Dispersion relation - Definition and Overview

The group velocity of a wave is the velocity with which the overall shape of the wave's amplitude (known as the envelope of the wave) propagates through space. The group velocity is defined in terms of the wave's angular frequency ω and wave number k by

<math>v_g \equiv \frac{\partial \omega}{\partial k}<math>

The group velocity is often thought of as the velocity at which energy or information is conveyed along a wave. In most cases this is accurate, and the group velocity can be thought of as the signal velocity of the waveform. However, it is possible to design experiments where the group velocity of laser light pulses sent through specially prepared materials significantly exceeds the signal velocity. (It is also possible to stop the laser pulse.)

The function ω(k), which gives ω as a function of k, is known as the dispersion relation. If ω is directly proportional to k, then the group velocity is exactly equal to the phase velocity. Otherwise, the envelope of the wave will become distorted as it propagates. This "group velocity dispersion" is an important effect in the propagation of signals through optical fibers and in the design of short pulse lasers.

For light, group velocity and phase velocity are related by the formula

<math>v_g v_p = c^2<math>

where c is the speed of light in a vacuum.

See also

• Dispersion (optics) for a full discussion of wave velocities