Electromagnetic Waves in Unmagnetized Plasmas

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 . Because the plasma is assumed to be electrically neutral, each species must have the same equilibrium number density, . Given that the electrons are much less massive than the ions (i.e., ), but have the same charge (modulo a sign), it follows from Equation (9.21) 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 while 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

Given that all of the electrons oscillate according to Equation (9.21) (with and ), we obtainWe saw earlier, in Section 6.7, that the -directed propagation of a plane electromagnetic wave, linearly polarized in the -direction, through a dielectric medium is governed by (see Appendix C)

Thus, writing in the form (9.19), in the form where is the effective impedance of the plasma, and in the form (9.23), Equations (9.24) and (9.25) yield the nonlinear dispersion relation (see Exercise 3) where is the velocity of light in vacuum, and the so-called (electron)The expression (9.30) 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 that 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 phase velocity of the wave, , 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 Equation (9.30), a sinusoidal electromagnetic wave of angular frequency propagates through the plasma at the superluminal phase velocity

Is this really unphysical? As is well known, Einstein's special theory of relativity forbids information from traveling faster than the velocity of light in vacuum, because this would violate causality (i.e., it would be possible to transform to a valid frame of reference in which an effect occurs prior to its cause) (Rindler 1997). However, a sinusoidal wave with a unique frequency, and an infinite spatial extent, does not transmit any information. (Recall, for instance, from Section 8.5, that the carrier wave in an AM radio signal transmits no information.) At what speed do electromagnetic waves propagating through a plasma transmit information? The most obvious way of using such waves to transmit information would be to send a message via Morse code. In other words, we could transmit a message by means of short wave pulses, of varying lengths and inter-pulse spacings, that are made to propagate through the plasma. The pulses in question would definitely transmit information, so the velocity of information propagation must be the same as that of the pulses; that is, the group velocity, . Differentiating the dispersion relation (9.27) with respect to , we obtain(9.32) |

(9.33) |

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 Equations (9.24) and (9.25) of the form,

It follows from Equations (9.20) and (9.22) that Substitution into Equations (9.24) and (9.25) reveals that (9.36) and (9.37) are indeed the correct solutions when , and also yields as well as (See Exercise 4.) Furthermore, the mean -directed electromagnetic energy flux becomes(9.41) |

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

where is the vacuum wavenumber. Here, is the amplitude of an electromagnetic wave of frequency that is normally incident on the plasma, whereas is the amplitude of the reflected wave, and the phase of this wave with respect to the incident wave. The wave electric and magnetic fields in the plasma are written(9.44) | ||

(9.45) |

(9.46) |

(9.53) |

The outer regions of the Earth's atmosphere consist of a tenuous gas that 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 (Pain 1999). 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 that 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 9.1. This diagram explains many of the characteristic features of radio reception. For instance, because of the curvature of the Earth's surface, VHF reception is only possible when the receiving antenna lies 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 toward the Earth's surface. Moreover, long range MW reception improves at night, because 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.