Sellmeier equation

Analytically, this process is based on approximating the underlying optical resonances as dirac delta functions, followed by the application of the Kramers-Kronig relations.

[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 where εr is the relative permittivity of the medium.

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

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

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.