Friday, February 8, 2008

Group velocity

From Wikipedia, the free encyclopedia

Jump to: navigation, search

The group velocity of a wave is the velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope of the wave) propagate through space. For example, imagine what happens if you throw a stone into the middle of a very still pond. When the stone hits the surface of the water, a circular pattern of waves appears. It soon turns into a circular ring of waves with a quiescent center. The ever expanding ring of waves is the group, within which one can discern individual wavelets of differing wavelengths traveling at different speeds. The longer waves travel faster than the group as a whole, but they die out as they approach the leading edge. The shorter waves travel slower and they die out as they emerge from the trailing boundary of the group.

The group velocity vg is defined by the equation

v_g \ \equiv\ \frac{\partial \omega}{\partial k}\,

where:

ω is the wave's angular frequency;
k is the wave number.

Note: The above definition of group velocity is only useful for wavepackets, which is a pulse that is localized in both real space and frequency space. Because waves at different frequencies propagate at differing phase velocities in dispersive media, for a large frequency range (a narrow envelope in space) the observed pulse would change shape while traveling, making group velocity an unclear or useless quantity.

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, if the wave is travelling through an absorptive medium, this does not always hold. Since the 1980s, various experiments have verified that it is possible for the group velocity of laser light pulses sent through specially prepared materials to significantly exceed the speed of light in vacuum. However, superluminal communication is not possible in this case, since the signal velocity remains less than the speed of light. It is also possible to reduce the group velocity to zero, stopping the pulse, or have negative group velocity, making the pulse appear to propagate backwards. However, in all these cases, photons continue to propagate at the expected speed of light in the medium. [1] [2] [3] [4]

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 high-power, short-pulse lasers.

Anomalous dispersion happens in areas of rapid spectral variation with respect to the refractive index. Therefore, negative values of the group velocity will occur in these areas. Anomalous dispersion plays a fundamental role in achieving backward propagating and superluminal light. Anomalous dispersion can also be used to produce group and phase speeds that are in different directions [ 2 ]. Materials that exhibit large anomalous dispersion allow the group velocity of the light to exceed c and/or become negative [ 4 ].

The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877[5].

Contents

[hide]

[edit] Matter wave group velocity

Albert Einstein first explained the wave-particle duality of light in 1905. Louis de Broglie hypothesized that any particle should also exhibit such a duality. The velocity of a particle, he concluded then (but may be questioned today, see above), should always equal the group velocity of the corresponding wave. De Broglie deduced that if the duality equations already known for light were the same for any particle, then his hypothesis would hold. This means that

v_g = \frac{\partial \omega}{\partial k} = \frac{\partial (E/\hbar)}{\partial (p/\hbar)} = \frac{\partial E}{\partial p}

where

E is the total energy of the particle,
p is its momentum,
\hbar is Dirac's constant.

Using special relativity, we find that

\begin{align} v_g &= \frac{\partial E}{\partial p} = \frac{\partial}{\partial p} \left( \sqrt{p^2c^2+m^2c^4} \right)\\ &= \frac{pc^2}{\sqrt{p^2c^2 + m^2c^4}}\\ &= \frac{p}{m\sqrt{(p/(mc))^2+1}}\\ &= \frac{p}{m\gamma}\\ &= \frac{mv\gamma}{m\gamma}\\ &= v. \end{align}

where

m is the rest mass of the particle,
c is the speed of light in a vacuum,
γ is the Lorentz factor.
and v is the velocity of the particle regardless of wave behavior.

Quantum mechanics has very accurately demonstrated this hypothesis, and the relation has been shown explicitly for particles as large as molecules.

[edit] See also

[edit] References

  1. ^ George M. Gehring, Aaron Schweinsberg, Christopher Barsi, Natalie Kostinski, Robert W. Boyd, “Observation of a Backward Pulse Propagation Through a Medium with a Negative Group Velocity”, Science. 312, 895-897 (2006).
  2. ^ Gunnar Dolling, Christian Enkrich, Martin Wegener, Costas M. Soukoulis, Stefan Linden, “Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial”, Science. 312, 892-894 (2006).
  3. ^ A. Schweinsberg, N. N. Lepeshkin, M.S. Bigelow, R.W. Boyd, S. Jarabo, “Observation of superluminal and slow light propagation in erbium-doped optical fiber”, Europhysics Letters. 73, 218-224 (2005).
  4. ^ Matthew S Bigelow, Nick N Lepeshkin, Heedeuk Shin, Robert W Boyd, “Propagation of a smooth and discontinuous pulses through materials with very large or very small group velocities”, Journal of Physics: Condensed Matter. 18, 3117-3126 (2006)
  5. ^ Brillouin, Léon. Wave Propagation and Group Velocity. Academic Press Inc., New York (1960)
  • Tipler, Paul A. and Ralph A. Llewellyn (2003). Modern Physics. 4th ed. New York; W. H. Freeman and Company. ISBN 0-7167-4345-0. 223 p.

[edit] External links

Posted by at
Labels:

No comments:

Post a Comment

Subscribe to: Post Comments (Atom)