Phase vs. Group Velocity:
The commonly used formula v = c / n where c is the speed of light and n is the index of refraction of a material refers to the phase velocity of a wave (i.e. the velocity at which the phase of any one frequency component of the wave propagates). This speed must always be ≤c. However, there is another velocity, group velocity (vg), which deals with the speed at which the modulation envelope advances. This velocity is often associated with the velocity at which energy or information is conveyed and is given mathematically by
Using the fact that ω = 2πf and ω = kv, it immediately follows that
For a nondispersive media in which v is independent of k, the second term in the equation above vanishes. In contrast, for a nondispersive medium in vacuum, it follows that
It should be noted that the group velocity may be greater than, equal to, or less than the speed of light.
Group Velocity Dispersion:
Qualitatively, the term group velocity dispersion (GVD) is used to refer to the spreading in time of short pulses; since group velocity depends on the index of refraction of a material as shown in the equation above (which in turn depends on the frequency of the light), different frequency components of the pulse will travel at different velocities. Therefore GVD is a measure of the frequency dependence of the group velocity in a medium. This temporal spreading of the pulse is of particular concern in the optical communications industry; a high dispersion will cause the group of pulses comprising a bit stream to merge together, making the information unintelligible. Consequently, GVD greatly limits the length of fiber through which a signal can be sent without regeneration.
Quantitatively, GVD is the derivative of the inverse group velocity with respect to either the angular frequency or the wavelength, which gives