in the same way as the deviations of two astigmatic image surfaces of the image plane of the axis point are represented as functions of the angles of the field of view.
The final form of a practical system consequently rests on compromise; enlargement of the aperture results in a diminution of the available field of view, and vice versa. The following may be regarded as typical:—(1) Largest aperture; necessary corrections are—for the axis point, and sine condition; errors of the field of view are almost disregarded; example—high power microscope objectives. (2) Largest field of view; necessary corrections are—for astigmatism, curvature of field and distortion; errors of the aperture only slightly regarded; examples—photographic widest angle objectives and oculars. Between these extreme examples stands the ordinary photographic objective: the portrait objective is corrected more with regard to aperture; objectives for groups more with regard to the field of view. (3) Telescope objectives have usually not very large apertures, and small fields of view; they should, however, possess zones as small as possible, and be built in the simplest manner. They are the best for analytical computation.
(b) Chromatic or Colour Aberration.
In optical systems composed of lenses, the position, magnitude and errors of the image depend upon the refractive indices of the glass employed (see Lens, and above, “Monochromatic Aberration”). Since the index of refraction varies with the colour or wave length of the light (see Dispersion), it follows that a system of lenses (uncorrected) projects images of different colours in somewhat different places and sizes and with different aberrations; i.e. there are “chromatic differences” of the distances of intersection, of magnifications, and of monochromatic aberrations. If mixed light be employed (e.g. white light) all these images are formed; and since they are all ultimately intercepted by a plane (the retina of the eye, a focussing screen of a camera, &c.), they cause a confusion, named chromatic aberration; for instance, instead of a white margin on a dark background, there is perceived a coloured margin, or narrow spectrum. The absence of this error is termed achromatism, and an optical system so corrected is termed achromatic. A system is said to be “chromatically under-corrected” when it shows the same kind of chromatic error as a thin positive lens, otherwise it is said to be “over-corrected.”
If, in the first place, monochromatic aberrations be neglected —in other words, the Gaussian theory be accepted—then every reproduction is determined by the positions of the focal planes, and the magnitude of the focal lengths, or if the focal lengths, as ordinarily happens, be equal, by three constants of reproduction. These constants are determined by the data of the system (radii, thicknesses, distances, indices, &c., of the lenses); therefore their dependence on the refractive index, and consequently on the colour, are calculable (the formulae are given in Czapski-Eppenstein, Grundzüge der Theorie der optischten Instrumente (1903, p. 166). The refractive indices for different wave lengths must be known for each kind of glass made use of. In this manner the conditions are maintained that any one constant of reproduction is equal for two different colours, i.e. this constant is achromatized. For example, it is possible, with one thick lens in air to achromatize the position of a focal plane of the magnitude of the focal length. If all three constants of reproduction be achromatized, then the Gaussian image for all distances of objects is the same for the two colours, and the system is said to be in “stable achromatism.”
In practice it is more advantageous (after Abbe) to determine the chromatic aberration (for instance, that of the distance of intersection) for a fixed position of the object, and express it by a sum in which each component contains the amount due to each refracting surface (see Czapski-Eppenstein, op. cit. p. 170; A. König in M. v. Rohr’s collection, Die Bilderzeugung, p. 340). In a plane containing the image point of one colour, another colour produces a disk of confusion; this is similar to the confusion caused by two “zones” in spherical aberration. For infinitely distant objects the radius of the chromatic disk of confusion is proportional to the linear aperture, and independent of the focal length (vide supra, “Monochromatic Aberration of the Axis Point”); and since this disk becomes the less harmful with an increasing image of a given object, or with increasing focal length, it follows that the deterioration of the image is proportional to the ratio of the aperture to the focal length, i.e. the relative aperture.” (This explains the gigantic focal lengths in vogue before the discovery of achromatism.)
Examples.—(a) In a very thin lens, in air, only one constant of reproduction is to be observed, since the focal length and the distance of the focal point are equal. If the refractive index for one colour be n, and for another n+dn, and the powers, or reciprocals of the focal lengths, be φ and φ+dφ, then (1) dφ/φ=dn/(n−1)=1/ν; dn is called the dispersion, and ν the dispersive power of the glass.
(b) Two thin lenses in contact: let φ1 and φ2 be the powers corresponding to the lenses of refractive indices n1 and n2 and radii r ′1, r ′′1, and r ′2, r ′′2 respectively; let φ denote the total power, and dφ, dn1, d2 the changes of φ, n1, and n2 with the colour. Then the following relations hold:
(2) φ=φ1+φ2 = (n1 − 1)(1/r1 − 1/r ′′1)+(n2 − 1)(1/r ′2 − 1/r ′′2) = (n1 − 1)k1+(n2 − 1)k2; and
(3) dφ = k1dn1 + k2dn2. For achromatism dφ = 0, hence, from (3),
(4) k1/k2 = −dn2/dn1, or φ1/φ2 = −ν1/ν2. Therefore φ1 and φ2 must have different algebraic signs, or the system must be composed of a collective and a dispersive lens. Consequently the powers of the two must be different (in order that φ be not zero (equation 2)), and the dispersive powers must also be different (according to 4).
Newton failed to perceive the existence of media of different dispersive powers required by achromatism; consequently he constructed large reflectors instead of refractors. James Gregory and Leonhard Euler arrived at the correct view from a false conception of the achromatism of the eye; this was determined by Chester More Hall in 1728, Klingenstierna in 1754 and by Dollond in 1757, who constructed the celebrated achromatic telescopes. (See Telescope.)
Glass with weaker dispersive power (greater ν) is named “crown glass”; that with greater dispersive power, “flint glass.” For the construction of an achromatic collective lens (φ positive) it follows, by means of equation (4), that a collective lens I. of crown glass and a dispersive lens II. of flint glass must be chosen; the latter, although the weaker, corrects the other chromatically by its greater dispersive power. For an achromatic dispersive lens the converse must be adopted.
Fig. 10.
This is, at the present day, the ordinary type, e.g., of telescope objective (fig. 10); the values of the four radii must satisfy the equations (2) and (4). Two other conditions may also be postulated; one is always the elimination of the aberration on the axis; the second either the Herschel” or “Fraunhofer condition,” the latter being the best (vide supra, “Monochromatic Aberation”). In practice, however, it is often more useful to avoid the second condition by making, the lenses have contact, i.e. equal radii. According to P. Rudolph (Eder’s Jahrb. f. Photog., 1891, 5, p. 225; 1893, 7, p. 221), cemented objectives of thin lenses permit the elimination of spherical aberration on the axis, if, as above, the collective lens has a smaller refractive index; on the other hand, they permit the elimination of astigmatism and curvature of the field, if the collective lens has a greater refractive index (this follows from the Petzval equation; see L. Seidel, Astr. Nachr., 1856, p. 289). Should the cemented system be positive, then the more powerful lens must be positive; and, according to (4), to the greater power belongs the weaker dispersive power (greater ν), that is to say, crown glass; consequently the crown glass must have the greater refractive index for astigmatic and plane images. In all earlier kinds of glass, however, the dispersive power increased with the refractive index; that is, ν decreased as n increased; but some of the Jena glasses by E. Abbe and O. Schott were crown
