Sellmeier Equation - The Equation

The Equation

The usual form of the equation for glasses is

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

where n is the refractive index, λ is the wavelength, and 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). A different form of the equation is sometimes used for certain types of materials, e.g. crystals.

As an example, the coefficients for a common borosilicate crown glass known as BK7 are shown below:

Coefficient	Value
B₁	1.03961212
B₂	0.231792344
B₃	1.01046945
C₁	6.00069867×10−3 μm2
C₂	2.00179144×10−2 μm2
C₃	1.03560653×102 μm2

The Sellmeier coefficients for many common optical materials can be found in the online database of RefractiveIndex.info.

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. 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 non-physical values of =±∞, 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 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. Other variants of the Sellmeier equation exist that can account for a material's refractive index change due to temperature, pressure, and other parameters.

Read more about this topic: Sellmeier Equation

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