Abbe number
In physics and optics, the Abbe number, also known as the V-number or constringence of a transparent material, is a measure of the material's dispersion (variation of refractive index with wavelength) in relation to the refractive index, with high values of V indicating low dispersion (low chromatic aberration). It is named after Ernst Abbe (1840–1905), the German physicist who defined it.
The Abbe number,[2][3] VD, of a material is defined as
where nD, nF and nC are the refractive indices of the material at the wavelengths of the Fraunhofer D-, F- and C- spectral lines (589.3 nm, 486.1 nm and 656.3 nm respectively).
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, and 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, and for other wavelengths, or for higher precision work, the full dispersion relation (refractive index as a function of wavelength) is used.
Due to the difficulty and inconvenience in producing sodium and hydrogen lines, alternate definitions of the Abbe number are used in some contexts (ISO 7944).[4] The value Vd is given by
which defines the Abbe number with respect to the yellow Fraunhofer d (or D3) helium line at 587.5618 nm wavelength. It can also be defined using the green mercury E-line at 546.073 nm:
where F' and C' are the blue and red cadmium lines at 480.0 nm and 643.8 nm, respectively.
An Abbe diagram is produced by plotting the Abbe number Vd of a material versus its refractive index nd. Glasses can then be categorised by their composition and position on the diagram. This can be a letter-number code, as used in the Schott Glass catalogue, or a 6-digit glass code.
Abbe numbers are used to calculate the necessary focal lengths of achromatic doublet lenses to minimize chromatic aberration.
The following table lists standard wavelengths at which n is usually determined, indicated by subscripts.[5] For example, nD is measured at 589.3 nm:
| λ in nm | Fraunhofer's symbol | Light source | Color |
|---|---|---|---|
| 365.01 | i | Hg | UV |
| 404.66 | h | Hg | violet |
| 435.84 | g | Hg | blue |
| 479.99 | F' | Cd | blue |
| 486.13 | F | H | blue |
| 546.07 | e | Hg | green |
| 587.56 | d | He | yellow |
| 589.3 | D | Na | yellow |
| 643.85 | C' | Cd | red |
| 656.27 | C | H | red |
| 706.52 | r | He | red |
| 768.2 | A' | K | IR |
| 852.11 | s | Cs | IR |
| 1013.98 | t | Hg | IR |
See also
- Abbe prism
- Abbe refractometer
- Calculation of glass properties, including Abbe number
- Glass code
- List of dimensionless quantities
References
- ^ Abbe number calculation of glasses
- ^ Hovestadt, H. (1902). Jena Glass and Its Scientific and Industrial Applications. London: Macmillan and Co. pp. 1–81.
- ^ Bergmann, Ludwig; Clemens Schaefer (1999). Optics of Waves and Particles. Berlin: Walter de Gruyter. pp. 198–201. ISBN 3-11-014318-6.
- ^ Meister, Darryl. "Understanding Reference Wavelengths" (PDF). Carl Zeiss Vision. Retrieved 2013-03-13.
- ^ L. D. Pye, V. D. Frechette, N. J. Kreidl: "Borate Glasses"; Plenum Press, New York, 1977