# 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.

## Cut-off frequency and critical density[edit]

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 wave[edit]

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 wave[edit]

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 is reached. As the density is further increased, the wave is evanescent until the resonance at the upper hybrid frequency . Then it can propagate again until the second cut-off at . 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 wave[edit]

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 relations[edit]

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 also[edit]

## References[edit]

**^**Chen, Francis (1984).*Introduction to Plasma Physics and Controlled Fusion, Volume 1*(2nd ed.). Plenum Publishing Corporation. p. 116. ISBN 978-0-306-41332-2.**^**Chen, Francis (1984).*Introduction to Plasma Physics and Controlled Fusion, Volume 1*(2nd ed.). Plenum Publishing Corporation. p. 127. ISBN 978-0-306-41332-2.**^**Chen, Francis (1984).*Introduction to Plasma Physics and Controlled Fusion, Volume 1*(2nd ed.). Plenum Publishing Corporation. p. 131. ISBN 978-0-306-41332-2.