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# Group Index

Definition: the ratio of the vacuum velocity of light to the group velocity in a medium

Alternative term: group refractive index

German: Gruppenindex

Units: (dimensionless)

Formula symbol: <$n_\textrm{g}$>

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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:

$${n_{\rm{g}}} = \frac{c}{{{\upsilon _{\rm{g}}}}} = c\frac{{\partial k}}{{\partial \omega }} = \frac{\partial }{{\partial \omega }}\left( {\omega \;n(\omega )} \right) = n(\omega ) + \omega \;\frac{{\partial n}}{{\partial \omega }}$$

For calculating this, one obviously needs to know not only the refractive index at the wavelength of interest, but also its frequency dependence.

The group index is used, for example, for calculating time delays for ultrashort pulses propagating in a medium, or the free spectral range of a resonator containing a dispersive medium.

For optical crystals or glasses, the group index in the visible or near-infrared spectral range is typically 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., a very large group index. Figure 1: Refractive index (solid lines) and group index (dotted lines) of silica versus wavelength at temperatures of 0 °C (blue), 100 °C (black) and 200 °C (red). Data taken from M. Medhat et al., J. Opt. A: Pure Appl. Opt. 4, 174 (2002); https://doi.org/10.1088/1464-4258/4/2/309.

Just as the normal refractive index, the group index depends somewhat on the material's temperature; see Figure 1 as an example.

Note that for optical fibers and other waveguides, one uses the so-called effective refractive index instead of the ordinary refractive index in order 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.

## Questions and Comments from Users

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?

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 geometrical 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.

2020-07-21

Which kind of index should be used to calculate refraction angles in the crystal?

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?

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?

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 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?

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?

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.

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