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Sellmeier Formula

Definition: an equation for calculating the wavelength-dependent refractive index of a medium

German: Sellmeier-Formel

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For the specification of a wavelength-dependent refractive index of a transparent optical material (e.g. an optical glass), it is common to use a so-called Sellmeier formula [1] (also called Sellmeier equation or Sellmeier dispersion formula, after Wolfgang von Sellmeier). This is typically of the form

$$n(\lambda ) = \sqrt {1 + \sum\limits_j {\frac{{{A_j}{\lambda ^2}}}{{{\lambda ^2} - {B_j}}}} }$$

with the coefficients <$A_j$> and <$B_j$>. That form results from a relatively simple physical model with damped oscillators driven by the light field. Such a model is accurate only to the wavelength region where the absorption is negligible.

As an example, the refractive index of fused silica can be calculated as [2]

$$n(\lambda ) = \sqrt {1 + \frac{{{\rm{0}}{\rm{.6961663}}\;{\lambda ^2}}}{{{\lambda ^2} - {\rm{0}}{\rm{.068404}}{{\rm{3}}^2}}} + \frac{{{\rm{0}}{\rm{.4079426}}\;{\lambda ^2}}}{{{\lambda ^2} - {\rm{0}}{\rm{.116241}}{{\rm{4}}^2}}} + \frac{{{\rm{0}}{\rm{.8974794}}\;{\lambda ^2}}}{{{\lambda ^2} - {\rm{9}}{\rm{.89616}}{{\rm{1}}^2}}}}$$

where the wavelength in micrometers has to be inserted.

The Sellmeier coefficients are usually obtained by a least-square fitting procedure, applied to refractive indices measured in a wide wavelength range.

Applications

Sellmeier equations are very useful, as they make it possible to describe fairly accurately the refractive index in a wide wavelength range with only a few so-called Sellmeier coefficients. Sellmeier coefficients for many optical materials are available in databases. Some caution is advisable when applying Sellmeier equations in extreme wavelength regions; unfortunately, the validity range of available data is often not indicated.

Sellmeier data can also be used for evaluating the chromatic dispersion of a material. This involves frequency derivatives, which can be performed analytically with Sellmeier data even for high orders of dispersion, whereas numerical differentiation on the basis of tabulated index data is sensitive to noise.

Another frequent application of Sellmeier data is the calculation of phase-matching configurations for nonlinear frequency conversion. Here, it is often critical to have Sellmeier data which are valid in a wide wavelength range.

Modified Equations

The literature contains a great variety of modified Sellmeier equations which are also often called Sellmeier formulas. Extensions to the simple form give above can enlarge the wavelength range of validity, or make it possible to include the temperature dependence of refractive indices. This can be important, for example, for calculating phase-matching configurations for nonlinear frequency conversion.

Alternatives to Sellmeier Equations

There are various other kinds of equations for refractive indices. For example, there is the old Cauchy formula, which is a bit simpler than the Sellmeier formula and still fits the refractive indices of many materials in the visible spectral region quite well, as long as the material has no absorption in the visible region. In the near infrared, however, substantially higher accuracy is achieved with the Sellmeier formula. Other equations have been presented by authors like Schott, Hartmann, Conrady, Kettler–Drude, and Herzberger. For example, the Schott formula is power series for calculating <$n^2$>.

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Bibliography

 [1] W. Sellmeier, Ann. Phys. Chem. 219 (6), 272 (1871); https://doi.org/10.1002/andp.18712190612 [2] I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica”, J. Opt. Soc. Am. 55 (10), 1205 (1965); https://doi.org/10.1364/JOSA.55.001205 [3] D. Smith and P. Baumeister, “Refractive index of some oxide and fluoride coating materials”, Appl. Opt. 18 (1), 111 (1979); https://doi.org/10.1364/AO.18.000111 [4] G. Ghosh, “Sellmeier coefficients and dispersion of thermo-optic coefficients for some optical glasses”, Appl. Opt. 36 (7), 1540 (1997); https://doi.org/10.1364/AO.36.001540 [5] D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate”, Opt. Lett. 22 (20), 1553 (1997); https://doi.org/10.1364/OL.22.001553