Sellmeier equation

The Sellmeier equation describes how the refractive index of a transparent material changes with wavelength. It is the standard form in which glass catalogs and crystal data sheets publish dispersion:

$$n^2(\lambda) = 1 + \sum_i \frac{B_i\,\lambda^2}{\lambda^2 - C_i} ,$$

with dimensionless strengths $B_i$ and resonance terms $C_i$ in µm² when $\lambda$ is in µm. Three terms usually reproduce measured indices within their measurement uncertainty across a glass's transparent range. For fused silica, Malitson's 1965 coefficients are $B$ = 0.6961663, 0.4079426, 0.8974794 and $C$ = 0.0684043², 0.1162414², 9.896161² µm², which give $n$ = 1.4570 at 633 nm, 1.4496 at 1064 nm and 1.4440 at 1550 nm.

Why the form works

Each term is the contribution of one Lorentz oscillator with its damping neglected: a bound charge with a resonance at wavelength $\sqrt{C_i}$ adds a polarization that grows as the light approaches that resonance. For silica the first two resonances sit in the ultraviolet, at 0.068 and 0.116 µm (electronic transitions), and the third in the infrared, at 9.90 µm (lattice vibrations of the Si–O bond). Between them, the ultraviolet terms raise the index toward short wavelengths and the infrared term pulls it down toward long ones. At 1550 nm the infrared term is already negative, −0.023 in $n^2$, and lowers the index from 1.4518 to 1.4440. The Kramers–Kronig relations make this picture exact in principle: the index at each wavelength follows from the absorption at all others.

Because damping is omitted, the formula diverges at each $C_i$ and fails inside absorption bands, where the index can rise with wavelength (anomalous dispersion). Malitson's fit is stated for 0.21–3.71 µm, the range of his measurements, at room temperature; extrapolating outside it, or using it at cryogenic or high temperature without a thermo-optic correction, gives errors of uncertain size.

Group index and dispersion

Derivatives of the Sellmeier equation give the quantities that matter for pulses and signals. The group index is

$$n_g = n - \lambda\,\frac{dn}{d\lambda} .$$

At 1550 nm, $dn/d\lambda$ = −0.01198 µm⁻¹ for fused silica, so $n_g$ = 1.4440 + 1.55 × 0.01198 = 1.4626, the value that sets the speed of light in fiber for the bulk glass. The second derivative gives the material chromatic dispersion:

$$D_M = -\frac{\lambda}{c}\,\frac{d^2 n}{d\lambda^2} .$$

For fused silica this is +21.9 ps/(nm·km) at 1550 nm and passes through zero at 1273 nm, the bulk zero-dispersion wavelength; the group index has its minimum at the same wavelength. Expressed as group velocity dispersion, the same coefficients give +36.2 fs²/mm at 800 nm, the figure used for pulse-compressor design. A guided mode adds a waveguide contribution, so fiber and waveguide dispersion require a mode solver fed with Sellmeier indices for core and cladding.

Numerical derivatives of a Sellmeier fit are reliable well inside its range. Near the ends, small coefficient errors are amplified in $d^2n/d\lambda^2$, so dispersion quoted outside the measured range deserves less trust than the index itself.

Other dispersion formulas

The Cauchy formula $n = A + B/\lambda^2 + \ldots$ is an expansion of the ultraviolet resonance terms for wavelengths well above them. It works over the visible but has no infrared term: fitted to fused silica at 486.1 and 587.6 nm, its two-term form predicts 1.4498 at 1550 nm, 0.0058 above the Sellmeier value. Older glass catalogs used the Schott power series in $\lambda^2$ and $\lambda^{-2}$, which fits well within its range but extrapolates poorly; most manufacturers now publish Sellmeier coefficients. Crystals often use extended forms with an extra constant or a pole term, and temperature-dependent versions for nonlinear-optics design. The Abbe number, 67.8 for fused silica, condenses visible dispersion into one figure computed from three indices.

The Refractive Index Calculator evaluates published Sellmeier equations for several common materials, with group index and dispersion.

Pitfalls

Coefficient sets differ in convention: some list $C_i$ as wavelengths in µm (to be squared), others as squared values in µm², and some use nm. Entering a wavelength in nm into a µm fit returns the limit $\sqrt{1+\sum B_i}$, 1.73 for silica, at every wavelength. Different melts and suppliers of the "same" glass can differ in the fourth decimal place, which matters for interferometry and precise focal-length work.

Common questions

What is the Sellmeier equation used for?

To compute the refractive index at any wavelength within a material's fitted range, and from its derivatives the group index and dispersion needed for lens design, pulse propagation and fiber links.

What is the difference between the Sellmeier and Cauchy equations?

Cauchy is a power series that approximates the ultraviolet resonance alone and holds in the visible. Sellmeier includes each resonance explicitly, including the infrared ones, and stays accurate into the near and mid infrared.

What are the Sellmeier coefficients of fused silica?

$B_1$ = 0.6961663, $B_2$ = 0.4079426, $B_3$ = 0.8974794, with resonance wavelengths 0.0684043, 0.1162414 and 9.896161 µm (square these for $C_i$), valid from 0.21 to 3.71 µm.

References: I. H. Malitson, "Interspecimen comparison of the refractive index of fused silica," Journal of the Optical Society of America 55, 1205 (1965); M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); G. P. Agrawal, Nonlinear Fiber Optics, 6th ed. (Academic Press, 2019).