Electromagnetic Waves in Magnetized Plasmas

Let us extend the analysis of the previous section to consider a general electromagnetic wave propagating through a uniform plasma with an equilibrium magnetic field of strength, ${\bf B} = B_0\,{\bf e}_z$. The plasma is assumed to consist of two species: electrons of mass $m_e$ and electric charge $-e$, and ions of mass $m_i$ and electric charge $+e$. The plasma is also assumed to be electrically neutral, so that the equilibrium number density of the ions is the same as that of the electrons; namely, $n_e$ (Stix 1962). The equations of motion of a constituent ion of the plasma are written

$$m_i\,\frac{d^{\,2}x_i}{dt^{\,2}} = e\,E_x+e\,B_0\,\frac{dy_i}{dt},$$ (9.92)

$$m_i\,\frac{d^{\,2} y_i}{dt^{\,2}} =e\,E_y-e\,B_0\,\frac{dx_i}{dt},$$ (9.93)

$$m_i\,\frac{d^{\,2} z_i}{dt^{\,2}} =e\,E_z,$$ (9.94)

where $x_i$, $y_i$, and $z_i$ are the wave-induced displacements of the ion along the three Cartesian axes. (Here, we are including ion motion in our analysis because such motion is important in certain frequency ranges.) As before, the former terms on the right-hand sides of the previous equations represent the forces exerted on the ion by the wave electric field, ${\bf E}$, whereas the latter terms represent the forces exerted by the equilibrium magnetic field when the ion moves (Fitzpatrick 2008). (As before, we can neglect any forces due to the wave magnetic field, as long as the particle motion remains non-relativistic.) The equations of motion of a constituent electron take the form

$$m_e\,\frac{d^{\,2}x_e}{dt^{\,2}} = -e\,E_x-e\,B_0\,\frac{dy_e}{dt},$$ (9.95)

$$m_e\,\frac{d^{\,2} y_e}{dt^{\,2}} =-e\,E_y+e\,B_0\,\frac{dx_e}{dt},$$ (9.96)

$$m_e\,\frac{d^{\,2} z_e}{dt^{\,2}} =-e\,E_z.$$ (9.97)

The Cartesian components of the electric dipole moment per unit volume are

$$P_x =e\,n_e\,(x_i-x_e),$$ (9.98)

$$P_y =e\,n_e\,(y_i-y_e),$$ (9.99)

$$P_z =e\,n_e\,(z_i-z_e).$$ (9.100)

Finally, the electric displacement is written

$${\bf D}= \epsilon_0\,{\bf E}+ {\bf P}.$$ (9.101)

Consider a right-hand circularly polarized (with respect to the direction of the equilibrium magnetic field) wave whose electric field takes the form

$$E_x({\bf r},t) = E_R\,\cos(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.102)

$$E_y({\bf r},t) = E_R\,\sin(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.103)

$$E_z({\bf r},t) =0.$$ (9.104)

Let us write

$$x_i({\bf r},t) = \hat{x}_i\,\cos(\omega\,t-{\bf k}\cdot {\bf r}),$$ (9.105)

$$y_i({\bf r},t) = \hat{y}_i\,\sin(\omega\,t-{\bf k}\cdot {\bf r}),$$ (9.106)

$$z_i({\bf r},t) =0,$$ (9.107)

$$x_e({\bf r},t) = \hat{x}_e\,\cos(\omega\,t-{\bf k}\cdot {\bf r}),$$ (9.108)

$$y_e({\bf r},t) = \hat{y}_e\,\sin(\omega\,t-{\bf k}\cdot {\bf r}),$$ (9.109)

$$z_e({\bf r},t) =0,$$ (9.110)

$$P_x({\bf r},t) =\hat{P}_x\,\cos(\omega\,t-{\bf k}\cdot {\bf r}),$$ (9.111)

$$P_y({\bf r},t) = \hat{P}_y\,\sin(\omega\,t-{\bf k}\cdot {\bf r}),$$ (9.112)

$$P_z({\bf r},t) =0.$$ (9.113)

Equations (9.92)–(9.113) yield

$$\hat{x}_i=\hat{y_i} = -\frac{e\,E_R}{m_i\,\omega\,(\omega+{\mit\Omega}_i)},$$ (9.114)

$$\hat{x}_e=\hat{y_e} = \frac{e\,E_R}{m_e\,\omega\,(\omega-{\mit\Omega}_e)},$$ (9.115)

$$\hat{P}_x=\hat{P}_y =-\epsilon_0\left[ \frac{\omega_{p\,i}^{\,2}}{\omega\,(\omega+{\mit\Omega}_i)}+\frac{\omega_{p\,e}^{\,2}}{\omega\,(\omega-{\mit\Omega}_e)}\right],$$ (9.116)

where

$${\mit\Omega}_i = \frac{e\,B_0}{m_i},$$ (9.117)

$${\mit\Omega}_e = \frac{e\,B_0}{m_e},$$ (9.118)

$$\omega_{p\,e} = \left(\frac{n_e\,e^{\,2}}{\epsilon_0\,m_e}\right)^{1/2},$$ (9.119)

$$\omega_{p\,i} = \left(\frac{n_e\,e^{\,2}}{\epsilon_0\,m_i}\right)^{1/2}.$$ (9.120)

Here, ${\mit\Omega}_i$ is termed the ion cyclotron frequency, and is the frequency at which ions gyrate in the plane perpendicular to the equilibrium magnetic field (Stix 1962). Moreover, ${\mit\Omega}_e$ is the electron cyclotron frequency, and is the frequency at which electrons gyrate in the plane perpendicular to the equilibrium magnetic field (ibid). Finally, $\omega_{p\,i}$ and $\omega_{p\,e}$ are termed the ion plasma frequency, and the electron plasma frequency, respectively (ibid). Of course, ${\mit\Omega}_e\gg {\mit\Omega}_i$ and $\omega_{p\,e}\gg \omega_{p\,i}$, because $m_i\gg m_e$. Finally, it follows from Equations (9.101)–(9.104), (9.113), and (9.116), that the electric displacement of a right-hand circularly polarized wave propagating through a magnetized plasma has the components

$$D_x({\bf r},t) = \epsilon_0\,R\,E_R\,\cos(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.121)

$$D_y({\bf r},t) = \epsilon_0\,R\,E_R\,\sin(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.122)

$$D_z({\bf r},t) =0,$$ (9.123)

where

$$R = 1 -\frac{\omega_{p\,i}^{\,2}}{\omega^{\,2}}\left(\frac{\omega... ...a_{p\,e}^{\,2}}{\omega^{\,2}}\left(\frac{\omega}{\omega-{\mit\Omega}_e}\right).$$ (9.124)

Consider a left-hand circularly polarized (with respect to the direction of the equilibrium magnetic field) wave whose electric field takes the form

$$E_x({\bf r},t) = E_L\,\cos(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.125)

$$E_y({\bf r},t) = -E_L\,\sin(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.126)

$$E_z({\bf r},t) =0.$$ (9.127)

By repeating the previously described analysis (with appropriate modifications), we deduce that

$$D_x({\bf r},t) = \epsilon_0\,L\,D_L\,\cos(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.128)

$$D_y({\bf r},t) = -\epsilon_0\,L\,D_L\,\sin(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.129)

$$D_z({\bf r},t) =0,$$ (9.130)

where

$$L= 1-\frac{\omega_{p\,i}^{\,2}}{\omega^{\,2}}\left(\frac{\omega}{... ...a_{p\,e}^{\,2}}{\omega^{\,2}}\left(\frac{\omega}{\omega+{\mit\Omega}_e}\right).$$ (9.131)

Finally, consider a wave whose electric field is polarized parallel to the equilibrium magnetic field, so that

$$E_x({\bf r},t) =0,$$ (9.132)

$$E_y({\bf r},t) =0,$$ (9.133)

$$E_z({\bf r},t) =E_P\,\cos(\omega\,t-{\bf k}\cdot{\bf r}).$$ (9.134)

Again, repeating the previous analysis (with suitable modifications), we obtain

$$D_x({\bf r},t) =0,$$ (9.135)

$$D_y({\bf r},t) =0,$$ (9.136)

$$D_z({\bf r},t) =\epsilon_0\,P\,\cos(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.137)

where

$$P= 1-\frac{\omega_{p\,i}^{\,2}}{\omega^{\,2}}-\frac{\omega_{p\,e}^{\,2}}{\omega^{\,2}}.$$ (9.138)

Now, the equations that govern electromagnetic wave propagation through a dielectric media are (see Appendix C)

$$\frac{\partial D_x}{\partial t} = \frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z},$$ (9.139)

$$\frac{\partial D_y}{\partial t} = \frac{\partial H_x}{\partial z}-\frac{\partial H_z}{\partial x},$$ (9.140)

$$\frac{\partial D_z}{\partial t} = \frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y},$$ (9.141)

$$\frac{\partial B_x}{\partial t} = \frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y},$$ (9.142)

$$\frac{\partial B_y}{\partial t} = \frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z},$$ (9.143)

$$\frac{\partial B_z}{\partial t} =\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}.$$ (9.144)

Consider an electromagnetic wave with a general polarization (with respect to the equilibrium magnetic field). Such a wave can be written as a linear combination of a right-hand circularly polarized wave, a left-hand circularly polarized wave, and a wave with parallel polarization. In other words,

$$E_x({\bf r},t) = (E_R+E_L)\,\cos(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.145)

$$E_y({\bf r},t) = (E_R-E_L)\,\sin(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.146)

$$E_z({\bf r},t) =E_P\,\cos(\omega\,t-{\bf k}\cdot{\bf r}).$$ (9.147)

[See Equations (9.102)–(9.104), (9.125)–(9.127), and (9.132)–(9.134).] It follows, from the previous analysis, that

$$D_x({\bf r},t) =\epsilon_0\, (R\,E_R+L\,E_L)\,\cos(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.148)

$$D_y({\bf r},t) = \epsilon_0\,(R\,E_R-L\,E_L)\,\sin(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.149)

$$D_z({\bf r},t) =\epsilon_0\,P\,E_P\,\cos(\omega\,t-{\bf k}\cdot{\bf r}).$$ (9.150)

[See Equations (9.121)–(9.123), (9.128)–(9.130), and (9.135)–(9.137).] Suppose that

$$H_x({\bf r},t) = \hat{H}_x\,\sin(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.151)

$$H_y({\bf r},t) = \hat{H}_y\,\cos(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.152)

$$H_z({\bf r},t) =\hat{H}_z\,\sin(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.153)

which implies that

$$B_x({\bf r},t) = \mu_0\,\hat{H}_x\,\sin(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.154)

$$B_y({\bf r},t) = \mu_0\,\hat{H}_y\,\cos(\omega\,t-{\bf k}\cdot{\bf r}),$$ (9.155)

$$B_z({\bf r},t) =\mu_0\,\hat{H}_z\,\sin(\omega\,t-{\bf k}\cdot{\bf r}).$$ (9.156)

Finally, let

$${\bf k} = k\,(\sin\theta,\,0,\,\cos\theta),$$ (9.157)

which means that the wavevector lies in the $x$-$z$ plane, and subtends an angle $\theta$ with the equilibrium magnetic field.

Equations (9.139)–(9.157) yield

$$\epsilon_0\,\omega\,(R\,E_R+L\,E_L) = k\,\cos\theta\,\hat{H}_y,$$ (9.158)

$$\epsilon_0\,\omega\,(R\,E_R-L\,E_L) =-k\,\cos\theta\,\hat{H}_x+k\,\sin\theta\,\hat{H}_z,$$ (9.159)

$$\epsilon_0\,\omega\,P\,E_P = - k\,\sin\theta\,\hat{H}_y,$$ (9.160)

$$\mu_0\,\omega\,\hat{H}_x = -k\,\cos\theta\,(E_R-E_L),$$ (9.161)

$$\mu_0\,\omega\,\hat{H}_y = -k\,\sin\theta\,E_P + k\,\cos\theta\,(E_R+E_L),$$ (9.162)

$$\mu_0\,\omega\,\hat{H}_z = k\,\sin\theta\,(E_R-E_L),$$ (9.163)

which can be combined to give

$$\left( \begin{array}{ccc} (\omega/c)^{\,2}\,R-k^{\,2}\,\cos^2... ...ft(\begin{array}{c}0\\ [0.5ex] 0\\ [0.5ex] 0\end{array}\right).$$ (9.164)

The previous equation determines the frequencies and polarizations of an electromagnetic wave of wavenumber $k$ that propagates through a magnetized plasma, and whose direction of propagation subtends an angle $\theta$ with the magnetic field.

Suppose, finally, that

$$E_x(x,z,t) = \hat{E}_x\,\cos(\omega\,t-k\,\sin\theta\,x-k\,\cos\theta\,z),$$ (9.165)

$$E_y(x,z,t) = \hat{E}_y\,\sin(\omega\,t-k\,\sin\theta\,x-k\,\cos\theta\,z),$$ (9.166)

$$E_z(x,z,t) = \hat{E}_z\,\cos(\omega\,t-k\,\sin\theta\,x-k\,\cos\theta\,z).$$ (9.167)

It follows, from Equations (9.145)–(9.147), that $E_R=(\hat{E}_x+\hat{E}_y)/2$, $E_L=(\hat{E}_x-\hat{E}_y)/2$, and $E_P=\hat{E}_z$. Hence, Equation (9.164) transforms to give the eigenmode equation

$$\left( \begin{array}{ccc} S-n^{\,2}\cos^2\theta,& D,&n^{\,2}\... ...ft(\begin{array}{c}0\\ [0.5ex] 0\\ [0.5ex] 0\end{array}\right).$$ (9.168)

Here,

$$n = \frac{c\,k}{\omega}$$ (9.169)

is the effective refractive index of the plasma, whereas

$$S = \frac{R+L}{2},$$ (9.170)

$$D =\frac{R-L}{2}.$$ (9.171)