Sellmeier equation

In 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 [ [http://schott.com/optics_devices/english/download/tie-29_refractive_index_v2.pdf Refractive index and dispersion] . Schott technical information document TIE-29 (2005).] is

:$n^2(lambda)\; =\; 1\; +\; frac\{B\_1\; lambda^2\; \}\{\; lambda^2\; -\; C\_1\}+\; frac\{B\_2\; lambda^2\; \}\{\; lambda^2\; -\; C\_2\}+\; frac\{B\_3\; lambda^2\; \}\{\; lambda^2\; -\; C\_3\},$

where "B"_1,2,3 and "C"_1,2,3 are experimentally determined "Sellmeier coefficients". These coefficients are usually quoted for λ in micrometres. Note that this λ is the vacuum wavelength; not that in the material itself, which is λ/"n"(λ).

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.

The equation was deduced in 1871 by W. Sellmeier, and was a development of the work of Augustin Cauchy on Cauchy's equation for modelling dispersion.

As an example, the coefficients for a common borosilicate crown glass known as "BK7" are shown below:

Coefficient	Value
B1	1.03961212
B2	2.31792344x10−1
B3	1.01046945
C1	6.00069867x10−3 μm2
C2	2.00179144x10−2 μm2
C3	1.03560653x102 μm2

The Sellmeier coefficients for many common optical glasses can be found in the Schott Glass [http://schott.com/optics_devices/english/download/index.html catalogue] , or in the Ohara [http://oharacorp.com/html/catalog.html catalogue] .

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 [http://oharacorp.com/html/o2.html] , which is of the order of the homogeneity of a glass sample [http://oharacorp.com/html/o7.html] . Additional terms are sometimes added to make the calculation even more precise. In its most general form, the Sellmeier equation is given as:$n^2(lambda)\; =\; 1\; +\; sum\_i\; frac\{B\_i\; lambda^2\}\{lambda^2\; -\; C\_i\},$with each term of the sum representing an absorption resonance of strength "B"_i at a wavelength √"C"_i. For example, the coefficients for BK7 above correspond to two absorption resonances in the ultraviolet, and one in the mid-infrared region. Close to each absorption peak, the equation gives non-physical values of "n"=±∞, 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:where ε_r is the relative dielectric constant of the medium.

The Sellmeier equation can also be given in another form::$n^2(lambda)\; =\; A\; +\; frac\{B\_1\; lambda^2\}\{lambda^2\; -\; C\_1\}\; +\; frac\{\; B\_2\; lambda^2\}\{lambda^2\; -\; C\_2\}.$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, pressure, and other parameters.

Coefficients

See also

*Dispersion (optics)
*Cauchy's equation
*Kramers–Kronig relation

References

*W. Sellmeier, Annalen der Physik und Chemie 143, 271 (1871)

External links

* [http://cvilaser.com/Common/PDFs/Dispersion_Equations.pdf A PDF giving Sellmeier coefficients for several common glasses and optical materials]
* [http://www.schott.com/optics_devices/german/download/opticalglassdatasheetsv101007.xls An XLS file with dispersion coefficients and other optical properties of all Schott] glasses.
* [http://www.calctool.org/CALC/phys/optics/sellmeier A browser-based calculator giving refractive index from Sellmeier coefficients.]
* [http://gallica.bnf.fr/ark:/12148/cb34462944f/date Annalen der Physik] - free Access, ditizied by the french national library