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(This page is under development, and might be completed Winter 2002).
The discovery by Newton of this phenomenon and its dependence on the ultraviolet (uv) electronic and infrared (ir) vibrational features was discussed at the beginning of this review. In the visible region of a colorless transparent substance, the refractive index n is given by the Sellmeier dispersion formula:
n2 - 1 = aL2(L2 - A2)-l + bL2(L2 - B2)-l + . . .
where L represents lambdam, and A, B. . . . are the wavelength of the individual ir and uv absorptions seen in Fig. 2 and a, b. . . . are constants representing the strengths of these absorptions. Two or three terms are usually enough for an excellent fit in the visible region.
If either the refractive-index variation or the coefficientof-absorption variation is known for all wavelengths, then the other one can be calculated by using the "Kramers-Kronig dispersion relationships." We usually tend to think of the absorption as the "cause" and the dispersion as the "effect," but the two are inextricably connected, and one cannot exist without the other. Only a vacuum has no absorption and no dispersion.
When there is a light absorption in an otherwise transparent medium, then anomalous dispersion results. Instead of n it is necessary to use the complex refractive index N = n + ik, where i is the imaginary squre root of - 1 and k is the absorption coefficient. The variation of n and k in a glass having a violet color derived from an absorption in the green part of spectrum is shown in Fig. 29.
In the region of absorption of Fig. 29, the natural resonating frequency of the absorbers interacts with the vibration of the light in a complex manner involving the phase velocity and the phase angle to produce a speeding up of the light, thus giving a lower n on one side of the absorption, and a slowing down and a higher n on the other side. In the region of the absorption, the refractive index increases with the wavelength, instead of decreasing; this is difficult to observe since it occurs just where the light is most strongly absorbed.
If a beam of light is passed through a thin prism cut out of the colorless glass of Fig. 2, then the sequence of colors seen is the normal spectral sequence shown at the top of Fig. 30. For the green-absorbing violet glass of Fig. 29, however, the sequence in the lower half of Fig. 30 applies. The red to yellow-green sequence at (a) follows normal behavior as at the right in Fig. 29, as does the blue-green to violet sequence at (c), corresponding to the left region in Fig. 29. The yellow- green to blue-green sequence at (b) in Fig. 30 is reversed; green itself is not included here since it is absorbed. The overall color sequence that would be observed from this prism is shown at (d) in this figure; compared to the normal spectrum this is truly "anomalous." Also note how much wider the anomalous spectrum is than the normal one.
In addition to the spectrum produced by a prism, there are several analogous color-producing phenomena. In a wellfaceted gemstone, a ray of light passing into the top of the stone is totally internally reflected and returned to the eye as the "brilliance." Since the geometry of the path corresponds to that in a prism, the reflected rays are also refracted, leading to flashes of color, the "fire" in a stone. The amount of fire depends on the magnitude of the dispersion; diamond is paramount among gemstones.
The refracted paths through a raindrop produce the primary and secondary rainbows; higher orders can be seen in the laboratory but have not been observed in nature. The refracted paths through hexagonal ice crystals produce the 22° and the 45° halos around the sun and moon, the parhelia or moondogs, as well as a variety of other effects.
Last, there is the green flash seen rarely at the setting of the sun. Here the density gradient of the atmosphere acts as a prism, separating the colors as shown in Fig. 3 1. Since the violet and blue rays are scattered (see below), a green image is seen under favorable circumstances for just a few seconds.
A more complicated case than dispersive refraction is "double refraction," which provides color when optically anisotropic crystals are viewed between crossed polarizers. Here the polarized light, on entering the crystal, is separated into ordinary and extraordinary rays moving at different velocities through the crystal; these rays are then recombined in the second polarizer to produce color by interference (see below).
FIG. 29. Anomalous dispersion of a violet crown glass having an absorption band in the green at 550 nm.
FIG. 30. Color sequences produced by dispersion in the colorless crown glass of FIG. 2 (above) and the violet crown glass of FIG. 29 (below).
