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. It is an electrostatic wave, thus it doesn't have an oscillating magnetic field. A plasma consists of charged particles which react to electric fields, in contrast with dielectric matter. When electrons in a uniform, homogeneous plasma are perturbed from their equilibrium position, a charge separation occurs creating an electric field which acts as restorting force on the electrons. Since electrons have inertia the system behaves as a harmonic osillator, where the electrons oscillate at a frequency ω, called electron plasma frequency. These oscillations does not propagate -- the group velocity is 0. When the thermal motion  of the electrons is taken into account a shift in frequency from the electron plasma frequency ω occurs. Now the electron pressure gradient acts as the restorting force, creating a propagating wave analogous to a sound wave in non-ionized gases. Combining these two restorting forces (from the electric field and electron pressure gradient) a type of wave, named Langmuir wave, is excited. The dispersion relation is: formula_1 The first term on the right hand side of the dispersion relation is the electron plasma oscillation related to the electric field force and the second term is related to the thermal  motion of the electrons, where C is the electron thermal speed and k is the wave vector. Electromagnetic wave. In an unmagnetized plasma, waves above the plasma frequency propagate through the plasma according to the dispersion relation: formula_2 In an unmagnetized plasma for the high frequency or low electron density limit, i.e. for formula_3 or formula_4 where ω 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. This  density is known as the critical density for the angular frequency ω of that wave and is given by 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, formula_2 . It is plane polarized with E || B. 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: formula_7 Where formula_8. It is partly transverse (with E⊥B) and partly longitudinal; the E-field is of the form formula_9 Where formula_10 refer to the Stix notation. As the density is increased, the phase velocity rises from c" until the cut-off at formula_11 is reached. As the density is further increased, the wave is evanescent until the resonance at the upper hybrid frequency formula_12. Then it can propagate again until the second cut-off at formula_13. The cut-off frequencies are given by where formula_15 is the electron cyclotron resonance  frequency, and formula_16 is the electron plasma frequency. The resonant frequencies for the X-wave are: formula_17 where formula_18 and formula_19. 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 (hence the designation of this frequency) and a resonance at ω. The L wave has a cut-off at ω and no resonance. R waves at frequencies below ω/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).