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 color, are calculable (the formulae are given in CzapskiEppenstein, Grundzuge der Theorie der optischen Instrumente (1903, p. 166).
The refractive indices for different wavelengths 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 colors, 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 colors, 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 conlins the amount due to each refracting surface (see CzapskiEppenstein, op. cit. p. 170; A. Konig in M. v. Rohr's collection, Die Bilderzeugung, p. 340).
In a plane containing the image point of one color, 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 color be n, and for another n+dn, and the powers, or reciprocals of the focal lengths, be f and f + df, then (1) df/f = dn/(n1) = 1/n; dn is called the dispersion, and n the dispersive power of the glass.
 (b) Two thin lenses in contact: let f1 and f2 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 f denote the total power, and df, dn1, dn2 the changes of f, n1, and n2 with the color. Then the following relations hold: —

 (2) f = f1f2== (n1  1)(1/r'11/r1) +(n21)(1/r'2  1/r2) = (n1  1)k1 + (n2  1)k2; and

 (3) df = k1dn1 + k2dn2. For achromatism df = 0, hence, from (3),

 (4) k1/k2 = dn2 / dn1, or f1/f2 = n1/n2. Therefore f1 and f2 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 f 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 v) is named crown glass; that with greater dispersive power, flint glass. For the construction of an achromatic collective lens (f 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. 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 Aberration). 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 v), 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, v decreased as n increased; but some of the Jena glasses by E. Abbe and O. Schott were crown glasses of high refractive index, and achromatic systems from such crown glasses, with flint glasses of lower refractive index, are called the new achromats, and were employed by P. Rudolph in the first anastigmats (photographic objectives).
Instead of making df vanish, a certain value can be assigned to it which will produce, by the addition of the two lenses, any desired chromatic deviation, e.g. sufficient to eliminate one present in other parts of the system. If the lenses I. and II. be cemented and have the same refractive index for one color, then its effect for that one color is that of a lens of one piece; by such decomposition of a lens it can be made chromatic or achromatic at will, without altering its spherical effect. If its chromatic effect (df/f) be greater than that of the same lens, this being made of the more dispersive of the two glasses employed, it is termed hyperchromatic.
For two thin lenses separated by a distance D the condition for achromatism is D = v1f1+v2f2; if v1=v2 (e.g. if the lenses be made of the same glass), this reduces to D= 1/2 (f1+f2), known as the condition for oculars.
If a constant of reproduction, for instance the focal length, be made equal for two colors, then it is not the same for other colors, if two different glasses are employed. For example, the condition for achromatism (4) for two thin lenses in contact is fulfilled in only one part of the spectrum, since dn2/dn1 varies within the spectrum. This fact was first ascertained by J. Fraunhofer, who defined the colors by means of the dark lines in the solar spectrum; and showed that the ratio of the dispersion of two glasses varied about 20% from the red to the violet (the variation for glass and water is about 50%). If, therefore, for two colors, a and b, fa = fb = f, then for a third color, c, the focal length is different; that is, if c lies between a and b, then fc
In fig. 11, taken from M. von Rohr's Theorie und Geschichte des photographischen Objectivs, the abscissae are focal lengths, and the ordinates wavelengths. The Fraunhofer lines used are shown in the table to the right of the figure.
C  656.3 nm
D  589.3 nm
Green Hg.  546.1 nm
F  486.2 nm
G'  454.1 nm
Violet Hg.  405.1 nm
Moreover, this region of the spectrum is that which appears brightest to the human eye, and consequently this curve of the secondary on spectrum, obtained by making fc = fF, is, according to the experiments of Sir G. G. Stokes (Proc. Roy. Soc., 1878), the most suitable for visual instruments (optical achromatism,). In a similar manner, for systems used in photography, the vertex of the color curve must be placed in the position of the maximum sensibility of the plates; this is generally supposed to be at G'; and to accomplish this the F and violet mercury lines are united.
This artifice is specially adopted in objectives for astronomical photography (pure actinic achromatism). For ordinary photography, however, there is this disadvantage: the image on the focusingscreen and the correct adjustment of the photographic sensitive plate are not in register; in astronomical photography this difference is constant, but in other kinds it depends on the distance of the objects. On this account the lines D and G' are united for ordinary photographic objectives; the optical as well as the actinic image is chromatically inferior, but both lie in the same place; and consequently the best correction lies in F (this is known as the actinic correction or freedom from chemical focus).
Should there be in two lenses in contact the same focal lengths for three colours a, b, and c, i.e. fa = fb = fc = f, then the relative partial dispersion (ncnb) (nanb) must be equal for the two kinds of glass employed. This follows by considering equation (4) for the two pairs of colors ac and bc. Until recently no glasses were known with a proportional degree of absorption; but R. Blair (Trans. Edin. Soc., 1791, 3, p. 3), P. Barlow, and F. S. Archer overcame the difficulty by constructing fluid lenses between glass walls.
Fraunhofer prepared glasses which reduced the secondary spectrum; but permanent success was only assured on the introduction of the Jena glasses by E. Abbe and O. Schott. In using glasses not having proportional dispersion, the deviation of a third colour can be eliminated by two lenses, if an interval be allowed between them; or by three lenses in contact, which may not all consist of the old glasses. In uniting three colors an achromatism of a higher order is derived; there is yet a residual tertiary spectrum, but it can always be neglected.
The Gaussian theory is only an approximation; monochromatic or spherical aberrations still occur, which will be different for different colors; and should they be compensated for one color, the image of another color would prove disturbing. The most important is the chromatic difference of aberration of the axis point, which is still present to disturb the image, after paraxial rays of different colors are united by an appropriate combination of glasses. If a collective system be corrected for the axis point for a definite wavelength, then, on account of the greater dispersion in the negative components — the flint glasses, — overcorrection will arise for the shorter wavelengths (this being the error of the negative components), and undercorrection for the longer wavelengths (the error of crown glass lenses preponderating in the red).
This error was treated by Jean le Rond d'Alembert, and, in special detail, by C. F. Gauss. It increases rapidly with the aperture, and is more important with medium apertures than the secondary spectrum of paraxial rays; consequently, spherical aberration must be eliminated for two colors, and if this be impossible, then it must be eliminated for those particular wavelengths which are most effectual for the instrument in question (a graphical representation of this error is given in M. von Rohr, Theorie und Geschichte des photographischen Objectivs).
The condition for the reproduction of a surface element in the place of a sharply reproduced point — the constant of the sine relationship must also be fulfilled with large apertures for several colors. E. Abbe succeeded in computing microscope objectives free from error of the axis point and satisfying the sine condition for several colors, which therefore, according to his definition, were aplanatic for several colors; such systems he termed apochromatic. While, however, the magnification of the individual zones is the same, it is not the same for red as for blue; and there is a chromatic difference of magnification.
This is produced in the same amount, but in the opposite sense, by the oculars, which Abbe used with these objectives (compensating oculars), so that it is eliminated in the image of the whole microscope. The best telescope objectives, and photographic objectives intended for threecolor work, are also apochromatic, even if they do not possess quite the same quality of correction as microscope objectives do. The chromatic differences of other errors of reproduction have seldom practical importances.
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