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 Wolfgang Sellmeier and was a development of the work of Augustin Cauchy on Cauchy's equation for modelling dispersion.^[1]

The equation

In its original and the most general form, the Sellmeier equation is given as

${\displaystyle n^{2}(\lambda )=1+\sum _{i}{\frac {B_{i}\lambda ^{2)){\lambda ^{2}-C_{i))))$,

where n is the refractive index, λ is the wavelength, and B_i and C_i 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 B_i at a wavelength √C_i. 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

${\displaystyle {\begin{matrix}n\approx {\sqrt {1+\sum _{i}B_{i))}\approx {\sqrt {\varepsilon _{r))}\end{matrix)),}$

where ε_r is the relative permittivity of the medium.

For characterization of glasses the equation consisting of three terms is commonly used:^[2][3]

${\displaystyle n^{2}(\lambda )=1+{\frac {B_{1}\lambda ^{2)){\lambda ^{2}-C_{1))}+{\frac {B_{2}\lambda ^{2)){\lambda ^{2}-C_{2))}+{\frac {B_{3}\lambda ^{2)){\lambda ^{2}-C_{3))},}$

As an example, the coefficients for a common borosilicate 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^[4] of 365 nm to 2.3 μm, which is of the order of the homogeneity of a glass sample.^[5] Additional terms are sometimes added to make the calculation even more precise.

Sometimes the Sellmeier equation is used in two-term form:^[6]

${\displaystyle n^{2}(\lambda )=A+{\frac {B_{1}\lambda ^{2)){\lambda ^{2}-C_{1))}+{\frac {B_{2}\lambda ^{2)){\lambda ^{2}-C_{2))}.}$

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