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.
Description In its original and the most general form, the Sellmeier equation is given as
n 2 ( λ ) = 1 + ∑ i B i λ 2 λ 2 − C i {\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 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 represents an absorption resonance of strength Bi at a wavelength √Ci. 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 n2 = ±∞, and in these wavelength regions a more precise model of dispersion such as the Lorentz oscillator model 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
n ≈ 1 + ∑ i B i ≈ ε r , {\displaystyle {\begin{matrix}n\approx {\sqrt {1+\sum _{i}B_{i}}}\approx {\sqrt {\varepsilon _{r}}}\end{matrix}},}
where εr is the relative permittivity of the medium. For characterization of glasses the equation consisting of three terms is commonly used:
n 2 ( λ ) = 1 + B 1 λ 2 λ 2 − C 1 + B 2 λ 2 λ 2 − C 2 + B 3 λ 2 λ 2 − C 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 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:
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