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.

Waves in an unmagnetized plasma

Langmuir Wave

The Langmuir wave is a purely longitudinal wave, that is, the wave vector is in the same direction as the E-field. This wave does not propagate -- the group velocity is 0. The dispersion relation is $1-{\frac {\omega _{pe}^{2}}{\omega ^{2}}}=0$ .

Electromagnetic wave

In an unmagnetized plasma, waves above the plasma frequency propagate through the plasma according to the dispersion relation:

$1-{\frac {\omega _{pe}^{2}}{\omega ^{2}}}-{\frac {k^{2}c^{2}}{w^{2}}}=0\rightarrow \omega ^{2}=c^{2}k^{2}+\omega _{pe}^{2}$

In an unmagnetized plasma for the high frequency or low electron density limit, i.e. for $\omega \gg \omega _{pe}=(n_{e}e^{2}/m_{e}\epsilon _{0})^{1/2}$ or $n_{e}\ll m_{e}\omega ^{2}\epsilon _{0}\,/\,e^{2}$ 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]

n_{c}={\frac {\varepsilon _{o}\,m_{e}}{e^{2}}}\,\omega ^{2}

(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

The O wave is the "ordinary" wave in the sense that its dispersion relation is the same as that in an unmagnetized plasma, that is,

$1-{\frac {\omega _{pe}^{2}}{\omega ^{2}}}-{\frac {k^{2}c^{2}}{\omega ^{2}}}=0\rightarrow \omega ^{2}=c^{2}k^{2}+\omega _{pe}^{2}$ ^[2]

. It is plane polarized with E₁ || B₀. It has a cut-off at the plasma frequency.

X wave

The X wave is the "extraordinary" wave because it has a more complicated dispersion relation ^[3]:

$n^{2}={\frac {[(\omega +\omega _{ci})(\omega -\omega _{ce})-\omega _{p}^{2}][(\omega -\omega _{ci})(\omega +\omega _{ce})-\omega _{p}^{2}]}{(\omega ^{2}-\omega _{ci}^{2})(\omega ^{2}-\omega _{ce}^{2})+\omega _{p}^{2}(\omega _{ce}\omega _{ci}-\omega ^{2})}}$

Where $\omega _{p}^{2}=\omega _{pe}^{2}+\omega _{pi}^{2}$ .

It is partly transverse (with E₁⊥B₀) and partly longitudinal; the E-field is of the form

$(E_{x},-j{\frac {S}{D}}E_{x},0)$

Where $S,D$ refer to the Stix notation.

As the density is increased, the phase velocity rises from c until the cut-off at $\omega _{R}$ is reached. As the density is further increased, the wave is evanescent until the resonance at the upper hybrid frequency $\omega _{h}^{2}=\omega _{p}^{2}+\omega _{c}^{2}$ . Then it can propagate again until the second cut-off at $\omega _{L}$ . The cut-off frequencies are given by ^[4]

{\begin{aligned}\omega _{R}&={\frac {1}{2}}\left[\omega _{c}+\left(\omega _{c}^{2}+4\omega _{p}^{2}\right)^{\frac {1}{2}}\right]\\\omega _{L}&={\frac {1}{2}}\left[-\omega _{c}+\left(\omega _{c}^{2}+4\omega _{p}^{2}\right)^{\frac {1}{2}}\right]\end{aligned}}

where $\omega _{c}$ is the electron cyclotron resonance frequency, and $\omega _{p}$ is the electron plasma frequency.

The resonant frequencies for the X-wave are:

$\omega ^{2}={\frac {\omega _{e}^{2}+\omega _{i}^{2}}{2}}\pm {\sqrt {({\frac {\omega _{e}^{2}-\omega _{i}^{2}}{2}})^{2}+\omega _{pe}^{2}\omega _{pi}^{2}}}$

where $\omega _{a}^{2}=\omega _{pa}^{2}+\omega _{ca}^{2}$ and $a=e,i$ .

R wave and L wave

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. ^[5]

Dispersion relations

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).

Summary of electromagnetic electron waves
Conditions	Dispersion relation	Name
${\vec {B}}_{0}=0$	$\omega ^{2}=\omega _{p}^{2}+k^{2}c^{2}$	Light wave
${\vec {k}}\perp {\vec {B}}_{0},\ {\vec {E}}_{1}\\|{\vec {B}}_{0}$	${\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}}{\omega ^{2}}}$	O wave
${\vec {k}}\perp {\vec {B}}_{0},\ {\vec {E}}_{1}\perp {\vec {B}}_{0}$	${\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}}{\omega ^{2}}}\,{\frac {\omega ^{2}-\omega _{p}^{2}}{\omega ^{2}-\omega _{h}^{2}}}$	X wave
${\vec {k}}\\|{\vec {B}}_{0}$ (right circ. pol.)	${\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}/\omega ^{2}}{1-\omega _{c}/\omega }}$	R wave (whistler mode)
${\vec {k}}\\|{\vec {B}}_{0}$ (left circ. pol.)	${\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}/\omega ^{2}}{1+\omega _{c}/\omega }}$	L wave

References

^ 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.
^ D. Gary Swanson, Plasma Waves, Second Edition, Academic Press, 1989.
^ D. Gary Swanson, Plasma Waves , Academic Press, 1989.
^ 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.

[Chen-1] 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.

[2] D. Gary Swanson, Plasma Waves, Second Edition, Academic Press, 1989.

[3] D. Gary Swanson, Plasma Waves , Academic Press, 1989.

[Chen2-4] 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.

[Chen3-5] 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.

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