# Electromagnetic electron wave

In plasma physics, an **electromagnetic electron wave** is a wave in a plasma which has a magnetic field component and in which primarily the electrons oscillate.

In an unmagnetized plasma, an electromagnetic electron wave is simply a light wave modified by the plasma. In a magnetized plasma, there are two modes perpendicular to the field, the O and X modes, and two modes parallel to the field, the R and L waves.

## Contents

## Cut-off frequency and critical densityEdit

In an unmagnetized plasma for the high frequency or low electron density limit, i.e. for
or
where ω_{pe} is the plasma frequency, the wave speed is the speed of light in vacuum. As the electron density increases, the phase velocity increases and the group velocity decreases until the **cut-off frequency** where the light frequency is equal to ω_{pe}. This density is known as the **critical density** for the angular frequency ω of that wave and is given by ^{[1]}

- (SI units)

If the critical density is exceeded, the plasma is called **over-dense**.

In a magnetized plasma, except for the O wave, the cut-off relationships are more complex.

## O waveEdit

The **O wave** is the *ordinary* wave in the sense that its dispersion relation is the same as that in an unmagnetized plasma. It is plane polarized with
**E**_{1} || **B**_{0}. It has a cut-off at the plasma frequency.

## X waveEdit

The **X wave** is the "extraordinary" wave because it has a more complicated dispersion relation. It is partly transverse (with **E**_{1}⊥**B**_{0})
and partly longitudinal. As the density is increased, the phase velocity rises from *c* until the cut-off at ω_{R} is reached. As the density is further increased, the wave is evanescent until the resonance at the upper hybrid frequency ω_{h}. Then it can propagate again until the second cut-off at ω_{L}. The cut-off frequencies are given by ^{[2]}

where is the electron cyclotron resonance frequency, and is the electron plasma frequency.

## R wave and L waveEdit

The **R wave** and the **L wave** are right-hand and left-hand circularly polarized, respectively. The R wave has a cut-off at ω_{R} (hence the designation of this frequency) and a resonance at ω_{c}. The L wave has a cut-off at ω_{L} and no resonance. R waves at frequencies below ω_{c}/2 are also known as **whistler modes**. ^{[3]}

## Dispersion relationsEdit

The dispersion relation can be written as an expression for the frequency (squared), but it is also common to write it as an expression for the index of refraction *ck*/ω (squared).

Conditions | Dispersion relation | Name |
---|---|---|

Light wave | ||

O wave | ||

X wave | ||

(right circ. pol.) | R wave (whistler mode) | |

(left circ. pol.) | L wave |

## See alsoEdit

## ReferencesEdit

**^**Chen, Francis (1984).*Introduction to Plasma Physics and Controlled Fusion, Volume 1*(2nd ed.). Plenum Publishing Corporation. p. 116. ISBN 0-306-41332-9.**^**Chen, Francis (1984).*Introduction to Plasma Physics and Controlled Fusion, Volume 1*(2nd ed.). Plenum Publishing Corporation. p. 127. ISBN 0-306-41332-9.**^**Chen, Francis (1984).*Introduction to Plasma Physics and Controlled Fusion, Volume 1*(2nd ed.). Plenum Publishing Corporation. p. 131. ISBN 0-306-41332-9.