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) does 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 1872 by Wolfgang Sellmeier and was a development of the work of Augustin Cauchy on Cauchy's equation for modelling dispersion.

The equation

In its original and the most general form, the Sellmeier equation is given as

$n^{2}(\lambda )=1+\sum _{i}{\frac {B_{i}\lambda ^{2}}{\lambda ^{2}-C_{i}}}$,

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

Each term of the sum representing an absorption resonance of strength B_i at a wavelength √C_i. For example, the coefficients for BK7 below 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 permittivity of the medium.

For characterization of glasses the equation consisting of three terms is commonly used:

$n^{2}(\lambda )=1+{\frac {B_{1}\lambda ^{2}}{\lambda ^{2}-C_{1}}}+{\frac {B_{2}\lambda ^{2}}{\lambda ^{2}-C_{2}}}+{\frac {B_{3}\lambda ^{2}}{\lambda ^{2}-C_{3}}},$

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

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.

Sometimes the Sellmeier equation is used in two-term 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.

Coefficients

Table of coefficients of Sellmeier equation

Material	B1	B2	B3	C1, μm2	C2, μm2	C3, μm2
borosilicate crown glass (known as BK7)	1.03961212	0.231792344	1.01046945	6.00069867×10−3	2.00179144×10−2	103.560653
sapphire (for ordinary wave)	1.43134930	0.65054713	5.3414021	5.2799261×10−3	1.42382647×10−2	325.017834
sapphire (for extraordinary wave)	1.5039759	0.55069141	6.5927379	5.48041129×10−3	1.47994281×10−2	402.89514
fused silica	0.696166300	0.407942600	0.897479400	4.67914826×10−3	1.35120631×10−2	97.9340025
Magnesium fluoride	0.48755108	0.39875031	2.3120353	0.001882178	0.008951888	566.13559

See also

• Cauchy's equation

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