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 aswhere n is the refractive index, λ is the wavelength, and Bi and Ci 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 Bi 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 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
where εr is the relative permittivity of the medium.
For characterization of glasses the equation consisting of three terms is commonly used:
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:
Here the coefficient A is an approximation of the short-wavelength 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.
The real part of the refractive index comes from applying the Kramers-Kronig relations to the imaginary part:
Plugging in the first equation above for the imaginary component:
The order of summation and integration can be swapped. When evaluated, this gives the following, where is the Heaviside function:
Since the domain is assumed to be far from any resonances, evaluates to 1 and a familiar form of the Sellmeier equation is obtained:
By rearranging terms, the constants and can be substituted into the equation above to give the Sellmeier equation.