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.

The equation

In its original and the most general form, the Sellmeier equation is given as where 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: 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 (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: 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. When evaluated, this gives the following, where H is the Heaviside function: 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: By rearranging terms, the constants B_i and C_i can be substituted into the equation above to give the Sellmeier equation.

Coefficients

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