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.

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.

Description

In its original and the most general form, the Sellmeier equation is given as
:$n^2(\lambda) = 1 + \sum_i \frac$,
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 . 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. However, close to each absorption peak, the equation gives non-physical values of ''n''² = ±∞, 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
:$\begin n \approx \sqrt \approx \sqrt \end,$
where ε_r is the relative permittivity of the medium.

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

:$n^2(\lambda) = 1 + \frac + \frac + \frac,$

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

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:
:$n^2(\lambda) = A + \frac + \frac.$
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 ($$) can be modeled as a delta function.

From the last point, the complex refractive index (and the electric susceptibility) becomes:
:$\chi_i(\omega) = \sum_i A_i \delta(\omega-\omega_i)$

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

:$n^2 = 1 + \chi_r(\omega) = 1 + \frac\int_0^\infty \fracd\omega$

Plugging in the first equation above for the imaginary component:

:$n^2 = 1 + \frac\int_0^\infty \sum_i A_i \delta(\omega-\omega_i) \fracd\omega$

The order of summation and integration can be swapped. When evaluated, this gives the following, where $H$ is the Heaviside function:
:$n^2 = 1 + \frac \sum_i A_i \int_0^\infty \delta(\omega-\omega_i) \fracd\omega = 1 + \frac \sum_i A_i \frac$

Since the domain is assumed to be far from any resonances (assumption 2 above), $H(\omega_i)$ evaluates to 1 and a familiar form of the Sellmeier equation is obtained:
:$n^2 = 1 + \frac \sum_i A_i \frac$

By rearranging terms, the constants $B_i$ and $C_i$ can be substituted into the equation above to give the Sellmeier equation.

Coefficients

See also

* Cauchy's equation

References

{{Reflist

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

Eponymous equations of physics
Optics