# 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 Wolfgang Sellmeier and was a development of the work of Augustin Cauchy on Cauchy's equation for modelling dispersion.

## Description

In its original and the most general form, the Sellmeier equation is given as
    $n^{2}(\lambda )=1+\sum _{i}{\frac {B_{i}\lambda ^{2}}{\lambda ^{2}-C_{i}}}$,
where n is the refractive index, λ is the wavelength, and B$_i$ and C$_i$ are experimentally determined Sellmeier coefficients. These coefficients are usually quoted for λ in micrometres. 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. crystals.
Each term of the sum representing an absorption resonance of strength B$_i$ at a wavelength √C$_i$. 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. However, 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
    ${\begin{matrix}n\approx {\sqrt {1+\sum _{i}B_{i}}}\approx {\sqrt {\varepsilon _{r}}}\end{matrix}},$
where ε$_r$ is the relative permittivity of the medium.
For characterization of glasses the equation consisting of three terms is commonly used:
    $n^{2}(\lambda )=1+{\frac {B_{1}\lambda ^{2}}{\lambda ^{2}-C_{1}}}+{\frac {B_{2}\lambda ^{2}}{\lambda ^{2}-C_{2}}}+{\frac {B_{3}\lambda ^{2}}{\lambda ^{2}-C_{3}}},$
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$^{−3}$ μm$^2$ |
| C$_2$ | 2.00179144×10$^{−2}$ μm$^2$ |
| C$_3$ | 1.03560653×10$^2$ μm$^2$ |
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 of 365 nm to 2.3 μm, which is of the order of the homogeneity of a glass sample. Additional terms are sometimes added to make the calculation even more precise.
Sometimes the Sellmeier equation is used in two-term form:
    $n^{2}(\lambda )=A+{\frac {B_{1}\lambda ^{2}}{\lambda ^{2}-C_{1}}}+{\frac {B_{2}\lambda ^{2}}{\lambda ^{2}-C_{2}}}.$
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 (${\chi _{i}}$) can be modeled as a delta function.
From the last point, the complex refractive index (and the electric susceptibility) becomes:
    $\chi _{i}(\omega )=\sum _{i}A_{i}\delta (\omega -\omega _{i})$
The real part of the refractive index comes from applying the Kramers-Kronig relations to the imaginary part:
    $n^{2}=1+\chi _{r}(\omega )=1+{\frac {2}{\pi }}\int _{0}^{\infty }{\frac {\omega \chi _{i}(\omega )}{\omega ^{2}-\Omega ^{2}}}d\omega$
Plugging in the first equation above for the imaginary component:
    $n^{2}=1+{\frac {2}{\pi }}\int _{0}^{\infty }\sum _{i}A_{i}\delta (\omega -\omega _{i}){\frac {\omega }{\omega ^{2}-\Omega ^{2}}}d\omega$
The order of summation and integration can be swapped. When evaluated, this gives the following, where $H$ is the Heaviside function:
    $n^{2}=1+{\frac {2}{\pi }}\sum _{i}A_{i}\int _{0}^{\infty }\delta (\omega -\omega _{i}){\frac {\omega }{\omega ^{2}-\Omega ^{2}}}d\omega =1+{\frac {2}{\pi }}\sum _{i}A_{i}{\frac {\omega _{i}H(\omega _{i})}{\omega _{i}^{2}-\Omega ^{2}}}$
Since the domain is assumed to be far from any resonances (assumption 2 above), $H(\omega _{i})$ evaluates to 1 and a familiar form of the Sellmeier equation is obtained:
    $n^{2}=1+{\frac {2}{\pi }}\sum _{i}A_{i}{\frac {\omega _{i}}{\omega _{i}^{2}-\Omega ^{2}}}$
By rearranging terms, the constants $B_{i}$ and $C_{i}$ can be substituted into the equation above to give the Sellmeier equation.

## Coefficients

Table of coefficients of Sellmeier equation :
| Material | B$_1$ | B$_2$ | B$_3$ | C$_1$, μm$^2$ | C$_2$, μm$^2$ | C$_3$, μm$^2$ |
| 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 |