Sellmeier equation


The Sellmeier equation is an empirical relationship between refractive index and wavelength for a particular transparent medium. The equation is used to determine the dispersion of light in the medium.
It was first proposed in 1872 by Wilhelm 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
where n is the refractive index, λ is the wavelength, and Bi and Ci 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. A different form of the equation is sometimes used for certain types of materials, e.g. crystals.
Each term of the sum representing an absorption resonance of strength Bi at a wavelength. For example, the coefficients for BK7 below 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 n2 = ±∞, 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 εr is the relative dielectric constant of the medium.
For characterization of glasses the equation consisting of three terms is commonly used:
As an example, the coefficients for a common borosilicate crown glass known as BK7 are shown below:
CoefficientValue
B11.03961212
B20.231792344
B31.01046945
C16.00069867×10−3 μm2
C22.00179144×10−2 μm2
C31.03560653×102 μm2

The Sellmeier coefficients for many common optical materials can be found in the online database of .
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:
Here the coefficient A is an approximation of the short-wavelength 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