Sellmeier equation

The Sellmeier equation in optics is an empirically determined, functional description of the dependence of the refractive index of a light-permeable medium on the wavelength of light. The equation was named after Wolfgang von Sellmeier , who published it in 1871 based on the Cauchy equation and Kramers-Kronig relation . It is mainly used in technical optics to describe the dispersion of optical glass and other optical materials. ${\displaystyle n}$ ${\displaystyle \lambda }$

Mathematical description

Example: Coefficients for borosilicate glass BK7

coefficient	value
B 1	1.03961212
B 2	0.231792344
B 3	1.01046945
C 1	6.00069867 10 −3 μm 2
C 2	2.00179144 10 −2 μm 2
C 3	103.560653 μm 2

Representation of the refractive index of borosilicate glass (BK7) against the wavelength. In the diagram, the measured values ​​and the corresponding parametric adjustments of the Cauchy or Sellmeier equation are compared with one another.

The Sellmeier equation can be understood as an extension of the Cauchy equation , it reads:

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

with B _1,2,3 and C _1,2,3 as experimentally determined Sellmeier coefficients . The B _1,2,3 are dimensionless and the C _1,2,3 are usually given in μm² .

The accuracy in the visible range is usually better than . ${\displaystyle \pm 5\cdot 10^{-6}}$

The right term of the equation can also be expanded to include additional summands for greater accuracy:

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

If one sets , then they can be explained as resonance wavelengths of absorption lines or bands. ${\displaystyle C_{i}=\lambda _{i}^{2}}$ ${\displaystyle \lambda _{i}}$

See also

• Achromat
• Apochromat
• Chromatic aberration .

Individual evidence

1. ↑ Dirk Poelman, Philippe Frederic Smet: Methods for the determination of the optical constants of thin films from single transmission measurements: a critical review . In: Journal of Physics D: Applied Physics . tape 36 , no. 15 , 2003, p. 1850–1857 , doi : 10.1088 / 0022-3727 / 36/15/316 .
2. ↑ Wolfgang von Sellmeier: To explain the abnormal color sequence in the spectrum of some substances . In: Annals of Physics and Chemistry . tape 143 , 1871, pp. 272–282 , doi : 10.1002 / andp.18712190612 ( digitized on Gallica ).