Sellmeier equation

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.

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.

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.^[1]

The equation

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

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

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. Analytically, this process is based on approximating the underlying optical resonances as dirac delta functions, followed by the application of the Kramers-Kronig relations. This results in real and imaginary parts of the refractive index which are physically sensible.^[2] However, 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

${\displaystyle {\begin{matrix}n\approx {\sqrt {1+\sum _{i}B_{i}}}\approx {\sqrt {\varepsilon _{r}}}\end{matrix}},}$

where ε_r is the relative permittivity of the medium.

For characterization of glasses the equation consisting of three terms is commonly used:^[3][4]

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

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^[5] of 365 nm to 2.3 μm, which is of the order of the homogeneity of a glass sample.^[6] Additional terms are sometimes added to make the calculation even more precise.

Sometimes the Sellmeier equation is used in two-term form:^[7]

${\displaystyle n^{2}(\lambda )=A+{\frac {B_{1}\lambda ^{2}}{\lambda ^{2}-C_{1}}}+{\frac {B_{2}\lambda ^{2}}{\lambda ^{2}-C_{2}}}.}$

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.

Derivation

Analytically, the Sellmeier equation models the refractive index as due to a series of optical resonances within the bulk material. Its derivation from the Kramers-Kronig relations requires a few assumptions about the material, from which any deviations will affect the model's accuracy:

• There exists a number of resonances, and the final refractive index can be calculated from the sum over the contributions from all resonances.
• All optical resonances are at wavelengths far away from the wavelengths of interest, where the model is applied.
• At these resonant frequencies, the imaginary component of the susceptibility (${\displaystyle {\chi _{i}}}$) can be modeled as a delta function.

From the last point, the complex refractive index (and the electric susceptibility) becomes:

${\displaystyle \chi _{i}(\omega )=\sum _{i}A_{i}\delta (\omega -\omega _{i})}$

The real part of the refractive index comes from applying the Kramers-Kronig relations to the imaginary part:

${\displaystyle n^{2}=1+\chi _{r}(\omega )=1+{\frac {2}{\pi }}\int _{0}^{\infty }{\frac {\omega \chi _{i}(\omega )}{\omega ^{2}-\Omega ^{2}}}d\omega }$

Plugging in the first equation above for the imaginary component:

${\displaystyle n^{2}=1+{\frac {2}{\pi }}\int _{0}^{\infty }\sum _{i}A_{i}\delta (\omega -\omega _{i}){\frac {\omega }{\omega ^{2}-\Omega ^{2}}}d\omega }$

The order of summation and integration can be swapped. When evaluated, this gives the following, where ${\displaystyle H}$ is the Heaviside function:

${\displaystyle n^{2}=1+{\frac {2}{\pi }}\sum _{i}A_{i}\int _{0}^{\infty }\delta (\omega -\omega _{i}){\frac {\omega }{\omega ^{2}-\Omega ^{2}}}d\omega =1+{\frac {2}{\pi }}\sum _{i}A_{i}{\frac {\omega _{i}H(\omega _{i})}{\omega _{i}^{2}-\Omega ^{2}}}}$

Since the domain is assumed to be far from any resonances (assumption 2 above), ${\displaystyle H(\omega _{i})}$ evaluates to 1 and a familiar form of the Sellmeier equation is obtained:

${\displaystyle n^{2}=1+{\frac {2}{\pi }}\sum _{i}A_{i}{\frac {\omega _{i}}{\omega _{i}^{2}-\Omega ^{2}}}}$

By rearranging terms, the constants ${\displaystyle B_{i}}$ and ${\displaystyle C_{i}}$ can be substituted into the equation above to give the Sellmeier equation.^[2]

Coefficients

Table of coefficients of Sellmeier equation^[8]

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

References

1. Sellmeier, W. (1872). "Ueber die durch die Aetherschwingungen erregten Mitschwingungen der Körpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien (II. Theil)". Annalen der Physik und Chemie. 223 (11): 386–403. doi:10.1002/andp.18722231105.
2. "2.7: Kramers-Kroenig Relations". Engineering LibreTexts. 2021-04-06. Retrieved 2024-07-09.
3. Refractive index and dispersion. Schott technical information document TIE-29 (2007).
4. Paschotta, Dr. Rüdiger. "Encyclopedia of Laser Physics and Technology - Sellmeier formula, refractive index, Sellmeier equation, dispersion formula". www.rp-photonics.com. Retrieved 2018-09-14.
5. "Optical Properties".
6. "Guarantee of Quality".
7. Ghosh, Gorachand (1997). "Sellmeier Coefficients and Dispersion of Thermo-Optic coefficients for some optical glasses". Applied Optics. 36 (7): 1540–6. Bibcode:1997ApOpt..36.1540G. doi:10.1364/AO.36.001540. PMID 18250832.
8. "Archived copy". Archived from the original on 2015-10-11. Retrieved 2015-01-16.{{cite web}}: CS1 maint: archived copy as title (link)

Internal 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