SELLMEIER EQUATION

In optics, the 'Sellmeier equation' is an empirical relationship between refractive index ''n'' and wavelength ''λ'' for a particular transparent medium. The usual form of the equation for glasses[1] is
:$$
n^2(lambda) = 1
+ rac{B_1 lambda^2 }{ lambda^2 - C_1}
+ rac{B_2 lambda^2 }{ lambda^2 - C_2}
+ rac{B_3 lambda^2 }{ lambda^2 - C_3},

where ''B''_1,2,3 and ''C''_1,2,3 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''(λ).
The equation is used to determine the dispersion of light in a refracting medium. A different form of the equation is sometimes used for certain types of materials, e.g. crystals.
The equation was deduced in 1871 by W. Sellmeier, and was a development of the work of Augustin Cauchy on Cauchy's equation for modelling dispersion.
As an example, the coefficients for a common borosilicate crown glass known as ''BK7'' are shown below:

Coefficient	Value
B1	1.03961212
B2	2.31792344x10−1
B3	1.01046945
C1	6.00069867x10−3 μm2
C2	2.00179144x10−2 μm2
C3	1.03560653x102 μm2

The Sellmeier coefficients for many common optical glasses can be found in the Schott Glass catalogue, or in the Ohara catalogue.
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[2], which is of the order of the homogeneity of a glass sample [3]. Additional terms are sometimes added to make the calculation even more precise. In its most general form, the Sellmeier equation is given as
:$$
n^2(lambda) = 1 + sum_i rac{B_i lambda^2}{lambda^2 - C_i},

with each term of the sum representing an absorption resonance of strength ''B''_i at a wavelength √''C''_i. For example, the coefficients for BK7 above 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 ''n''=±∞, 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
:
n pprox sqrt{1 + sum_i B_i } pprox sqrt{arepsilon_r}
end{matrix},
where ε_r is the relative dielectric constant of the medium.
The Sellmeier equation can also be given in another form:
:$$
n^2(lambda) = A + rac{B_1 lambda^2}{lambda^2 - C_1} + rac{ B_2 lambda^2}{lambda^2 - C_2}.

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.

Contents

Coefficients

External links

Coefficients

Table of coefficients of Sellmeier equation[4]
Material	B1	B2	B3	C1	C2	C3
borosilicate crown glass (known as ''BK7'')	1.03961212	2.31792344x10−1	1.01046945	6.00069867x10−3µm2	2.00179144x10−2µm2	1.03560653x102µm2
sapphire (for ordinary wave)	1.43134930	6.5054713x10−1	5.3414021	5.2799261x10−3µm2	1.42382647x10−2µm2	3.25017834x102µm2
sapphire (for extraordinary wave)	1.5039759	5.5069141x10−1	6.5937379	5.48041129x10−3µm2	1.47994281x10−2µm2	4.0289514x102µm2

External links

★ A PDF giving Sellmeier coefficients for several common glasses and optical materials