Electromagnetic electron wave

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

In an unmagnetized plasma in the high frequency or low density limit, i.e. for ω > > (4πn_ee² / m_e)^{1 / 2} or $n_e << m_e\omega^2\,/\,4\pi e^2$, the wave speed is the speed of light in vacuum. As the density increases, the phase velocity increases and the group velocity decreases until the cut-off frequency where the light frequency is equal to the plasma frequency. This density is known as the critical density for the angular frequency ω of that wave and is given by

$n_c = \frac{\varepsilon_o\,m_e}{e^2}\,\omega^2$.

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. 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. It is partly transverse (with E₁||B₀) 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

$\omega_R = \frac{1}{2}\left[ \omega_c + (\omega_c^2+4\omega_p^2)^{1/2} \right]$

$\omega_L = \frac{1}{2}\left[ -\omega_c + (\omega_c^2+4\omega_p^2)^{1/2} \right]$

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.

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
$\vec B_0=0$	$\omega^2=\omega_p^2+k^2c^2$	light wave
$\vec k\perp\vec B_0,\ \vec E_1\|\vec B_0$	$\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2}$	O wave
$\vec k\perp\vec B_0,\ \vec E_1\perp\vec B_0$	$\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2}\, \frac{\omega^2-\omega_p^2}{\omega^2-\omega_h^2}$	X wave
$\vec k\|\vec B_0$ (right circ. pol.)	$\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2/\omega^2}{1-(\omega_c/\omega)}$	R wave (whistler mode)
$\vec k\|\vec B_0$ (left circ. pol.)	$\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2/\omega^2}{1+(\omega_c/\omega)}$	L wave

See also

• Appleton-Hartree equation