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 dφ 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 colour, then its effect for that one colour 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 (dφ/φ) be greater than that of the same lens, this being made of the more dispersive of the two glasses employed, it is termed “hyper-chromatic.”
For two thin lenses separated by a distance D the condition for achromatism is D＝(v1f1+v2f2) (v1+v2); if v1＝v2 (e.g. if the lenses be made of the same glass), this reduces to D＝12 (f1+f2), known as the “condition for oculars.”
If a constant of reproduction, for instance the focal length, be made equal for two colours, then it is not the same for other colours, 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 colours 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 colours, a and b, fa＝fb＝f, then for a third colour, c, the focal length is different, viz. if c lie between a and b, then fc <f, and vice versa; these algebraic results follow from the fact that towards the red the dispersion of the positive crown glass preponderates, towards the violet that of the negative flint. These chromatic errors of systems, which are achromatic for two colours, are called the “secondary spectrum,” and depend upon the aperture and focal length in the same manner as the primary chromatic errors do.
In fig. 11, taken from M. von Rohr’s Theorie und Geschichte des photographischen Objectivs, the abscissae are focal lengths, and the ordinates wave-lengths; of the latter the Fraunhofer lines used are—
| A′ | C | D | Green Hg. | F | G′ | Violet Hg. |
| 767·7 | 656·3 | 589·3 | 546·1 | 486·2 | 454·1 | 405·1 μμ, |
Fig. 11.—Secondary Spectrum of the combination O. 1726, O. 108. Optical correction fc＝ff＝100mm. The ordinates give the wave-length in μμ. The abscissae give fλ - fc in 0·01mm., commencing at fc ff.
(From M. v. Rohr, op. cit.)
and the focal lengths are made equal for the lines C and F. In the neighbourhood of 550 μμ the tangent to the curve is parallel to the axis of wave-lengths; and the focal length varies least over a fairly large range of colour, therefore in this neighbourhood the colour union is at its best. Moreover, this region of the spectrum is that which appears brightest to the human eye, and consequently this curve of the secondary 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 colour 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 focussing-screen 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 (nc−nb) (na−nb) must be equal for the two kinds of glass employed. This follows by considering equation (4) for the two pairs of colours 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 colours 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 colours; and should they be compensated for one colour, the image of another colour 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 par-axial rays of different colours are united by an appropriate combination of glasses. If a collective system be corrected for the axis point for a definite wave-length, then, on account of the greater dispersion in the negative components—the flint glasses,—over-correction will arise for the shorter wavelengths (this being the error of the negative components), and under-correction for the longer wave-lengths (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 par-axial rays; consequently, spherical aberration must be eliminated for two colours, and if this be impossible, then it must be eliminated for those particular wave-lengths 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 relation—must also be fulfilled with large apertures for several colours. E. Abbe succeeded in computing microscope objectives free from error of the axis point and satisfying the sine condition for several colours, which therefore, according to his definition, were “aplanatic for several colours”; 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 are 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 three-colour 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.
