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.
In its original and the most general form, the Sellmeier equation is given as
where n is the refractive index, û is the wavelength, and B<sub>i</sub> and C<sub>i</sub> are experimentally determined Sellmeier coefficients. These coefficients are usually quoted for û in micrometres. Note that this û is the vacuum wavelength, not that in the material itself, which is û/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<sub>i</sub> 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. Analytically, this process is based on approximating the underlying optical resonances as dirac delta functions, followed by the application of the Kramers-Kronig relations. This results in real and imaginary parts of the refractive index which are physically sensible. However, close to each absorption peak, the equation gives non-physical values of n<sup>2</sup> = ñâÂÂ, 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 õ<sub>r</sub> is the relative permittivity 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:
For common optical glasses, the refractive index calculated with the three-term Sellmeier equation deviates from the actual refractive index by less than 5ÃÂ10<sup>âÂÂ6</sup> over the wavelengths' range of 365 nm to 2.3 üm, 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 (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.
Analytically, the Sellmeier equation models the refractive index as due to a series of optical resonances within the bulk material. Its derivation from the Kramers-Kronig relations requires a few assumptions about the material, from which any deviations will affect the model's accuracy:
From the last point, the complex refractive index (and the electric susceptibility) becomes:
The real part of the refractive index comes from applying the Kramers-Kronig relations to the imaginary part:
Plugging in the first equation above for the imaginary component:
The order of summation and integration can be swapped. When evaluated, this gives the following, where is the Heaviside function:
Since the domain is assumed to be far from any resonances (assumption 2 above), evaluates to 1 and a familiar form of the Sellmeier equation is obtained:
By rearranging terms, the constants and can be substituted into the equation above to give the Sellmeier equation.