Sellmeier Equation

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## The equation

## Coefficients

## See also

## References

## External links

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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.^{[1]}

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

- ,

where *n* is the refractive index, *?* is the wavelength, and *B*_{i} and *C*_{i} are experimentally determined *Sellmeier coefficients*. These coefficients are usually quoted for ? in micrometres. Note that this ? is the vacuum wavelength, not that in the material itself, which is ?/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 . 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 *n*^{2} = ±?, 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 ?_{r} is the relative permittivity of the medium.

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

As an example, the coefficients for a common borosilicate crown glass known as *BK7* are shown below:

Coefficient | Value |
---|---|

B_{1} |
1.03961212 |

B_{2} |
0.231792344 |

B_{3} |
1.01046945 |

C_{1} |
6.00069867×10^{−3} ?m^{2} |

C_{2} |
2.00179144×10^{−2} ?m^{2} |

C_{3} |
1.03560653×10^{2} ?m^{2} |

The Sellmeier coefficients for many common optical materials can be found in the online database of RefractiveIndex.info.

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 ?m, 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]}

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.

Material | B_{1} |
B_{2} |
B_{3} |
C_{1}, ?m^{2} |
C_{2}, ?m^{2} |
C_{3}, ?m^{2} |
---|---|---|---|---|---|---|

borosilicate crown glass (known as BK7) |
1.03961212 | 0.231792344 | 1.01046945 | 6.00069867×10^{−3} |
2.00179144×10^{−2} |
103.560653 |

sapphire (for ordinary wave) |
1.43134930 | 0.65054713 | 5.3414021 | 5.2799261×10^{−3} |
1.42382647×10^{−2} |
325.017834 |

sapphire (for extraordinary wave) |
1.5039759 | 0.55069141 | 6.5927379 | 5.48041129×10^{−3} |
1.47994281×10^{−2} |
402.89514 |

fused silica | 0.696166300 | 0.407942600 | 0.897479400 | 4.67914826×10^{−3} |
1.35120631×10^{−2} |
97.9340025 |

Magnesium fluoride | 0.48755108 | 0.39875031 | 2.3120353 | 0.001882178 | 0.008951888 | 566.13559 |

**^**Sellmeier, W. (1872). "Ueber die durch die Aetherschwingungen erregten Mitschwingungen der Körpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien (II. Theil)".*Annalen der Physik und Chemie*.**223**(11): 386-403. doi:10.1002/andp.18722231105.**^**Refractive index and dispersion. Schott technical information document TIE-29 (2007).**^**Paschotta, Dr. Rüdiger. "Encyclopedia of Laser Physics and Technology - Sellmeier formula, refractive index, Sellmeier equation, dispersion formula".*www.rp-photonics.com*. Retrieved .**^**"Optical Properties".**^**"Guarantee of Quality".**^**Ghosh, Gorachand (1997). "Sellmeier Coefficients and Dispersion of Thermo-Optic coefficients for some optical glasses".*Applied Optics*.**36**(7): 1540-6. Bibcode:1997ApOpt..36.1540G. doi:10.1364/AO.36.001540. PMID 18250832.**^**"Archived copy". Archived from the original on 2015-10-11. Retrieved .CS1 maint: archived copy as title (link)

- RefractiveIndex.INFO Refractive index database featuring Sellmeier coefficients for many hundreds of materials.
- A browser-based calculator giving refractive index from Sellmeier coefficients.
- Annalen der Physik - free Access, digitized by the French national library
- Sellmeier coefficients for 356 glasses from Ohara, Hoya, and Schott

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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