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.

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

n^2(\lambda) = 1 + \sum_i \frac{B_i \lambda^2}{\lambda^2 - C_i} $$, 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 $\sqrt{C_{i}}|undefined$. 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
 * $$\begin{matrix}

n \approx \sqrt{1 + \sum_i B_i } \approx \sqrt{\varepsilon_r} \end{matrix},$$ where εr is the relative permittivity of the medium.

For characterization of glasses the equation consisting of three terms is commonly used:



n^2(\lambda) = 1 + \frac{B_1 \lambda^2 }{ \lambda^2 - C_1} + \frac{B_2 \lambda^2 }{ \lambda^2 - C_2} + \frac{B_3 \lambda^2 }{ \lambda^2 - C_3}, $$

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:

n^2(\lambda) = A + \frac{B_1\lambda^2}{\lambda^2 - C_1} + \frac{ 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.

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:
 * There exists a number of resonances, and the final refractive index can be calculated from the sum over the contributions from all resonances.
 * All optical resonances are at wavelengths far away from the wavelengths of interest, where the model is applied.
 * At these resonant frequencies, the imaginary component of the susceptibility ($${\chi_i}$$) can be modeled as a delta function.

From the last point, the complex refractive index (and the electric susceptibility) becomes:
 * $$\chi_i(\omega) = \sum_i A_i \delta(\omega-\omega_i)$$

The real part of the refractive index comes from applying the Kramers-Kronig relations to the imaginary part:


 * $$ n^2 = 1 + \chi_r(\omega) = 1 + \frac{2}{\pi}\int_0^\infty \frac{\omega \chi_i(\omega)}{\omega ^2 - \Omega ^2}d\omega$$

Plugging in the first equation above for the imaginary component:


 * $$ n^2 = 1 + \frac{2}{\pi}\int_0^\infty \sum_i A_i \delta(\omega-\omega_i) \frac{\omega}{\omega ^2 - \Omega ^2}d\omega$$

The order of summation and integration can be swapped. When evaluated, this gives the following, where $$H$$ is the Heaviside function:
 * $$ n^2 = 1 + \frac{2}{\pi} \sum_i A_i \int_0^\infty \delta(\omega-\omega_i) \frac{\omega}{\omega ^2 - \Omega ^2}d\omega = 1 + \frac{2}{\pi} \sum_i A_i \frac{\omega_i H(\omega_i)}{\omega_i^2-\Omega^2}$$

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:
 * $$ n^2 = 1 + \frac{2}{\pi} \sum_i A_i \frac{\omega_i}{\omega_i^2-\Omega^2}$$

By rearranging terms, the constants $$B_i$$ and $$C_i$$ can be substituted into the equation above to give the Sellmeier equation.

Internal links

 * RefractiveIndex.INFO Refractive index database featuring Sellmeier coefficients for many hundreds of materials.
 * A browser-based calculator giving refractive index from Sellmeier coefficients.
 * Annalen der Physik - free Access, digitized by the French national library
 * Sellmeier coefficients for 356 glasses from Ohara, Hoya, and Schott