Sellmeier equation

Refractive index vs. wavelength for BK7 glass, showing measured points (blue crosses) and the Sellmeier equation (red line).

Same as the graph above, but with Cauchy's equation (blue line) for comparison. While Cauchy's equation (blue line) deviates significantly from the measured refractive indices outside of the visible region (which is shaded red), the Sellmeier equation (green dashed line) do not.

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 1871 by Wilhelm Sellmeier, and was a development of the work of Augustin Cauchy on Cauchy's equation for modelling dispersion.

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.^[1] 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 μm^2
C2	2.00179144×10^−2 μm^2
C3	1.03560653×10^2 μm^2

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^[2] of 365 nm to 2.3 µm, which is of the order of the homogeneity of a glass sample.^[3] 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 n² = ±∞, 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:^[4]

$n^2(\lambda) = A + \frac{B_1\lambda^2}{\lambda^2 - C_1} + \frac{ B_2 \lambda^2}{\lambda^2 - C_2}.$

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.

Coefficients

Table of coefficients of Sellmeier equation^[5]

Material	B1	B2	B3	C1	C2	C3
borosilicate crown glass (known as BK7)	1.03961212	0.231792344	1.01046945	6.00069867×10^−3µm^2	2.00179144×10^−2µm^2	1.03560653×10^2µm^2
sapphire (for ordinary wave)	1.43134930	0.65054713	5.3414021	5.2799261×10^−3µm^2	1.42382647×10^−2µm^2	3.25017834×10^2µm^2
sapphire (for extraordinary wave)	1.5039759	0.55069141	6.5927379	5.48041129×10^−3µm^2	1.47994281×10^−2µm^2	4.0289514×10^2µm^2
fused silica	0.696166300	0.407942600	0.897479400	4.67914826×10^−3µm^2	1.35120631×10^−2µm^2	97.9340025 µm^2

See also

• Cauchy's equation

References

1. Refractive index and dispersion. Schott technical information document TIE-29 (2007).
2. http://oharacorp.com/o2.html
3. http://oharacorp.com/o7.html
4. Ghosh, Gorachand. "Sellmeier Coefficients and Dispersion of Thermo-Optic coefficients for some optical glasses" (PDF). Applied Optics. Retrieved 18 February 2014.
5. http://www.lacroixoptical.com/sites/default/files/content/LaCroix%20Dynamic%20Material%20Selection%20Data%20Tool%20vJanuary%202015.xlsm

• W. Sellmeier, Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen, Annalen der Physik und Chemie 219, 272-282 (1871).

External links

• RefractiveIndex.INFO Refractive index database featuring Sellmeier coefficients for many hundreds of materials.
• A browser-based calculator giving refractive index from Sellmeier coefficients.
• Annalen der Physik - free Access, digitized by the French national library
• Sellmeier coefficients for 356 glasses from Ohara, Hoya, and Schott