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The Sellmeier equation is an empirical relationship between
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, ...
and
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
for a particular transparent
medium Medium may refer to: Science and technology Aviation * Medium bomber, a class of war plane * Tecma Medium, a French hang glider design Communication * Media (communication), tools used to store and deliver information or data * Medium ...
. The equation is used to determine the
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variatio ...
of
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
in the medium. It was first proposed in 1872 by Wolfgang Sellmeier and was a development of the work of Augustin Cauchy on Cauchy's equation for modelling dispersion.


The equation

In its original and the most general form, the Sellmeier equation is given as : n^2(\lambda) = 1 + \sum_i \frac , where ''n'' is the refractive index, ''λ'' is the wavelength, and ''B''i and ''C''i are experimentally determined ''Sellmeier
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s''. These coefficients are usually quoted for λ in
micrometre The micrometre (American and British English spelling differences#-re, -er, international spelling as used by the International Bureau of Weights and Measures; SI symbol: μm) or micrometer (American and British English spelling differences# ...
s. Note that this λ is the vacuum wavelength, not that in the material itself, which is λ/n. A different form of the equation is sometimes used for certain types of materials, e.g.
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
s. Each term of the sum representing an
absorption Absorption may refer to: Chemistry and biology *Absorption (biology), digestion **Absorption (small intestine) *Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials *Absorption (skin), a route by which s ...
resonance of strength ''B''i at a wavelength . For example, the coefficients for BK7 below correspond to two absorption resonances in the
ultraviolet Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nm (with a corresponding frequency around 30  PHz) to 400 nm (750  THz), shorter than that of visible light, but longer than X-rays. UV radiation ...
, and one in the mid-
infrared Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of Light, visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from ...
region. Close to each absorption peak, the equation gives non-physical values of ''n''2 = ±∞, and in these wavelength regions a more precise model of dispersion such as Helmholtz's must be used. If all terms are specified for a material, at long wavelengths far from the absorption peaks the value of ''n'' tends to :\begin n \approx \sqrt \approx \sqrt \end, where εr is the relative permittivity of the medium. For characterization of glasses the equation consisting of three terms is commonly used: : n^2(\lambda) = 1 + \frac + \frac + \frac, As an example, the coefficients for a common
borosilicate Borosilicate glass is a type of glass with silica and boron trioxide as the main glass-forming constituents. Borosilicate glasses are known for having very low coefficients of thermal expansion (≈3 × 10−6 K−1 at 20 °C), ...
crown glass known as ''BK7'' are shown below: For common optical glasses, the refractive index calculated with the three-term Sellmeier equation deviates from the actual refractive index by less than 5×10−6 over the wavelengths' range of 365 nm to 2.3 μm, which is of the order of the homogeneity of a glass sample. Additional terms are sometimes added to make the calculation even more precise. Sometimes the Sellmeier equation is used in two-term form: : n^2(\lambda) = A + \frac + \frac. Here the coefficient ''A'' is an approximation of the short-wavelength (e.g., ultraviolet) absorption contributions to the refractive index at longer wavelengths. Other variants of the Sellmeier equation exist that can account for a material's refractive index change due to
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
,
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
, and other parameters.


Coefficients


See also

* Cauchy's equation


References

{{Reflist


External links


RefractiveIndex.INFO
Refractive index database featuring Sellmeier coefficients for many hundreds of materials.
A browser-based calculator giving refractive index from Sellmeier coefficients.Annalen der Physik
- free Access, digitized by the French national library
Sellmeier coefficients for 356 glasses from Ohara, Hoya, and Schott
Optics Equations