in the reproduction of O, i.e. the angle u, is generally determined by the margin of one of the lenses or by a hole in a thin plate placed between, before, or behind the lenses of the system. This hole is termed the “stop” or “diaphragm”; Abbe used the term “aperture stop” for both the hole and the limiting margin of the lens. The component S1 of the system, situated between the aperture stop and the object O, projects an image of the diaphragm, termed by Abbe the “entrance pupil”; the “exit pupil” is the image formed by the component S2, which is placed behind the aperture stop. All rays which issue from O and pass through the aperture stop also pass through the entrance and exit pupils, since these are images of the aperture stop. Since the maximum aperture of the pencils issuing from O is the angle u subtended by the entrance pupil at this point, the magnitude of the aberration will be determined by the position and diameter of the entrance pupil. If the system be entirely behind the aperture stop, then this is itself the entrance pupil (“front stop”); if entirely in front, it is the exit pupil (“back stop”).
If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their “perpendicular height of incidence,” i.e. their distance from the axis. This distance replaces the angle u in the preceding considerations; and the aperture, i.e. the radius of the entrance pupil, is its maximum value.
 Fig. 5. |
(2) Aberration of elements, i.e. smallest objects at right angles to the axis.—If rays issuing from O (fig. 5) be concurrent, it does not follow that points in a portion of a plane perpendicular at O to the axis will be also concurrent, even if the part of the plane be very small. With a considerable aperture, the neighbouring point N will be reproduced, but attended by aberrations comparable in magnitude to ON. These aberrations are avoided if, according to Abbe, the “sine condition,” sin u′1/sin u1=sin u′2/sin u2, holds for all rays reproducing the point O. If the object point O be infinitely distant, u1 and u2 are to be replaced by h1 and h2, the perpendicular heights of incidence; the “sine condition” then becomes sin u′1/h1 = sin u′2/h2. A system fulfilling this condition and free from spherical aberration is called “aplanatic” (Greek α-, privative, πλáνη, a wandering). This word was first used by Robert Blair (d. 1828), professor of practical astronomy at Edinburgh University, to characterize a superior achromatism, and, subsequently, by many writers to denote freedom from spherical aberration. Both the aberration of axis points, and the deviation from the sine condition, rapidly increase in most (uncorrected) systems with the aperture.
 Fig. 6. |
(3) Aberration of lateral object points (points beyond the axis) with narrow pencils. Astigmatism.—A point O (fig. 6) at a finite distance from the, axis (or with an infinitely distant object, a point which subtends a finite angle at the system) is, in general, even then not sharply reproduced, if the pencil of rays issuing from it and traversing the system is made infinitely narrow by reducing the aperture stop; such a pencil consists of the rays which can pass from the object point through the now infinitely small entrance pupil. It is seen (ignoring exceptional cases) that the pencil does not meet he refracting or reflecting surface at right angles; therefore it is astigmatic (Gr. α-, privative, στίγμa, a point). Naming the central ray passing through the entrance pupil the “axis of the pencil,” or “principal ray,” we can say: the rays of the pencil intersect, not in one point, but in two focal lines, which we can assume to be at right angles to the principal ray; of these, one lies in the plane containing the principal ray and
- ↑ The investigations of E. Abbe on geometrical optics, originally published only in his university lectures, were first compiled by S. Czapski in 1893. See below, Authorities.
