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+ \frac+ \frac]

where B_1,2,3 and C_1,2,3 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

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:

[n^2(\lambda) = 1 + \sum_i \frac]

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:

[\beginn \approx \sqrt \approx \sqrt\end]

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 + \frac]

here the coefficient A is an approximation of the short-wavelength (e.g., ultraviolet) absorption contributions to the refractive index at longer wavelengths.

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