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| [[File:Sellmeier-equation.svg|thumb|right|Refractive index vs. wavelength for [[BK7 glass]], showing measured points (blue crosses) and the Sellmeier equation (red line).]]
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| 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.
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| It was first proposed in 1871 by Wilhelm Sellmeier, and was a development of the work of [[Augustin Louis Cauchy|Augustin Cauchy]] on [[Cauchy's equation]] for modelling dispersion.
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| ==The equation==
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| The usual form of the equation for glasses is
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| :<math>
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| n^2(\lambda) = 1
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| + \frac{B_1 \lambda^2 }{ \lambda^2 - C_1}
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| + \frac{B_2 \lambda^2 }{ \lambda^2 - C_2}
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| + \frac{B_3 \lambda^2 }{ \lambda^2 - C_3},
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| </math>
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| where ''n'' is the refractive index, ''λ'' is the wavelength, and ''B''<sub>1,2,3</sub> and ''C''<sub>1,2,3</sub> are experimentally determined ''Sellmeier [[coefficient]]s''.<ref>[http://www.us.schott.com/advanced_optics/english/download/schott_tie-29_refractive_index_v3_jan_2007_us.pdf Refractive index and dispersion]. Schott technical information document TIE-29 (2007).</ref> 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''(λ). A different form of the equation is sometimes used for certain types of materials, e.g. [[crystal]]s.
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| As an example, the coefficients for a common [[borosilicate glass|borosilicate]] [[Crown glass (optics)|crown glass]] known as ''BK7'' are shown below:
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| {| class="wikitable"
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| |-
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| ! Coefficient !! Value
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| | B<sub>1</sub> || 1.03961212
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| |-
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| | B<sub>2</sub> || 0.231792344
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| |-
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| | B<sub>3</sub> || 1.01046945
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| |-
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| | C<sub>1</sub> || 6.00069867×10<sup>−3</sup> μm<sup>2</sup>
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| |-
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| | C<sub>2</sub> || 2.00179144×10<sup>−2</sup> μm<sup>2</sup>
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| |-
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| | C<sub>3</sub> || 1.03560653×10<sup>2</sup> μm<sup>2</sup>
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| |}
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| The Sellmeier coefficients for many common optical materials can be found in the online database of [http://refractiveindex.info RefractiveIndex.info].
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| 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>http://oharacorp.com/o2.html</ref> of 365 nm to 2.3 µm, which is of the order of the homogeneity of a glass sample.<ref>http://oharacorp.com/o7.html</ref> Additional terms are sometimes added to make the calculation even more precise. In its most general form, the Sellmeier equation is given as
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| :<math> | |
| n^2(\lambda) = 1 + \sum_i \frac{B_i \lambda^2}{\lambda^2 - C_i},
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| </math>
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| with each term of the sum representing an [[absorption (optics)|absorption]] resonance of strength ''B''<sub>i</sub> at a wavelength √''C''<sub>i</sub>. For example, the coefficients for BK7 above 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 ''<math>n^2</math>''=±∞, and in these wavelength regions a more precise model of dispersion such as [[Helmholtz dispersion|Helmholtz's]] must be used.
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| If all terms are specified for a material, at long wavelengths far from the absorption peaks the value of ''n'' tends to
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| :<math>\begin{matrix}
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| n \approx \sqrt{1 + \sum_i B_i } \approx \sqrt{\varepsilon_r}
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| \end{matrix},</math>
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| where ε<sub>r</sub> is the relative [[dielectric constant]] of the medium.
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| The Sellmeier equation can also be given in another form:
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| :<math>
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| n^2(\lambda) = A + \frac{B_1}{\lambda^2 - C_1} + \frac{ B_2 \lambda^2}{\lambda^2 - C_2}.
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| </math>
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| 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.
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| ==Coefficients==
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| {| class="wikitable" style="text-align:center"
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| |+ Table of coefficients of Sellmeier equation<ref>http://cvimellesgriot.com/products/Documents/Catalog/Dispersion_Equations.pdf</ref>
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| !Material||B<sub>1||B<sub>2||B<sub>3||C<sub>1||C<sub>2||C<sub>3
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| |[[borosilicate glass|borosilicate]] [[glass|crown glass]]<br>(known as ''BK7'')||1.03961212||0.231792344||1.01046945||6.00069867×10<sup>−3</sup>µm<sup>2</sup>|| 2.00179144×10<sup>−2</sup>µm<sup>2</sup>||1.03560653×10<sup>2</sup>µm<sup>2</sup>
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| |sapphire<br>(for [[ordinary wave]])||1.43134930||0.65054713||5.3414021||5.2799261×10<sup>−3</sup>µm<sup>2</sup>|| 1.42382647×10<sup>−2</sup>µm<sup>2</sup>||3.25017834×10<sup>2</sup>µm<sup>2</sup>
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| |sapphire<br>(for [[extraordinary wave]])||1.5039759||0.55069141||6.5927379||5.48041129×10<sup>−3</sup>µm<sup>2</sup>|| 1.47994281×10<sup>−2</sup>µm<sup>2</sup>||4.0289514×10<sup>2</sup>µm<sup>2</sup>
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| |[[Fused quartz|fused silica]]||0.696166300||0.407942600||0.897479400||4.67914826×10<sup>−3</sup>µm<sup>2</sup>|| 1.35120631×10<sup>−2</sup>µm<sup>2</sup>||97.9340025 µm<sup>2</sup>
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| |}
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| == See also ==
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| *[[Cauchy's equation]]
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| *[[Kramers–Kronig relation]]
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| ==References==
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| <references />
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| *W. Sellmeier, Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen, ''Annalen der Physik und Chemie'' '''219''', 272-282 (1871).
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| ==External links==
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| *[http://RefractiveIndex.INFO/ RefractiveIndex.INFO] Refractive index database featuring Sellmeier coefficients for many hundreds of materials.
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| *[http://www.calctool.org/CALC/phys/optics/sellmeier A browser-based calculator giving refractive index from Sellmeier coefficients.]
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| *[http://gallica.bnf.fr/ark:/12148/cb34462944f/date Annalen der Physik] - free Access, digitized by the French national library
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| [[Category:Optics]]
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| [[Category:Equations]]
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vitiligo pictures is something that has specially made for the awareness with the Vitiligo skin disorder disease by our team and can be checked at our website. Special cells called melanocytes are to blame for the construction of melanin. Basically, melanin is a derivative of tyrosine, an amino acid. If you want to know more about vitiligo disease, vitiligo cure and vitiligo treatment then visit our site: how to treat vitiligo</a>, you can call us at our own webpage.