Electromagnetic Waves in Unmagnetized Plasmas

Consider a point particle of mass $m$ and electric charge $q$ interacting with a linearly polarized, sinusoidal, electromagnetic plane wave that propagates in the $z$-direction. Provided that the wave amplitude is not sufficiently large to cause the particle to move at relativistic speeds, the electric component of the wave exerts a much greater force on the particle than the magnetic component. [This follows, from standard electrodynamics, because the ratio of the magnetic to the electric force is of order $B_0\,v/E_0$, where $E_0$ is the amplitude of the wave electric field-strength, $B_0=E_0/c$ the amplitude of the wave magnetic field-strength, $v$ the particle velocity, and $c$ the velocity of light in vacuum. Hence, the ratio of the forces is approximately $v/c$ (Fitzpatrick 2008).] Suppose that the electric component of the wave oscillates in the $x$-direction, and takes the form

$$E_x(z,t) = E_0\,\cos(\omega\,t-k\,z),$$ (9.19)

where $k$ is the wavenumber, and $\omega$ the angular frequency. The equation of motion of the particle is thus (see Appendix C)

$$m\,\frac{d^{\,2} x}{dt^{\,2}} = q\,E_x,$$ (9.20)

where $x$ measures its wave-induced displacement in the $x$-direction. The previous equation can be solved to give

$$x = -\frac{q\,E_0}{m\,\omega^{\,2}}\,\cos(\omega\,t-k\,z).$$ (9.21)

Thus, the wave causes the particle to execute sympathetic simple harmonic oscillations, in the $x$-direction, with an amplitude that 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 $m_e$ and charge $-e$, and free ions, of mass $m_i$ and charge $+e$. Because the plasma is assumed to be electrically neutral, each species must have the same equilibrium number density, $n_e$. Given that the electrons are much less massive than the ions (i.e., $m_e\ll m_i$), 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 $x$-direction of the form $p_x = -e\,x$ (the other component of the dipole is a stationary ion of charge $+e$ located at $x=0$). Hence, the $x$-directed electric dipole moment per unit volume is

$$P_x = n_e\,p_x = -n_e\,e\,x.$$ (9.22)

Given that all of the electrons oscillate according to Equation (9.21) (with $q=-e$ and $m=m_e$), we obtain

$$P_x(z,t)= - \frac{n_e\,e^{\,2}\,E_0}{m_e\,\omega^{\,2}}\,\cos(\omega\,t-k\,z).$$ (9.23)

We saw earlier, in Section 6.7, that the $z$-directed propagation of a plane electromagnetic wave, linearly polarized in the $x$-direction, through a dielectric medium is governed by (see Appendix C)

$$\frac{\partial E_x}{\partial t} = - \frac{1}{\epsilon_0}\left(\frac{\partial P_x}{\partial t}+\frac{\partial H_y}{\partial z}\right),$$ (9.24)

$$\frac{\partial H_y}{\partial t} = -\frac{1}{\mu_0}\,\frac{\partial E_x}{\partial z}.$$ (9.25)

Thus, writing $E_x$ in the form (9.19), $H_y$ in the form

$$H_y(z,t) = Z^{\,-1}\,E_0\,\cos(\omega\,t-k\,z),$$ (9.26)

where $Z$ is the effective impedance of the plasma, and $P_x$ in the form (9.23), Equations (9.24) and (9.25) yield the nonlinear dispersion relation (see Exercise 3)

$$\omega^{\,2} = k^{\,2}\,c^{\,2}+\omega_{p\,e}^{\,2},$$ (9.27)

where $c=1/\sqrt{\epsilon_0\,\mu_0}$ is the velocity of light in vacuum, and the so-called (electron) plasma frequency,

$$\omega_{p\,e} = \left(\frac{n_e\,e^{\,2}}{\epsilon_0\,m_e}\right)^{1/2},$$ (9.28)

is the characteristic frequency of collective electron oscillations in the plasma (Stix 1962). Equations (9.24) and (9.25) also yield

$$Z = \frac{Z_0}{n},$$ (9.29)

where $Z_0=\sqrt{\mu_0/\epsilon_0}$ is the impedance of free space, and

$$n = \frac{k\,c}{\omega} = \left(1-\frac{\omega_{p\,e}^{\,2}}{\omega^{\,2}}\right)^{1/2}$$ (9.30)

the effective refractive index of the plasma. We, thus, conclude that sinusoidal electromagnetic waves propagating through an unmagnetized plasma have a nonlinear dispersion relation. Moreover, this nonlinearity arises because the effective refractive index of the plasma is frequency dependent.

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., $\omega>\omega_{p\,e}$), 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., $\omega<\omega_{p\,e}$), 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, $v_p=\omega/k = c/n$, becomes superluminal (i.e., $v_p>c$), 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, $\omega>\omega_{p\,e}$. According to Equation (9.30), a sinusoidal electromagnetic wave of angular frequency $\omega>\omega_{p\,e}$ propagates through the plasma at the superluminal phase velocity

$$v_p = \frac{\omega}{k} = \frac{c}{n}=\frac{c}{(1-\omega_{p\,e}^{\,2}/\omega^{\,2})^{1/2}}.$$ (9.31)

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, $v_g=d\omega/dk$. Differentiating the dispersion relation (9.27) with respect to $k$, we obtain

$$2\,\omega\,\frac{d\omega}{dk} = 2\,k\,c^{\,2},$$ (9.32)

or

$$\frac{\omega}{k}\,\frac{d\omega}{dk }= v_p\,v_g = c^{\,2}.$$ (9.33)

Thus, it follows, from Equation (9.31), that the group velocity of high-frequency electromagnetic waves in a plasma is

$$v_g = n\,c=(1-\omega_{p\,e}^{\,2}/\omega^{\,2})^{1/2}\,c.$$ (9.34)

The group velocity is sub-luminal (i.e., $v_g<c$). Hence, as long as we accept that high-frequency electromagnetic waves transmit information through a plasma at the group velocity, rather than the phase velocity, then there is no problem with causality. Incidentally, it follows, from this discussion, that the phase velocity of dispersive waves has very little physical significance. It is the group velocity that matters. For instance, according to Equations (6.128), (9.29), (9.30), and (9.34), the mean flux of electromagnetic energy in the $z$-direction due to a high-frequency sinusoidal wave propagating through a plasma is given by

$$\langle {\cal I}_z\rangle = \frac{1}{2}\,\epsilon_0\,E_0^{\,2}\,n\,c = \frac{1}{2}\,\epsilon_0\,E_0^{\,2}\,v_g,$$ (9.35)

because $Z_0=\sqrt{\mu_0/\epsilon_0}$ and $c=1/\sqrt{\epsilon_0\,\mu_0}$. Thus, if the group velocity is zero, as is the case when $\omega = \omega_{p\,e}$, then there is zero energy flux associated with the wave.

The fact that the energy flux and the group velocity of a sinusoidal wave propagating through a plasma both go to zero when $\omega = \omega_{p\,e}$ suggests that the wave ceases to propagate at all in the low-frequency limit, $\omega<\omega_{p\,e}$. This observation leads us to search for spatially decaying, standing wave solutions to Equations (9.24) and (9.25) of the form,

$$E_x(z,t) = E_0\,{\rm e}^{-k\,z}\,\cos(\omega\,t),$$ (9.36)

$$H_y(z,t) =Z^{\,-1}\,E_0\,{\rm e}^{-k\,z}\,\sin(\omega\,t).$$ (9.37)

It follows from Equations (9.20) and (9.22) that

$$P_x(z,t) = - \frac{n_e\,e^{\,2}\,E_0}{m_e\,\omega^{\,2}}\,{\rm e}^{-k\,z}\,\cos(\omega\,t).$$ (9.38)

Substitution into Equations (9.24) and (9.25) reveals that (9.36) and (9.37) are indeed the correct solutions when $\omega<\omega_{p\,e}$, and also yields

$$k\,c=\sqrt{\omega_{p\,e}^{\,2}-\omega^{\,2}},$$ (9.39)

as well as

$$Z = Z_0\,\frac{\omega}{k\,c}= Z_0\left(\frac{\omega_{p\,e}^{\,2}}{\omega^{\,2}}-1\right)^{-1/2}.$$ (9.40)

(See Exercise 4.) Furthermore, the mean $z$-directed electromagnetic energy flux becomes

$$\langle {\cal I}_z\rangle =\langle E_x\,H_y\rangle = E_0^{\,2}\,Z^{\,-1}\,{\rm e}^{-2\,k\,z}\,\langle\cos(\omega\,t)\,\sin(\omega\,t)\rangle=0.$$ (9.41)

The previous analysis demonstrates that a sinusoidal electromagnetic wave cannot propagate through a plasma when its frequency lies below the plasma frequency. Instead, the amplitude of the wave decays exponentially into the plasma. Moreover, the electric and magnetic components of the wave oscillate in phase quadrature (i.e., $\pi /2$ radians out of phase), and the wave consequentially has zero associated net energy flux. This suggests that a plasma reflects, rather than absorbs, an incident electromagnetic wave whose frequency is less than the plasma frequency (because if the wave were absorbed then there would be a net flux of energy into the plasma). Let us investigate what happens when a low-frequency electromagnetic wave is normally incident on a plasma in more detail.

Suppose that the region $z<0$ is a vacuum, and the region $z>0$ is occupied by a plasma of plasma frequency $\omega_{p\,e}$. Let the wave electric and magnetic fields in the vacuum region take the form

$$E_x(z,t) = E_i\,\cos[k_0\,(c\,t-z)] + E_r\,\cos[k_0\,(c\,t+z)+\phi_r],$$ (9.42)

$$H_y(z,t) = E_i\,Z_0^{\,-1}\,\cos[k_0\,(c\,t-z)] - E_r\,Z_0^{\,-1}\,\cos[k_0\,(c\,t+z)+\phi_r],$$ (9.43)

where $k_0=\omega/c$ is the vacuum wavenumber. Here, $E_i$ is the amplitude of an electromagnetic wave of frequency $\omega<\omega_{p\,e}$ that is normally incident on the plasma, whereas $E_r$ is the amplitude of the reflected wave, and $\phi_r$ the phase of this wave with respect to the incident wave. The wave electric and magnetic fields in the plasma are written

$$E_x(z,t) =E_t\,{\rm e}^{-k_0\,\alpha\,z}\,\cos(\omega\,t+\phi_t),$$ (9.44)

$$H_y(z,t) = E_t\,Z_0^{\,-1}\,\alpha\,{\rm e}^{-k_0\,\alpha\,z}\,\sin(\omega\,t+\phi_t),$$ (9.45)

where $E_t$ is the amplitude of the evanescent wave that penetrates into the plasma, $\phi_t$ is the phase of this wave with respect to the incident wave, and

$$\alpha =\frac{k\,c}{\omega}= \left(\frac{\omega_{p\,e}^{\,2}}{\omega^{\,2}}-1\right)^{1/2}.$$ (9.46)

The appropriate matching conditions are the continuity of $E_x$ and $H_y$ at the vacuum/plasma interface ($z=0$). (See Appendix C.) In other words,

$$E_i\,\cos(\omega\,t) + E_r\,\cos(\omega\,t+\phi_r) = E_t\,\cos(\omega\,t+\phi_t),$$ (9.47)

$$E_i\,\cos(\omega\,t) - E_r\,\cos(\omega\,t+\phi_r) = E_t\,\alpha\,\sin(\omega\,t+\phi_t).$$ (9.48)

These two equations, which must be satisfied at all times, can be solved to give

$$E_r =E_i,$$ (9.49)

$$\tan\phi_r = \frac{2\,\alpha}{1-\alpha^{\,2}},$$ (9.50)

$$E_t = \frac{2\,E_i}{(1+\alpha^{\,2})^{1/2}},$$ (9.51)

$$\tan\phi_t = \alpha.$$ (9.52)

(See Exercise 5.) Thus, the coefficient of reflection,

$$R = \left(\frac{E_r}{E_i}\right)^2 =1,$$ (9.53)

is unity, which implies that all of the incident wave energy is reflected by the plasma, and there is no energy absorption. The relative phase of the reflected wave varies from 0 (when $\omega = \omega_{p\,e}$) to $\pi$ (when $\omega\ll \omega_{p\,e}$) radians.

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 $\omega_{p\,e}\propto \sqrt{n_e}$). 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 $0.5$ 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.

Figure 9.1: Reflection and transmission of radio waves by the ionosphere.

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.