Electromagnetic Wave Propagation in Plasmas

where is the wavenumber, and the angular frequency. The equation of motion of the particle is thus

where measures its wave-induced displacement in the -direction. The above equation can easily be solved to give

Thus, the wave causes the particle to execute sympathetic simple harmonic oscillations, in the -direction, with an amplitude which is directly proportional to its charge, and inversely proportional to its mass.

Suppose that the wave is actually propagating through an unmagnetized electrically neutral *plasma* consisting
of free electrons, of mass and charge , and free ions, of mass
and charge . Since the plasma is assumed to be electrically neutral, each species must have the same equilibrium number density,
. Now, given that the electrons are much less massive than the ions (*i.e.*, ), but have the same charge (modulo a sign), it follows from (586) that the wave-induced oscillations of the electrons
are of much higher amplitude than those of the ions. In fact, to a first approximation,
we can say that the electrons oscillate whilst the ions remain stationary.
Assuming that the electrons and ions are evenly distributed throughout the
plasma, the wave-induced displacement of an individual electron generates an effective *electric
dipole moment* in the -direction of the form (the other component of the dipole is
a stationary ion of charge located at ).
Hence, the -directed
*electric dipole moment per unit volume* is

Now, we saw earlier, in Section 357, that the -directed propagation of an electromagnetic
wave, polarized in the -direction (*i.e.*, with its electric component
oscillating in the -direction), through a dielectric medium is
governed by

Thus, writing in the form (584), in the form

(591) |

where is the velocity of light in vacuum, and the so-called

is the characteristic frequency of

where is the impedance of free space, and

the effective

The expression (595) for the refractive index of a plasma has some rather
unusual properties. For wave frequencies lying above the plasma frequency (*i.e.*,
), it yields a real refractive index which is
*less than unity*. On the other hand, for wave frequencies lying below the plasma
frequency (*i.e.*,
), it yields an *imaginary* refractive index. Neither of these results makes much sense. The former result is problematic because if the
refractive index is less than unity then the wave *phase velocity*,
, becomes
*superluminal* (*i.e.*, ), and superluminal velocities are generally thought to be unphysical.
The latter result is problematic because an imaginary refractive index implies an
imaginary phase velocity, which seems utterly meaningless. Let us investigate further.

Consider, first of all, the high frequency limit,
. According to
(595), a sinusoidal electromagnetic wave of angular frequency
propagates through the plasma
at the superluminal phase velocity

(597) |

(598) |

Note that the group velocity is

(600) |

The fact that the energy flux and the group velocity of a sinusoidal wave propagating through a plasma both go to zero when
suggests that the wave ceases to propagate at all in the low frequency limit,
. This observation leads us to search for
spatially decaying standing wave solutions to (589) and (590) of the form,

It follows from (585) and (587) that

Substitution into Equations (589) and (590) reveals that (601) and (602) are indeed the correct solutions when (see Exercise 2), and also yields

as well as

Furthermore, the mean -directed electromagnetic energy flux becomes

(606) |

Suppose that the region is a vacuum, and the region is
occupied by a plasma of plasma frequency . Let the wave electric and
magnetic fields in the vacuum region take the form

Here, is the amplitude of an electromagnetic wave of frequency which is incident on the plasma, whereas is the amplitude of the reflected wave, and the phase of this wave with respect to the incident wave. Moreover, we have made use of the vacuum dispersion relation . The wave electric and magnetic fields in the plasma are written

(609) | |||

(610) |

where is the amplitude of the decaying wave which penetrates into the plasma, is the phase of this wave with respect to the incident wave, and

(611) |

These two equations, which must be satisfied at all times, can be solved to give (see Exercise 3)

Thus, the coefficient of reflection,

(618) |

The outer regions of the Earth's atmosphere consist of a tenuous gas which is
*partially ionized* by ultraviolet and X-ray radiation from the Sun, as well as by cosmic rays incident from outer space. This
region, which is known as the *ionosphere*, acts like a plasma
as far as its interaction with radio waves is concerned. The ionosphere
consists of many layers. The two most important, as far as radio
wave propagation is concerned, are the *E layer*, which lies at an altitude of
about 90 to 120 km above the Earth's surface, and the *F layer*, which
lies at an altitude of about 120 to 400 km. The plasma frequency in the
F layer is generally larger than that in the E layer, because of the greater
density of free electrons in the former (recall that
).
The free electron number density in the
E layer drops steeply after sunset, due to the lack of solar ionization combined with the gradual recombination of free electrons
and ions. Consequently, the plasma frequency in the E layer also drops steeply after sunset. Recombination in the F layer occurs at a much slower rate, so there is nothing like
as great a reduction in the plasma frequency of this layer at night.
Very High Frequency (VHF) radio signals (*i.e.*, signals with frequencies greater than 30 MHz), which include FM radio and TV signals, have frequencies well in excess
of the plasma frequencies of both the E and the F layers, and thus pass straight through
the ionosphere. Short Wave (SW) radio signals (*i.e.*, signals with frequencies in the
range 3 to 30 MHz) have frequencies in excess of the plasma
frequency of the E layer, but not of the F layer. Hence, SW signals pass through the
E layer, but are reflected by the F layer.
Finally, Medium Wave (MW) radio signals (*i.e.*, signals with frequencies in the range
to 3 MHz) have frequencies which lie below the plasma frequency of the F layer,
and also lie below the plasma frequency of the E layer during daytime, but not
during nighttime. Thus, MW signals are reflected by the E layer during the day,
but pass through the E layer, and are reflected by the F layer, during the night.

The reflection and transmission of the various different types of radio wave by the ionosphere is shown schematically in Figure 41. This diagram explains many of the features of radio reception. For instance, due to the curvature of the Earth's surface, VHF reception is only possible when the receiving antenna is in the line of sight of the transmitting antenna, and is consequently fairly local in nature. MW reception is possible over much larger distances, because the signal is reflected by the ionosphere back towards the Earth's surface. Moreover, long range MW reception improves at night, since the signal is reflected at a higher altitude. Finally, SW radio reception is possible over very large distances, because the signal is reflected at extremely high altitudes.