The Dispersion of Light
DispersionEach material through which light passes will have an Index of Refraction that varies with the wavelength of the incoming light. That is, the amount of bending will increase as the wavelength decreases. This is called Chromatic Dispersion and explains why white light, which contains all wavelengths in the Visible Spectrum, is split into a rainbow when it bends through a prism.
Even the air around us has Chromatic Dispersion properties. Atmospheric dispersion is usually very small, and requires telescopic observation;
but careful examination of the images of the Sun and Moon, when very near the horizon, may reveal a bluer upper rim, and a redder lower rim.
Below is an image of the Moon near the horizon, showing a greenish upper rim, and (if you look very closely) a reddish lower rim. The arc
above the top limb of the Moon is produced by unusual atmospheric conditions not discussed here, while the overall reddish tone is caused
by scattering of shorter wavelengths (extinction), which is proportional to the increase in refraction, so that the flatter the Moon
appears, the redder it appears, as well. (Laurent Laveder, apod050826)
So, Chromatic Dispersion is the change of index of refraction, for a given material, with wavelength. Generally the index decreases as wavelength increases, blue light traveling more slowly in the material than red light. In addition to separating colors in prisms, Dispersion is the phenomenon that also gives the generally undesirable chromatic aberration in lenses. You can find more information on that subject in Geometrical Optics.
The shape of the curve, that is, the precise function of Index vs. Wavelength, is given by the Sellmeier Equation, that is a function of six coefficients. These coefficients are supplied for each kind of optical glass for use by lens designers, so that they may appropriately account for, and correct if necessary, chromatic aberration.
Both of these important dispersion equations are described in more detail below.
Abbe's NumberThe Abbe Number was developed by Ernst Abbe. Ernst was a German physicist and a professor at the University of Jena. He was a professor of mathematics and physics. In 1868 he invented the apochromatic lens system for use in microscopes. The apochromatic lens system eliminated primary and secondary color distortions in microscopes. His work gained the interest of Carl Zeiss, who in 1866 hired Ernst to help with various optical problems and the manufacturing process of optical instruments. In 1888, Ernst became the sole owner of Carl Zeiss' company!
Abbe's work led him to conclude that there is a value, or number, that is a measure of a material's light dispersion in relation to the refractive index. As described above, dispersion is the scattering of light into its component colors by prisms or lenses. This number, or value, is sometimes referred to as refractive efficiency. This is the Abbe number. This value is also called the V-number. When light passes through a lens and gets dispersed, colors with shorter wavelengths travel more slowly than colors with longer wavelengths. The result of this is chromatic aberration.
Lenses that have a higher Abbe number will disperse light less. This produces less chromatic aberration. Lenses that have a lower Abbe number will have more chromatic aberration, and will disperse light more. The higher the refractive index of a lens, the lower the Abbe number will be.
Chromatic aberration can be broken down further into axial chromatic aberration and lateral chromatic aberration. Axial chromatic aberration is the measure of the difference in focus between the red and blue ends of the color spectrum. Lateral chromatic aberration is the measure of the prismatic deviation between the red and blue ends of the color spectrum.
Abbe numbers are used to classify glass and other optically transparent materials. For example flint glass has V < 50 and crown glass has V > 50. Typical values of V range from around 20 for very dense flint glass, around 30 for polycarbonate plastics, up to 65 for very light crown glass, and up to 85 for fluor-crown glass. Abbe numbers are only a useful measure of dispersion for visible light. For other wavelengths or for extremely precise work, the group velocity dispersion is used.
An Abbe Diagram is produced by plotting the Abbe Number of a material versus it's refractive index. Glasses can then be categorized by their composition and position on the diagram.
The Sellmeier EquationIn optics, the Sellmeier equation is an empirical relationship between refractive index n and wavelength λ for a particular transparent medium. The usual form of the equation for glasses is:
where B1,2,3 and C1,2,3 are experimentally determined Sellmeier coefficients. These coefficients are usually quoted for λ measured in micrometers. The equation is used to determine the dispersion of light in a refracting medium. A different form of the equation is sometimes used for certain types of materials, e.g. crystals.
See an extensive list of Optical Material Dispersion Properties in this Excel file