Sellmeier equation

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$A plot of the refractive index vs. wavelength using the Sellmeier equation for BK7 glass.$

A plot of the refractive index vs. wavelength using the Sellmeier equation for BK7 glass.

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:

$n^2(\lambda) = 1 + \frac{B_1 \lambda^2 }{ \lambda^2 - C_1} + \frac{B_2 \lambda^2 }{ \lambda^2 - C_2} + \frac{B_3 \lambda^2 }{ \lambda^2 - C_3}$

(definition used by Schott Glass)

where B_1,2,3 and C_1,2,3 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(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, or in the Ohara catalogue.

For common optical glasses, the refractive index calculated with the 3 terms Sellmeier equation only deviates from the actual refractive index by less than 5.10^-6 for wavelengths ranging fron 365nm to 2.3µm [1], which is of the order of the homogeneity of a glass sample [2]. Additional terms are sometimes added to make the calculation even more precise. In its most general form, the Sellmeier equation is given as:

$n^2(\lambda) = 1 + \sum_i \frac{B_i \lambda^2}{\lambda^2 - C_i}$

with each term of the sum representing an absorption resonance of strength B_i at a wavelength √C_i. 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:

$\begin{matrix} n \approx \sqrt{1 + \sum_i B_i } \approx \sqrt{\varepsilon_r} \end{matrix}$

where ε_r is the relative dielectric constant of the medium.

The Sellmeier equation can also be given in another 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.

[edit] Coefficients

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Table of coefficients of Sellmeier equation
Material	B₁	B₂	B₃	C₁	C₂	C₃
borosilicate crown glass (known as BK7)	1.03961212	2.31792344x10⁻¹	1.01046945	6.00069867x10⁻³µm²	2.00179144x10⁻²µm²	1.03560653x10²µm²
sapphire (for ordinary wave)	1.43134930	6.5054713x10⁻¹	5.3414021	5.2799261x10⁻³µm²	1.42382647x10⁻²µm²	3.25017834x10²µm²
sapphire (for extraordinary wave)	1.5039759	5.5069141x10⁻¹	6.5937379	5.48041129x10⁻³µm²	1.47994281x10⁻²µm²	4.0289514x10²µm²