Sellmeier equation
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In optics, the Sellmeier equation is an empirical relationship between refractive index n and wavelength λ for a particular transparent medium. The usual form of the equation for glasses is:
- <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>
where B1,2,3 and C1,2,3 are experimentally determined Sellmeier coefficients. These coefficients are usually quoted for λ measured in micrometres. Note that this λ is the vacuum wavelength; not that in the material itself, which is λ/n(λ).
The equation is used to determine the dispersion of light in a refracting medium. A different form of the equation is sometimes used for certain types of materials, e.g. crystals.
The equation was deduced in 1871 by W. Sellmeier, and was a development of the work of Augustin Cauchy on Cauchy's equation for modelling dispersion.
As an example, the coefficients for a common borosilicate crown glass known as BK7 are shown below:
| Coefficient | Value |
|---|---|
| B1 | 1.03961212 |
| B2 | 2.31792344x10−1 |
| B3 | 1.01046945 |
| C1 | 6.00069867x10−3 μm2 |
| C2 | 2.00179144x10−2 μm2 |
| C3 | 1.03560653x102 μm2 |
The Sellmeier coefficients for many common optical glasses can be found in the Schott Glass catalogue.
In its 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> with each term of the sum representing an absorption resonance of strength Bi at a wavelength √Ci. 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 unphysical values of n=±∞, and in these wavelength regions a more precise model of dispersion such as Helmholtz's must be used.
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 εr is the relative dielectric constant of the medium.
The Sellmeier equation can also be given in another form:
- <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.