Editing Sellmeier equation

{{Short description|Empirical relationship between refractive index and wavelength}}
[[File:Sellmeier-equation.svg|thumb|right|Refractive index vs. wavelength for [[BK7 glass]], showing measured points (blue crosses) and the Sellmeier equation (red line)]]

[[Image:Cauchy-equation-1.svg|thumb|Same as the graph above, but with Cauchy's equation (blue line) for comparison. While Cauchy's equation (blue line) deviates significantly from the measured refractive indices outside of the visible region (which is shaded red), the Sellmeier equation (green dashed line) does not.]]

The '''Sellmeier equation''' is an [[empirical relationship]] between [[refractive index]] and [[wavelength]] for a particular [[transparency (optics)|transparent]] [[optical medium|medium]]. The equation is used to determine the [[dispersion (optics)|dispersion]] of [[light]] in the medium.

It was first proposed in 1872 by [[Wolfgang Sellmeier]] and was a development of the work of [[Augustin Louis Cauchy|Augustin Cauchy]] on [[Cauchy's equation]] for modelling dispersion.<ref>
{{cite journal
|last=Sellmeier|first=W.
|title=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)
|journal=Annalen der Physik und Chemie
|volume=223|issue=11|pages=386–403
|date=1872
|doi=10.1002/andp.18722231105
|url=https://zenodo.org/record/1839719
}}</ref>

==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.

==Derivation==
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:
*There exists a number of resonances, and the final refractive index can be calculated from the sum over the contributions from all resonances.
*All optical resonances are at wavelengths far away from the wavelengths of interest, where the model is applied.
*At these resonant frequencies, the imaginary component of the susceptibility (<math>{\chi_i}</math>) can be modeled as a [[delta function]].

From the last point, the complex refractive index (and the [[electric susceptibility]]) becomes:
:<math>\chi_i(\omega) = \sum_i A_i \delta(\omega-\omega_i)</math>

The real part of the refractive index comes from applying the Kramers-Kronig relations to the imaginary part:

:<math> n^2 = 1 + \chi_r(\omega) = 1 + \frac{2}{\pi}\int_0^\infty \frac{\omega \chi_i(\omega)}{\omega ^2 - \Omega ^2}d\omega</math>

Plugging in the first equation above for the imaginary component:

:<math> n^2 = 1 + \frac{2}{\pi}\int_0^\infty \sum_i A_i \delta(\omega-\omega_i) \frac{\omega}{\omega ^2 - \Omega ^2}d\omega</math>

The order of summation and integration can be swapped. When evaluated, this gives the following, where <math>H</math> is the [[Heaviside function]]:
:<math> n^2 = 1 + \frac{2}{\pi} \sum_i A_i \int_0^\infty \delta(\omega-\omega_i) \frac{\omega}{\omega ^2 - \Omega ^2}d\omega = 1 + \frac{2}{\pi} \sum_i A_i \frac{\omega_i H(\omega_i)}{\omega_i^2-\Omega^2}</math>

Since the domain is assumed to be far from any resonances (assumption 2 above), <math>H(\omega_i)</math> evaluates to 1 and a familiar form of the Sellmeier equation is obtained:
:<math> n^2 = 1 + \frac{2}{\pi} \sum_i A_i \frac{\omega_i}{\omega_i^2-\Omega^2}</math> 

By rearranging terms, the constants <math>B_i</math> and <math>C_i</math> can be substituted into the equation above to give the Sellmeier equation.<ref name="b841"></ref>

==Coefficients==
{| class="wikitable" style="text-align:center"
|+ Table of coefficients of Sellmeier equation<ref>{{cite web |url=http://www.lacroixoptical.com/sites/default/files/content/LaCroix%20Dynamic%20Material%20Selection%20Data%20Tool%20vJanuary%202015.xlsm |title=Archived copy |access-date=2015-01-16 |url-status=dead |archive-url=https://web.archive.org/web/20151011033820/http://www.lacroixoptical.com/sites/default/files/content/LaCroix%20Dynamic%20Material%20Selection%20Data%20Tool%20vJanuary%202015.xlsm |archive-date=2015-10-11 }}</ref>
|-
!Material||B<sub>1</sub>||B<sub>2</sub>||B<sub>3</sub>||C<sub>1</sub>, μm<sup>2</sup>||C<sub>2</sub>, μm<sup>2</sup>||C<sub>3</sub>, μm<sup>2</sup>
|-
|[[borosilicate glass|borosilicate]] [[crown glass (optics)|crown glass]]<br />(known as ''BK7'')||1.03961212||0.231792344||1.01046945||6.00069867×10<sup>&minus;3</sup>|| 2.00179144×10<sup>&minus;2</sup>||103.560653
|-
|sapphire<br />(for [[ordinary wave]])||1.43134930||0.65054713||5.3414021||5.2799261×10<sup>&minus;3</sup>|| 1.42382647×10<sup>&minus;2</sup>||325.017834
|-
|sapphire<br />(for [[extraordinary wave]])||1.5039759||0.55069141||6.5927379||5.48041129×10<sup>&minus;3</sup>|| 1.47994281×10<sup>&minus;2</sup>||402.89514
|-
|[[Fused quartz|fused silica]]||0.6961663||0.4079426||0.8974794||0.004679148|| 0.01351206||97.934
|-
|[[Magnesium fluoride]]||0.48755108||0.39875031||2.3120353||0.001882178||0.008951888||566.13559
|}

== See also ==
*[[Cauchy's equation]]

==References==
{{Reflist}}

==External links==
*[http://RefractiveIndex.INFO/ RefractiveIndex.INFO] Refractive index database featuring Sellmeier coefficients for many hundreds of materials.
*[http://www.calctool.org/CALC/phys/optics/sellmeier A browser-based calculator giving refractive index from Sellmeier coefficients.]
*[http://gallica.bnf.fr/ark:/12148/cb34462944f/date Annalen der Physik] - free Access, digitized by the French national library
*[https://web.archive.org/web/20151011033820/http://www.lacroixoptical.com/sites/default/files/content/LaCroix%20Dynamic%20Material%20Selection%20Data%20Tool%20vJanuary%202015.xlsm Sellmeier coefficients for 356 glasses from Ohara, Hoya, and Schott]

[[Category:Eponymous equations of physics]]
[[Category:Optics]]