Theories of Dispersion.—The first attempt at a mathematical theory of dispersion was made by A. Cauchy and published in 1835. This was based on the assumption that the medium in which the light is propagated is discontinuous and molecular in character, the molecules being subject to a mutual attraction. Thus, if one molecule is disturbed from its mean position, it communicates the disturbance to its neighbours, and so a wave is propagated. The formula arrived at by Cauchy was
| n = A + | B | + | C | + .... |
| λ2 | λ4 |
n being the refractive index, λ the wave-length, and A, B, C, &c., constants depending on the material, which diminish so rapidly that only the first three as here written need be taken into account. If suitable values are chosen for these constants, the formula can be made to represent the dispersion of ordinary transparent media within the visible spectrum very well, but when extended to the infra-red region it often departs considerably from the truth, and it fails altogether in cases of anomalous dispersion. There are also grave theoretical objections to Cauchy’s formula.
The modern theory of dispersion, the foundation of which was laid by W. Sellmeier, is based upon the assumption that an interaction takes place between ether and matter. Sellmeier adopted the elastic-solid theory of the ether, and imagined the molecules to be attached to the ether surrounding them, but free to vibrate about their mean positions within a limited range. Thus the ether within the dispersive medium is loaded with molecules which are forced to perform oscillations of the same period as that of the transmitted wave. It can be shown mathematically that the velocity of propagation will be greatly increased if the frequency of the light-wave is slightly greater, and greatly diminished if it is slightly less than the natural frequency of the molecules; also that these effects become less and less marked as the difference in the two frequencies increases. This is exactly in accordance with the observed facts in the case of substances showing anomalous dispersion. Sellmeier’s theory did not take account of absorption, and cannot be applied to calculate the dispersion within a broad absorption band. H. von Helmholtz, working on a similar hypothesis, but with a frictional term introduced into his equations, obtained formulae which are applicable to cases of absorption. A modified form of Helmholtz’s equation, due to E. Ketteler and known as the Ketteler-Helmholtz formula, has been much used in calculating dispersion, and expresses the facts with remarkable accuracy. P. Drude has obtained a similar formula based on the electromagnetic theory, thus placing the theory of dispersion on a much more satisfactory basis. The fundamental assumption is that the medium contains positively and negatively charged ions or electrons which are acted on by the periodic electric forces which occur in wave propagation on Maxwell’s theory. The equations finally arrived at are
| n²(1 − κ²) = 1 + Σ | Dλ²(λ² − λm²) | , |
| (λ² − λm²)² + g²λ² |
| 2n²κ² = Σ | Dgλ³ | , |
| (λ² − λm²)² + g²λ² |
where λ is the wave-length in free ether of light whose refractive index is n, and λm the wave-length of light of the same period as the electron, κ is a coefficient of absorption, and D and g are constants. The sign of summation Σ is used in cases where there are several absorption bands, and consequently several similar terms on the right-hand side, each with a different value of λm. This would occur if there were several kinds of ions, each with its own natural period.
In a region where there is no absorption, we have κ = 0 and therefore g = 0, and we have only one equation, namely,
| n² = 1 + Σ | Dλ² | , |
| (λ² − λm²) |