Editing Sellmeier equation (section)

==Description==
In its original and the most general form, the Sellmeier equation is given as
:<math>
n^2(\lambda) = 1 + \sum_i \frac{B_i \lambda^2}{\lambda^2 - C_i}
</math>,
where ''n'' is the refractive index, ''λ'' is the wavelength, and ''B''<sub>i</sub> and ''C''<sub>i</sub> are experimentally determined ''Sellmeier [[coefficient]]s''. These coefficients are usually quoted for λ in [[micrometre]]s. 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. [[crystal]]s.

Each term of the sum representing an [[absorption (optics)|absorption]] resonance of strength ''B''<sub>i</sub> at a wavelength {{math|{{radical|''C''<sub>i</sub>}}}}. 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.<ref name="b841">{{cite web | title=2.7: Kramers-Kroenig Relations | website=Engineering LibreTexts | date=2021-04-06 | url=https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Book%3A_Ultrafast_Optics_(Kaertner)/02%3A_Maxwell-Bloch_Equations/2.07%3A_Kramers-Kroenig_Relations | access-date=2024-07-09}}</ref> However, close to each absorption peak, the equation gives non-physical values of ''n''<sup>2</sup> = ±∞, and in these wavelength regions a more precise model of dispersion such as [[Helmholtz dispersion|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
:<math>\begin{matrix}
n \approx \sqrt{1 + \sum_i  B_i } \approx \sqrt{\varepsilon_r}
\end{matrix},</math>
where ε<sub>r</sub> is the [[relative permittivity]] of the medium.

For characterization of glasses the equation consisting of three terms is commonly used:<ref>[http://www.schott.com/advanced_optics/english/download/schott_tie-29_refractive_index_and_dispersion_eng.pdf Refractive index and dispersion]. Schott technical information document TIE-29 (2007).</ref><ref>{{Cite web|url=https://www.rp-photonics.com/sellmeier_formula.html|title=Encyclopedia of Laser Physics and Technology - Sellmeier formula, refractive index, Sellmeier equation, dispersion formula|last=Paschotta|first=Dr. Rüdiger|website=www.rp-photonics.com|language=en|access-date=2018-09-14}}</ref>

:<math>
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},
</math>

As an example, the coefficients for a common [[borosilicate glass|borosilicate]] [[Crown glass (optics)|crown glass]] known as ''BK7'' are shown below:
{| class="wikitable"
|-
! Coefficient !! Value
|-
| B<sub>1</sub> || 1.03961212
|-
| B<sub>2</sub> || 0.231792344
|-
| B<sub>3</sub> || 1.01046945
|-
| C<sub>1</sub> || 6.00069867×10<sup>&minus;3</sup> μm<sup>2</sup>
|-
| C<sub>2</sub> || 2.00179144×10<sup>&minus;2</sup> μm<sup>2</sup>
|-
| C<sub>3</sub> || 1.03560653×10<sup>2</sup> μm<sup>2</sup>
|}

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<sup>−6</sup> over the wavelengths' range<ref>{{Cite web | url=http://oharacorp.com/o2.html |title = Optical Properties}}</ref> of 365&nbsp;nm to 2.3&nbsp;μm, which is of the order of the homogeneity of a glass sample.<ref>{{Cite web | url=http://oharacorp.com/o7.html |title = Guarantee of Quality}}</ref> Additional terms are sometimes added to make the calculation even more precise.

Sometimes the Sellmeier equation is used in two-term form:<ref>{{cite journal|last=Ghosh|first=Gorachand|title=Sellmeier Coefficients and Dispersion of Thermo-Optic coefficients for some optical glasses|journal=Applied Optics|volume=36|issue=7|pages=1540–6|url=https://www.researchgate.net/publication/5601154 |doi= 10.1364/AO.36.001540|pmid=18250832|bibcode=1997ApOpt..36.1540G|year=1997}}</ref>
:<math>
n^2(\lambda) = A + \frac{B_1\lambda^2}{\lambda^2 - C_1} + \frac{ B_2 \lambda^2}{\lambda^2 - C_2}.
</math>
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.