Editing Dispersion (optics) (section)

== Group-velocity dispersion ==
{{main|Group-velocity dispersion}}
[[File:Optical dispersion dynamics.gif|thumb|Time evolution of a short pulse in a hypothetical dispersive medium (''k''&nbsp;=&nbsp;''ω''<sup>2</sup>) showing that the longer-wavelength components travel faster than the shorter-wavelength components (positive GVD), resulting in chirping and pulse broadening]]

Beyond simply describing a change in the phase velocity over wavelength, a more serious consequence of dispersion in many applications is termed  [[group-velocity dispersion]] (GVD). While phase velocity ''v'' is defined as ''v'' = ''c''/''n'', this describes only one frequency component. When different frequency components are combined, as when considering a signal or a pulse, one is often more interested in the [[group velocity]], which describes the speed at which a pulse or information superimposed on a wave (modulation) propagates. In the accompanying animation, it can be seen that the wave itself (orange-brown) travels at a phase velocity much faster than the speed of the ''envelope'' (black), which corresponds to the group velocity. This pulse might be a communications signal, for instance, and its information only travels at the group velocity rate, even though it consists of wavefronts advancing at a faster rate (the phase velocity).

It is possible to calculate the group velocity from the refractive-index curve ''n''(''ω'') or more directly from the wavenumber ''k'' = ''ωn''/''c'', where ''ω'' is the radian frequency ''ω''&nbsp;=&nbsp;2''πf''. Whereas one expression for the phase velocity is ''v''<sub>p</sub>&nbsp;=&nbsp;''ω''/''k'', the group velocity can be expressed using the [[derivative]]: ''v''<sub>g</sub>&nbsp;=&nbsp;''dω''/''dk''. Or in terms of the phase velocity ''v''<sub>p</sub>,

: <math>v_\text{g} = \frac{v_\text{p}}{1 - \dfrac{\omega}{v_\text{p}} \dfrac{dv_\text{p}}{d\omega}}.</math>

When dispersion is present, not only the group velocity is not equal to the phase velocity, but generally it itself varies with wavelength. This is known as group-velocity dispersion and causes a short pulse of light to be broadened, as the different-frequency components within the pulse travel at different velocities. Group-velocity dispersion is quantified as the derivative of the ''reciprocal'' of the group velocity with respect to [[angular frequency]], which results in ''group-velocity dispersion''&nbsp;=&nbsp;''d''<sup>2</sup>''k''/''dω''<sup>2</sup>.

If a light pulse is propagated through a material with positive group-velocity dispersion, then the shorter-wavelength components travel slower than the longer-wavelength components. The pulse therefore becomes ''positively [[chirp]]ed'', or ''up-chirped'', increasing in frequency with time. On the other hand, if a pulse travels through a material with negative group-velocity dispersion, shorter-wavelength components travel faster than the longer ones, and the pulse becomes ''negatively chirped'', or ''down-chirped'', decreasing in frequency with time.

An everyday example of a negatively chirped signal in the acoustic domain is that of an approaching train hitting deformities on a welded track. The sound caused by the train itself is impulsive and travels much faster in the metal tracks than in air, so that the train can be heard well before it arrives. However, from afar it is not heard as causing impulses, but leads to a distinctive descending chirp, amidst reverberation caused by the complexity of the vibrational modes of the track. Group-velocity dispersion can be heard in that the volume of the sounds stays audible for a surprisingly long time, up to several seconds.