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.^{[1]}
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 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 √C_{i}. 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.^{[2]} 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
where ε_{r} is the relative permittivity of the medium.
For characterization of glasses the equation consisting of three terms is commonly used:^{[3]}^{[4]}
As an example, the coefficients for a common borosilicate crown glass known as BK7 are shown below:
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} | 1.03560653×10^{2} μm^{2} |
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^{[5]} of 365 nm to 2.3 μm, which is of the order of the homogeneity of a glass sample.^{[6]} Additional terms are sometimes added to make the calculation even more precise.
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. When evaluated, this gives the following, where is the Heaviside function:
Since the domain is assumed to be far from any resonances (assumption 2 above), 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.^{[2]}
Material | B_{1} | B_{2} | B_{3} | C_{1}, μm^{2} | C_{2}, μm^{2} | C_{3}, μm^{2} |
---|---|---|---|---|---|---|
borosilicate crown glass (known as BK7) |
1.03961212 | 0.231792344 | 1.01046945 | 6.00069867×10^{−3} | 2.00179144×10^{−2} | 103.560653 |
sapphire (for ordinary wave) |
1.43134930 | 0.65054713 | 5.3414021 | 5.2799261×10^{−3} | 1.42382647×10^{−2} | 325.017834 |
sapphire (for extraordinary wave) |
1.5039759 | 0.55069141 | 6.5927379 | 5.48041129×10^{−3} | 1.47994281×10^{−2} | 402.89514 |
fused silica | 0.696166300 | 0.407942600 | 0.897479400 | 4.67914826×10^{−3} | 1.35120631×10^{−2} | 97.9340025 |
Magnesium fluoride | 0.48755108 | 0.39875031 | 2.3120353 | 0.001882178 | 0.008951888 | 566.13559 |
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