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# Group Velocity

Author: the photonics expert

Definition: the velocity with which the envelope of a weak narrow-band optical pulse propagates in a medium

More general term: velocity of light

Units: m/s

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

The group velocity of light in a medium is defined as the inverse of the derivative of the wavenumber with respect to angular optical frequency:

$${\upsilon _{\rm{g}}} = {\left( {\frac{{\partial k}}{{\partial \omega }}} \right)^{ - 1}} = c{\left( {\frac{\partial }{{\partial \omega }}\left( {\omega \;n(\omega )} \right)} \right)^{ - 1}} = \frac{c}{{n(\omega ) + \omega \frac{{\partial n}}{{\partial \omega }}}} = \frac{c}{{{n_{\rm{g}}}(\omega )}}$$

where <$n(\omega )$> is the refractive index and <$n_\textrm{g}$> is called the group index. The wavenumber <$k$> can be considered as the change in spectral phase per unit length.

An alternative definition of the group velocity is the velocity with which the envelope of a pulse propagates in a medium. Unfortunately, these two definitions do not fully agree with each other. According to the first definition, the group velocity can be seen as a property of the medium alone (e.g., of an optical fiber), while the velocity of a pulse propagating in the medium may well depend on some properties of that pulse. However, the definitions agree for light pulses with a sufficiently narrow bandwidth (where higher-order chromatic dispersion is not relevant) and the absence of nonlinear effects (i.e., low enough optical intensities). In the following, we use the first definition.

The equation above shows that the group velocity has the same value as the phase velocity if the derivative of refractive index with respect to frequency (or wavelength) is zero – which is the case e.g. in vacuum, but normally not in optical media.

Figure 1 illustrates how the different frequency components combine to form a pulse, and how the different velocities arise. The gray lines indicate the wavefronts for some of the frequency components of the pulse (spatially offset for clarity). Due to chromatic dispersion, the higher-frequency components have somewhat lower phase velocities. The pulse maximum forms where the wavefronts coincide (constructive interference), and it propagates with the group velocity (which in this example is 80% of the medium phase velocity).

In the shown example, there is a reduced group velocity, but no temporal pulse broadening, since the group velocity is constant over the whole pulse spectrum, i.e., there is no group velocity dispersion.

Due to chromatic dispersion, the group velocity in a medium is in general different from the phase velocity (typically smaller than the latter), and it is frequency-dependent; this effect is called group velocity dispersion. The difference between group velocity and phase velocity also changes the carrier–envelope offset of the pulse.

In analogy with the refractive index, the group index (see the equation above) can be defined as the ratio of the group velocity in vacuum to the group velocity in the medium.

Under certain circumstances, the group velocity can be higher than the vacuum velocity of light. However, this does not allow for superluminal transmission of information, which would amount to a violation of causality. There are also cases with a strongly reduced group velocity dispersion (usually in the vicinity of some narrow resonance) (→ slow light).

## Group Velocity in a Waveguide

For light propagating in a waveguide such as an optical fiber, the group velocity can be calculated by replacing the wavenumber <$k$> with <$\beta$> (the imaginary part of the propagation constant) (or replacing the refractive index <$n$> with the effective refractive index) in the equation given above. The deviation of that result from the group velocity in a homogeneous medium can be interpreted as the influence of waveguide dispersion.

## Group Velocity in Nonlinear Propagation

In the literature, a certain group velocity is sometimes assigned even to a broadband pulse with complicated shape, or to a soliton pulse where an optical nonlinearity has an important influence (see e.g. Ref. [1]). However, that use of the term group velocity is questionable because it effectively redefines the term, giving it a different meaning, and can thus be misleading. Note that some frequently used relations involving the group velocity (as discussed above) are not valid in the nonlinear regime.

## More to Learn

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### Bibliography

 [1] H. A. Haus and E. P. Ippen, “Group velocity of solitons”, Opt. Lett. 26 (21), 1654 (2001); https://doi.org/10.1364/OL.26.001654

## Questions and Comments from Users

2023-10-06

Considering a short laser pulse: Is the group velocity given by the center wavelength? And shouldn't the bandwidth influence the group velocity? Especially in the case of very short pulses (broad spectrum)?