Electromagnetic electron 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,

${\displaystyle 1-{\frac {\omega _{pe}^{2}}{\omega ^{2}}}-{\frac {k^{2}c^{2}}{\omega ^{2}}}=0\rightarrow \omega ^{2}=c^{2}k^{2}+\omega _{pe}^{2}}$ [3]

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

The X wave is the "extraordinary" wave because it has a more complicated dispersion relation:[4]

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

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

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

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

Where ${\displaystyle S,D}$ refer to the Stix notation.

As the density is increased, the phase velocity rises from c until the cut-off at ${\displaystyle \omega _{R}}$ is reached. As the density is further increased, the wave is evanescent until the resonance at the upper hybrid frequency ${\displaystyle \omega _{h}^{2}=\omega _{p}^{2}+\omega _{c}^{2}}$. Then it can propagate again until the second cut-off at ${\displaystyle \omega _{L}}$. The cut-off frequencies are given by [5]

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

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

The resonant frequencies for the X-wave are:

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

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

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. [6]

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

• Appleton-Hartree equation
• List of plasma (physics) articles