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.

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.

It was first proposed in 1872 by Wilhelm Sellmeier and was a development of the work of Augustin Cauchy on Cauchy's equation for modelling dispersion.[1]

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 Ci. For example, the coefficients for BK7 below correspond to two absorption resonances in the ultraviolet, and one in the mid-infrared region. 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:[2][3]

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

CoefficientValue
B11.03961212
B20.231792344
B31.01046945
C16.00069867×103 μm2
C22.00179144×102 μm2
C31.03560653×102 μm2

The Sellmeier coefficients for many common optical materials can be found in the online database of RefractiveIndex.info.

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[4] of 365 nm to 2.3 μm, which is of the order of the homogeneity of a glass sample.[5] Additional terms are sometimes added to make the calculation even more precise.

Sometimes the Sellmeier equation is used in two-term form:[6]

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.

Coefficients

Table of coefficients of Sellmeier equation[7]
MaterialB1B2B3C1, μm2C2, μm2C3, μm2
borosilicate crown glass
(known as BK7)
1.039612120.2317923441.010469456.00069867×1032.00179144×102103.560653
sapphire
(for ordinary wave)
1.431349300.650547135.34140215.2799261×1031.42382647×102325.017834
sapphire
(for extraordinary wave)
1.50397590.550691416.59273795.48041129×1031.47994281×102402.89514
fused silica0.6961663000.4079426000.8974794004.67914826×1031.35120631×10297.9340025
Magnesium fluoride0.487551080.398750312.31203530.0018821780.008951888566.13559

See also

References

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