Tool · Optical materials

Refractive Index Calculator

What is the refractive index of a glass or crystal at the wavelength you work at, and how fast does a pulse travel through it? The calculator evaluates the published Sellmeier equation of six common optical materials, or one you enter, and gives the index, the group index and the material dispersion, with the Abbe number and the zero-dispersion wavelength. Background: refractive index, group index, chromatic dispersion, and Abbe number.

Material and wavelength

Material

Wavelength λ (nm)

Values are for the bulk material at room temperature. A waveguide or fiber mode has its own effective and group index, set by its geometry as well as its materials.

Index at standard wavelengths

Line	λ (nm)	n	n_g

Readouts

Index and group index against wavelength

index ngroup index n_g

Material dispersion

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. Each built-in fit is compared with catalogue or published index values, the closed-form derivatives are compared with finite differences, and the two forms of the group velocity dispersion are checked against each other.

Check	Expected	Computed	Tolerance

The expected values are the catalogue indices in the SCHOTT data sheets, the measured values in Malitson’s and Li’s papers, and the material zero-dispersion wavelength of silica given by Agrawal. The tolerance is the largest relative difference from Expected that still passes.

The model

Between its absorption bands a transparent material’s index follows the Sellmeier equation, a sum of terms each of which stands for a resonance at the wavelength $\sqrt{C_i}$ :

n^2(\lambda) = 1 + \sum_{i=1}^{3} \frac{B_i\,\lambda^2}{\lambda^2 - C_i}

with $\lambda$ in micrometres and $C_i$ in square micrometres. The coefficients are fitted to measured indices over a stated range, and outside it the equation can be badly wrong. A pulse travels at the group velocity $c/n_g$ , and its spread in arrival time per unit bandwidth and length is the dispersion parameter $D$ :

n_g = n - \lambda\frac{dn}{d\lambda}, \qquad D = -\frac{\lambda}{c}\frac{d^2 n}{d\lambda^2}, \qquad \beta_2 = \frac{\lambda^3}{2\pi c^2}\frac{d^2 n}{d\lambda^2}

The tool differentiates the Sellmeier equation in closed form. The Abbe number is $V_d = (n_d - 1)/(n_F - n_C)$ at 587.56, 486.13 and 656.27 nm. All values are for the bulk material at the temperature of the source data; the index changes by a few parts in a million per kelvin for N-BK7, about 10⁻⁵ per kelvin for fused silica and 1.8 × 10⁻⁴ per kelvin for silicon. The coefficients of the built-in materials, with their fit ranges:

Fused silica, 0.21 to 3.71 µm: Malitson (1965).
N-BK7 and N-SF11, 0.3 or 0.37 to 2.5 µm: SCHOTT data sheets.
Calcium fluoride, 0.23 to 9.7 µm: Malitson (1963).
Sapphire, ordinary ray, 0.2 to 5.5 µm: Malitson and Dodge (1972).
Silicon, 1.36 to 11 µm: a three-term fit to the data of Salzberg and Villa (1957), which agrees with Li’s recommended values to about 0.002 near 1.5 µm.

Worked example

N-BK7 has $n$ = 1.51680 at the d line, the catalogue value, and an Abbe number of 64.17. At 1550 nm its index falls to 1.50065, but the group index is 1.52007, so a pulse crosses a 10 mm window in 50.70 ps rather than the 50.06 ps the phase index suggests. The dispersion there is 19.31 ps/(nm·km), anomalous, and N-BK7’s material zero-dispersion wavelength lies at 1322 nm.

References: I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205 (1965). I. H. Malitson, “A redetermination of some optical properties of calcium fluoride,” Appl. Opt. 2, 1103 (1963). I. H. Malitson and M. J. Dodge, “Refractive index and birefringence of synthetic sapphire,” J. Opt. Soc. Am. 62, 1405 (1972). C. D. Salzberg and J. J. Villa, “Infrared refractive indexes of silicon, germanium and modified selenium glass,” J. Opt. Soc. Am. 47, 244 (1957). H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561 (1980). SCHOTT Optical Glass data sheets, N-BK7 and N-SF11. G. P. Agrawal, Fiber-Optic Communication Systems, 5th ed. (Wiley, 2021), Ch. 2.