Group Index
Author: the photonics expert Dr. Rüdiger Paschotta (RP)
Definition: the ratio of the vacuum velocity of light to the group velocity in a medium
Alternative term: group refractive index
Related: group velocityrefractive index
Units: (dimensionless)
Formula symbol: ($n_\textrm{g}$)
Page views in 12 months: 6101
DOI: 10.61835/2bh Cite the article: BibTex BibLaTex plain textHTML Link to this page! LinkedIn
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What is a Group Index?
In analogy with the refractive index, the group index (or group refractive index) ($n_\textrm{g}$) of a material can be defined as the ratio of the vacuum velocity of light to the group velocity in the medium:
$$n_{\textrm{g}} = \frac{c}{\upsilon_{\textrm{g}}}$$Using the definition of group velocity, this leads to:
$$n_{\textrm{g}} = \frac{c}{\upsilon_{\textrm{g}}} = c\frac{\partial k}{\partial \omega} = \frac{\partial }{\partial \omega}\left( \omega \;n(\omega ) \right) = n(\omega ) + \omega \;\frac{\partial n}{\partial \omega}$$For calculating this, one obviously needs to know not only the refractive index at the wavelength of interest, but also its optical frequency derivative. That derivative may be estimated from multiple refractive index values at different wavelengths, or calculated from a Sellmeier formula.
Group Index and Time Delays
The group index is used, for example, for calculating time delays for ultrashort pulses propagating in a medium, as pulse envelopes travel with the group velocity. Also, the free spectral range of a resonator containing a dispersive medium is determined by its group index.
Values of the Group Index
For optical crystals or glasses, the group index in the visible or near-infrared spectral range is typically somewhat larger than the ordinary refractive index: the group velocity is somewhat smaller than the phase velocity. In certain special (artificial) situations, one obtains dramatically reduced group velocities (→ slow light), i.e., an extremely large group index, while the refractive index stays in the “normal” region.
Just as the normal refractive index, the group index depends somewhat on the material's temperature; see Figure 1 as an example.
Effective Group Index of Fibers
Note that for optical fibers and other waveguides, one uses the so-called effective refractive index instead of the ordinary refractive index to calculate the group velocity, since waveguide dispersion has to be taken into account. Based on that, an effective group index of a fiber could be calculated.
Frequently Asked Questions
This FAQ section was generated with AI based on the article content and has been reviewed by the article’s author (RP).
What is the group index?
The group index ($n_\textrm{g}$) of a material is defined as the ratio of the speed of light in vacuum to the group velocity of light in that material. It is relevant for describing the propagation speed of the envelope of a light pulse.
How is the group index related to the refractive index?
The group index ($n_\textrm{g}$) depends not only on the refractive index ($n$) but also on its frequency derivative, according to the equation ($n_\textrm{g} = n(\omega) + \omega (\partial n/\partial\omega)$). In the visible and near-infrared region, the group index of optical materials is typically slightly larger than their refractive index.
What are typical applications of the group index?
The group index is used to calculate propagation time delays for ultrashort pulses in a medium, as their pulse envelopes travel with the group velocity. It also determines the free spectral range of an optical resonator containing a dispersive medium.
What is the effective group index of an optical fiber?
For waveguides like optical fibers, one defines an effective group index. This is calculated using the effective refractive index instead of the material's refractive index, thus taking into account the effects of waveguide dispersion.
Questions and Comments from Users
2020-07-21
Which kind of index should be used to calculate refraction angles in the crystal?
The author's answer:
The refractive index, not the group index.
2021-02-20
In an ideal Fabry–Perot resonator with R = 1, infinitely narrow linewidth and filled with a dispersive medium, is the free spectral range still determined by the group index?
The author's answer:
Sure, it is!
2021-11-12
Can I calculate the group refractive index knowing the effective refractive index, for example in a step index optical fiber can we derive the group index? Will the group index be (ncore2) / neff?
The author's answer:
No, the group index is not determined by the effective refractive index. Additional information would be needed concerning frequency derivatives.
2022-11-01
When determining the length difference of one arm of an MZI interferometer needed to get from one intensity minimum to the next, I have deltaL = lambda / (2 *n) — or should I use the group index instead?
The author's answer:
The normal refractive index is what counts here; we are not considering frequency derivatives but simply how fast the optical phase delay changes spatially. Anyway, usually we vary path length differences in air, where both ($n$) and ($n_\textrm{g}$) are close to 1.
2023-06-02
In an optical waveguide, what is the pulse delay for the fundamental TE mode with a given length? Is it L ng / c or L * neff / c?
The author's answer:
What is relevant for the time delay of a pulse is a mixture of both: the effective group index, as mentioned at the end of the article.
2023-11-01
If the refractive index is 1.46 and group index is 1.48, does it mean the light will refract based on n = 1.46 (Snell's law), and travel with a speed of c/1.48?
The refractive index can be smaller or larger than 1, or even negative. However, the group index can never be below 1, is that correct?
The author's answer:
First question: yes, and more precisely the “speed” is the phase velocity.
Second question: no, the group index can also be smaller than 1 in some situations, where, however, the velocity of information transfer is different from the group velocity and still below the vacuum velocity of light. See also the article on superluminal transmission.
2024-03-26
Is it possible to derive effective index from the group index?
The author's answer:
I don't think so, as they describe quite different aspects.
2024-04-24
Which refraction index should be used for calculating the focal length of a thin lens, the group index or the phase index?
The author's answer:
This is clearly the phase index (= refractive index).
2024-06-04
How to calculate the group index of a certain glass in the wavelength range of 1000 nm to 1300 nm? The formula for the group index does not reflect the range of light waves, only a single wavelength.
The author's answer:
The formula contains a frequency derivative of the refractive index, which you may estimate using differences of index values at different wavelengths.
2024-08-30
We use white-light interference to measure thickness. The OPD can be calculated by FFT, thickness = OPD/2n. Is this n is a refractive index or group index?
The author's answer:
The optical path length difference (OPD) is based on the refractive index, not the group index. However, with a white-light interferometer you can measure the complex transmission coefficient versus optical frequency, from which you can get both quantities.
2024-10-08
Optical fiber companies often provide information on the effective group index of refraction at a certain wavelength. What is this?
The author's answer:
I think it is just the group index as explained on this page.
2025-09-11
For a single wavelength continuous source: For calculating the recombination in let's say a ring resonator or MZI with unequal paths, should one use the group index or the normal refractive index?
The author's answer:
What matters here are just phase differences between the beams. So the refractive index is used, not the group index.
2026-02-05
Can you get lensing based on group index? I.e., could you build a metalens which operates in the narrowband regime by engineering the group index profile?
The author's answer:
For lensing effects, the normal refractive index is relevant, not the group index.
2020-04-01
I wonder what is determining the wavelengths of cavity resonances, e.g. in a silicon ring resonator — is it group index or the effective refractive index?
The author's answer:
In short: a combination of both!
The mode spacing (the frequency spacing of the resonator modes) is determined by the group delay for one resonator round-trip. For a silicon ring resonator, containing a waveguide, the group delay is proportional to the geometric round-trip length and to the effective group index. Here, “effective” means that we do not simply take a material property, but an effective value calculated for the waveguide structure. Further, “group index” means that we do not simply calculate the effective refractive index, which is relevant only for the phase delay, but the group index, which is relevant for the group delay. That calculation involves the use of frequency derivatives of propagation constants.