Electron cyclotron resonance heating is the simplest of the radio frequency heating methods. There is no evanescent region between the antenna and the plasma. However there may be cut-offs in the plasma. It turns out to provide remarkable advantages for heating of fusion plasma. An electromagnetic wave in the electron cyclotron resonance frequency range always propagates in the Gaussian beam in the free space as the light goes. The refraction index of the plasma doesn’t differ much from the free space refraction index as long as the plasma density is not too high. So the waves are almost perfectly matched to the plasma and experience a smooth change of the refraction index until they reach the resonant or cut-off layers. There is no impurity problems in the vicinity of in-vessel launching structures and the first wall. The impurity is very important for the future fusion devices.
The resonance layer for absorption of the electromagnetic waves can be easily controlled by proper adjustment of the magnetic induction inside the plasma. It ensures a well localized narrow power deposition profile. On the other hand, the absorption layer can be chosen at fixed magnetic induction by simply steering a well focused narrow Gaussian beam to the proper poloidal plasma position.
Wave propagation
In a homogeneous medium, the general solution of the Maxwell’s equation can be constructed as a superposition of plane waves:
\(\mathbf E(\mathbf r,t)=\mathbf E_k e^{i(\mathbf k \cdot \mathbf r-\omega t)}\)
In vacuum, the Maxwell’s equation is
\[\begin{eqnarray}
\epsilon_0 \nabla \cdot \mathbf E &=& \rho \\
\nabla \times \mathbf{E} &=& -\dot{\mathbf{B}} \\
\nabla \cdot \mathbf B &=& 0\\
\nabla \times \mathbf{B} &=& \mu_0 (\mathbf j+\epsilon_0 \dot{\mathbf E})
\end{eqnarray}\]
where \(\rho and \mathbf j\) are the free charge and current densities. The “bound” charge and current densities arising from the polarization and magnetization of the medium are included in the definition of the quantities D and H in terms of \(\epsilon\) and \(\mu\). In a plasma, the ions and electrons comprising the plasma the the equivalent of the “bound” charges and curretns. It is impossible to describe their effects into two constants because these charges move in a complicated way. Consequently, in plasma physics, one generally works with the vacuum equations in which \(\rho\) and \(\mu\) include all the charges and the currents, both the external and the internal.
We use \(\mathbf E\) and \(\mathbf B\) in the Maxwell’s equations rather than their counterparts \(\mathbf D\) and \(\mathbf H\) because the forces \(q \mathbf E\) and \(\mathbf j \times \mathbf B\) depend on \(\mathbf E \) and \(\mathbf B\).
Substitution of the plane wave solution into Maxwell’s equations yields:
\[\begin{eqnarray}
\mathbf k \times \mathbf B &=& -i \mu_0 \mathbf j -\frac{\omega}{c^2}\mathbf E \\
\mathbf k \times \mathbf E &=& \omega \mathbf B
\end{eqnarray}\]
In linear theory, the current is related to the electric field via
\[\mathbf j=\sigma \mathbf E\]
where the conductivity tensor \(\sigma\) is a function of both \(\mathbf k\) and \(\omega\). It specifies the plasma response.
We introduced the dielectric permittivity tensor just as we introduce the complex permittivity in a conductive media.
\[\mathbf k \times \mathbf B=-\frac{\omega}{c^2}\mathbf K \cdot \mathbf E\]
where the permittivity tensor \(\mathbf K\) is
\[\mathbf K=\mathbf I+\frac{i \mathbf{\sigma}}{\epsilon_0 \omega}\]
and \(\mathbf I\) is the identity tensor.
The ions and electrons are forced to move by the perturbed electric field and the induced magnetic field. If we denote the velocity by \(\mathbf v\), the current \(\mathbf j\) due to the particle motion is given by
\[\mathbf j=\sum_k n_k q_k \mathbf{v_k}\]
where n and q are the density and charge of one species, respectively.
The equation of motion of a single particle of a specie is
\[m \dot{\mathbf{v}}=q(\mathbf E+\mathbf v \times \mathbf{B})\]
Suppose \(B_0\) is along the Z axis. The linearized equation in these quantities is
\[-i \omega m \mathbf v=q(\mathbf E+\mathbf v \times \mathbf{B_0})\]
Expand the vector to 3 directions,
\[\begin{eqnarray}
-i m v_x \omega &=& q (B_0 v_y+E_x)\\
-i m v_y \omega &=& q (E_y-B_0 v_x)\\
-i m v_z \omega &=& q E_z
\end{eqnarray}\]
Solve them for \( v_x, v_y \) and \(v_z\),
\[\begin{eqnarray}
v_x &=& -\frac{-B_0 E_y q^2 + i E_x m q \omega}{B_0^2 q^2 – m^2 \omega^2}=\frac{ i \frac{E_x}{B_0} \frac{q B_0}{m} \omega-\frac{E_y}{B_0}\frac{B_0^2 q^2}{m^2}}{ \omega^2-\frac{B_0^2 q^2}{m^2} }=\frac{- i \frac{E_x}{B_0} \omega_c \omega-\frac{E_y}{B_0}\omega_c^2}{ \omega^2-\omega_c^2 }\\
v_y &=&-\frac{B_0 E_x q^2 + i E_y m q \omega}{B_0^2 q^2 – m^2 \omega^2}=\frac{\frac{E_x}{B_0}\frac{B_0^2 q^2}{m^2}+ i \frac{E_y}{B_0} \frac{q B_0}{m} \omega}{ \omega^2-\frac{B_0^2 q^2}{m^2} }=\frac{\frac{E_x}{B_0}\omega_c^2-i \frac{E_y}{B_0} \omega_c \omega}{ \omega^2-\omega_c^2}\\
v_z &=& \frac{i E_z q}{m \omega }=-i \frac{E_z}{B_0}\frac{\omega_c}{ \omega}
\end{eqnarray}\]
Where \(\omega_c=\frac{-q B_0}{m}\) is the cyclotron frequency of the charged particle, \(\omega_{ce}>0\) for electrons and \(\omega_{ci}<0\) for ions.
Write it into the matrix form
\[\mathbf v=\frac{\omega_c}{B_0}\begin{bmatrix}
\frac{-i \omega}{\omega^2-\omega_c^2} & \frac{-\omega_c}{\omega^2-\omega_c^2} & 0 \\
\frac{ \omega_c}{\omega^2-\omega_c^2} & \frac{-i \omega}{\omega^2-\omega_c^2} & 0 \\
0 & 0 & -\frac{i}{\omega}
\end{bmatrix}
\begin{bmatrix}
E_x \\
E_y \\
E_z
\end{bmatrix}
\]
The current density is
\[\mathbf j=\sigma \mathbf E=\sum_k n_k q_k \mathbf{v_k}=n q\frac{\omega_c}{B_0}\begin{bmatrix}
\frac{-i \omega}{\omega^2-\omega_c^2} & \frac{-\omega_c}{\omega^2-\omega_c^2} & 0 \\
\frac{ \omega_c}{\omega^2-\omega_c^2} & \frac{-i \omega}{\omega^2-\omega_c^2} & 0 \\
0 & 0 & -\frac{i}{\omega}
\end{bmatrix}
\begin{bmatrix}
E_x \\
E_y \\
E_z
\end{bmatrix}
=-\frac{n q^2 }{m}\begin{bmatrix}
\frac{-i \omega}{\omega^2-\omega_c^2} & \frac{-\omega_c}{\omega^2-\omega_c^2} & 0 \\
\frac{ \omega_c}{\omega^2-\omega_c^2} & \frac{-i \omega}{\omega^2-\omega_c^2} & 0 \\
0 & 0 & -\frac{i}{\omega}
\end{bmatrix}
\begin{bmatrix}
E_x \\
E_y \\
E_z
\end{bmatrix}
\]
So, the conductivity is
\[\mathbf \sigma=-\frac{n q^2 }{m}\begin{bmatrix}
\frac{-i \omega}{\omega^2-\omega_c^2} & \frac{-\omega_c}{\omega^2-\omega_c^2} & 0 \\
\frac{ \omega_c}{\omega^2-\omega_c^2} & \frac{-i \omega}{\omega^2-\omega_c^2} & 0 \\
0 & 0 & -\frac{i}{\omega}
\end{bmatrix}\]
The dielectric permittivity tenser,
\[\mathbf K=\mathbf I+\frac{i \mathbf{\sigma}}{\epsilon_0 \omega}=\mathbf I-\frac{i n q^2 }{m \epsilon_0\omega}\begin{bmatrix}
\frac{-i \omega}{\omega^2-\omega_c^2} & \frac{-\omega_c}{\omega^2-\omega_c^2} & 0 \\
\frac{ \omega_c}{\omega^2-\omega_c^2} & \frac{-i \omega}{\omega^2-\omega_c^2} & 0 \\
0 & 0 & -\frac{i}{\omega}
\end{bmatrix}=\mathbf I-\frac{i \omega_p^2 }{\omega}\begin{bmatrix}
\frac{-i \omega}{\omega^2-\omega_c^2} & \frac{-\omega_c}{\omega^2-\omega_c^2} & 0 \\
\frac{ \omega_c}{\omega^2-\omega_c^2} & \frac{-i \omega}{\omega^2-\omega_c^2} & 0 \\
0 & 0 & -\frac{i}{\omega}
\end{bmatrix}\]
\(=\begin{bmatrix}
1-\frac{\omega_p^2}{\omega^2-\omega_c^2} & \frac{i\omega_p^2}{\omega^2-\omega_c^2}\frac{\omega_c}{\omega} & 0 \\
\frac{-i\omega_p^2}{\omega^2-\omega_c^2}\frac{\omega_c}{\omega}&1-\frac{\omega_p^2}{\omega^2-\omega_c^2} & 0 \\
0 & 0 & 1-\frac{\omega_p^2}{\omega_c^2}
\end{bmatrix}\)
where \(\omega_p=\sqrt{\frac{n q^2}{\epsilon_0 m}}\) is called the plasma frequency. The dielectric permittivity tensor is simplified by introducing three quantities.
\[\mathbf K=\begin{bmatrix}
K_{\bot} & -i K_{\times} & 0 \\
i K_{\times} & K_{\bot} & 0 \\
0 & 0 & k_{\parallel}
\end{bmatrix}\]
where
\[\displaylines{
K_{\bot}=S=1-\sum_k \frac{\omega_p^2}{\omega^2-\omega_c^2} \\
K_{\times}=D=-\sum_k \frac{\omega_p^2}{\omega^2-\omega_c^2}\frac{\omega_c}{\omega} \\
K_{\parallel}=P=1-\sum_k \frac{\omega_p^2}{\omega^2}
}\]
According to the Strix notation, the following quantities are introduced：
\[R=S+D \,\,\,\, L=S-D\]
Here is a tool to calculate the dielectric permittivity tenser numerically.
Eliminating the induced magnetic field from the Maxwell’s equation, we obtain the Helmholtz equation,
\[\mathbf k \times (\mathbf k \times \mathbf E)+\frac{\omega^2}{c^2} \mathbf K \cdot \mathbf E=0\]
It is convenient to define a vector \(\mathbf n=\frac{\mathbf k c}{\omega}\) to cancel the constant in the previous equation in plasma physics. It has the same direction as the wave-vector and its magnitude n is called the refractive index. It is the ratio of the speed of light in vacuum to the phase-velocity.
\[\mathbf n \times (\mathbf n \times \mathbf E)+ \mathbf K \cdot \mathbf E=0\]
As \(\mathbf K\) is a tenser, the 2nd term has been expanded in the matrix form. The first term is expanded to the matrix form with the help of the matrix operation.Without loss of generality, we assume the angle between the \(\mathbf n\) and \(\mathbf{B_0}\) is denoted by \(\theta\) , (\mathbf{B_0}\) lies in the z axis and x axis is taken so than \(\mathbf n\) lies in the z,x plane. Then \(\mathbf n=n(\sin\theta,0,\cos\theta)\), the Helmholtz’s equation is
\[\begin{bmatrix}
K_{\bot}-n^2 \cos^2 \theta & -i K_{\times} & n^2 \sin\theta\cos\theta \\
i K_{\times} & K_{\bot}-n^2 & 0 \\
n^2\sin\theta\cos\theta & 0 & k_{\parallel}-n^2 \sin^2\theta
\end{bmatrix}
\begin{bmatrix}
E_x \\
E_y \\
E_z
\end{bmatrix}=0
\]
The condition for a nontrivial solution of the electric field is that the determinant of the square matrix be zero.
\(\displaylines{
M(\omega,\mathbf k)
= (K_{\bot}-n^2 \cos^2 \theta)(K_{\bot}-n^2)(k_{\parallel}-n^2 \sin^2\theta)-(-i K_{\times})(i K_{\times})(k_{\parallel}-n^2 \sin^2\theta) \\
-(n^2 \sin\theta\cos\theta)(K_{\bot}-n^2)(n^2\sin\theta\cos\theta)\\
=(K_{\bot}-n^2)[(K_{\bot}-n^2 \cos^2 \theta)(k_{\parallel}-n^2 \sin^2\theta)-\\(n^2 \sin\theta\cos\theta)(n^2\sin\theta\cos\theta)]-K_{\times}^2(k_{\parallel}-n^2 \sin^2\theta) \\
=(K_{\bot}-n^2)(K_{\bot}K_{\parallel}-n^2 K_{\parallel}\cos^2 \theta-K_{\bot} n^2 \sin^2 \theta)-K_{\times}^2K_{\parallel}+K_{\times}^2n^2 \sin^2\theta\\
=K_{\bot}^2K_{\parallel}-n^2K_{\bot} K_{\parallel}\cos^2 \theta-K_{\bot}^2 n^2 \sin^2 \theta\\
-n^2K_{\bot}K_{\parallel}+n^4 K_{\parallel}\cos^2 \theta+K_{\bot} n^4 \sin^2 \theta-K_{\times}^2K_{\parallel}+K_{\times}^2n^2 \sin^2\theta\\
=n^4(K_{\bot} \sin^2 \theta+K_{\parallel}\cos^2 \theta)-n^2[K_{\bot}^2 \sin^2 \theta-K_{\times}^2 \sin^2\theta+n_{\bot}K_{\parallel}(1+\cos^2 \theta)]+(K_{\bot}^2K_{\parallel}-K_{\times}^2K_{\parallel})\\
=n^4(K_{\bot} \sin^2 \theta+K_{\parallel}\cos^2 \theta)-n^2[(K_{\bot}^2 -K_{\times}^2) \sin^2\theta+K_{\bot}K_{\parallel}(1+\cos^2 \theta)]+(K_{\bot}^2-K_{\times}^2)K_{\parallel}=0
}\)
It is a quadratic in \(n^2\) and it has two roots.
\[An^4-Bn^2+C=0\]
where,\(A=K_{\bot} \sin^2 \theta+K_{\parallel}\cos^2 \theta\,\,\, B=(K_{\bot}^2 -K_{\times}^2) \sin^2\theta+K_{\bot}K_{\parallel}(1+\cos^2 \theta) \,\,\, C=(K_{\bot}^2-K_{\times}^2)K_{\parallel}=PRL\)
\[\begin{eqnarray}
\Delta &=& ((K_{\bot}^2 -K_{\times}^2) \sin^2\theta+K_{\bot}K_{\parallel}(1+\cos^2 \theta))^2-4(K_{\bot} \sin^2 \theta+K_{\parallel}\cos^2 \theta)((K_{\bot}^2-K_{\times}^2)K_{\parallel}) \\
&=&(R L \sin^2\theta+S P (1+\cos^2 \theta))^2-4(S \sin^2 \theta+P \cos^2 \theta)P R L \\
&=&( R L \sin^2 \theta +S P (2-\sin^2 \theta))^2-4 PRLS\sin^2 \theta-4P^2 RL\cos^2 \theta \\
&=&((RL-SP) \sin^2 \theta+2SP)^2-4 PRLS\sin^2 \theta-4P^2 RL\cos^2 \theta \\
&=&(RL-SP)^2 \sin^4 \theta+4 S^2 P^2 +4 PS (RL-PS) \sin^2 \theta-4 PRLS\sin^2 \theta-4P^2 RL\cos^2 \theta \\
&=&(RL-SP)^2 \sin^4 \theta+4 S^2 P^2 -4 P^2S^2 \sin^2 \theta-4P^2 RL\cos^2 \theta \\
&=&(RL-SP)^2 \sin^4 \theta+4 P^2 S^2 \cos^2 \theta-4P^2 RL\cos^2 \theta \\
&=&(RL-SP)^2 \sin^4 \theta+4P^2 \cos^2 \theta (S^2-RL) \\
&=&(RL-SP)^2 \sin^4 \theta+4P^2 D^2 \cos^2 \theta > 0
\end{eqnarray}\]
\(n^2\) is always real. n is either purely real or purely imaginary. The cold plasma dispersion relation describes waves which either propagate without evanescence, or decay without spatial oscillation.
In tokamak, the wave that propagation perpendicular to the magnetic field (\(\theta=\frac{\pi}{2})\) is more interesting,
\[n^2=\frac{RL+PS+RL-PS}{2 S}=\frac{RL}{S}\,\,\, or \,\,\, n^2=\frac{RL+PS-(RL-PS)}{2 S}=P\]
It is not clear for those modes. We start again from the original matrix in order to get a clear solution. When (\(\theta=\frac{\pi}{2})\), \(\mathbf k\) lies in the x-axis, the Helmholtz’s equation simplifies to
\[\begin{bmatrix}
S & -i D & 0 \\
i D & S-N^2 & 0 \\
0 & 0 & P-N^2
\end{bmatrix}
\begin{bmatrix}
E_x \\
E_y \\
E_z
\end{bmatrix}=0
\]
It is obvious that \(P-N^2=0\) is a solution with the eigenvector \((0,0, E_z)\). It is a transverse wave polarized with its electric field parallel to equilibrium magnetic field. Particle motions are along the magnetic field, so the mode dynamics are completely unaffected by the magnetic field. The wave is the same the electromagnetic plasma wave as that in the unmagnetized plasma. It is known as the ordinary or O-mode.
Here is another way to obtain the reflected index in special case by dividing \(\cos^2\theta\) on the original dispersion relation,
\[\begin{eqnarray}
0 &=&n^4(S \sin^2 \theta+P\cos^2 \theta)-n^2[(RL\sin^2\theta+SP(1+\cos^2 \theta)]+PRL \\
&=& n^4(S \tan^2 \theta+P)-n^2[(RL\tan^2\theta+SP(2+\tan^2 \theta)]+PRL(1+\tan^2\theta) \\
&=& n^4 S \tan^2 \theta+n^4P-n^2RL\tan^2\theta+2n^2SP+n^2SP\tan^2 \theta+PRL+PRL\tan^2\theta\\
&=& \tan^2\theta(n^4S-n^2RL-n^2SP+PRL)+(Pn^4-2SPn^2+PRL)\\
&=& \tan^2\theta(Sn^2-RL)(n^2-P)+P(n^2-R)(n^2-L)
\end{eqnarray}\]
As a result,
\[\tan^2\theta=-\frac{P(n^2-R)(n^2-L)}{(Sn^2-RL)(n^2-P)}\]
For the special case of wave propagation parallel to the magnetic field \(\theta=0\), it reduces to
\[P=0\,\,\,\, or \,\,\,\, n^2=R\,\,\,\, or \,\,\,\, n^2=L\]
Likewise, for the special case of propagation perpendicular to the filed \(\theta=\pi/2\), it yields
\[ n^2=\frac{RL}{S}\,\,\,\, or \,\,\,\, n^2=P\]
For certain values of the plasma parameters, \(n^2\) goes to zero or infinity. Reflection occurs wherever \(n^2\) goes through zero, and absorption takes place wherever \(n^2\) goes through infinity. The former case is called a wave cutoff, whereas the latter case is termed a wave resonance.
So, for the O-mode wave, reflection occurs when \(P=0\). As a result, we need \(\omega>\omega_{pe}\). As \(\omega_{pe}\) is proportional to the square root of the plasma density, it follows that O-mode electromagnetic radiation of a given frequency will only propagate in a plasma below a certain critical density.
The other solution is obtained by setting the \(2 \times 2\) determinant involving the x and y components of the electric field to zero. It is
\[S(S-n^2)-(-iD)(iD)=0\]
The dispersion relation reduces to
\[n^2=\frac{R L}{S} \]
with the associated eigenvector \(E_x(1,\frac{i D}{S},0)\). This mode is a little strange and called ex-ordinary or X-mode . It is both the transverse and longitudinal wave. It is almost right-circular polarized near the electron cyclone frequency.
Special wave
Here is a summery of the wave in the special case
| Special case | Dispersion relation | name | polarization |
| \[\theta=0\] | \[n^2=R\] | R-wave fast wave whistler wave | right circular |
| \[\theta=0\] | \[n^2=L\] | L-wave slow wave | left circular |
| \[\theta=\frac{\pi}{2}\] | \[n^2=P\] | O-wave | linear |
| \[\theta=\frac{\pi}{2}\] | \[n^2=\frac{RL}{S}\] | X-wave | elliptically |
The cutoff takes place at R=0 or L=0. The resonance takes place at S=0.
\[S=1- \frac{\omega_{pi}^2}{\omega^2-\omega_{ci}^2}- \frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2}=0\]
It is a quadratic equation of \(\omega^2\), the exact solution is very complicated. We usually use a good approximation to its roots. If the first two term are equated to zero, we obtain
\[\omega=\omega_{UH}=\sqrt{\omega_{pe}^2+\omega_{ce}^2}\]
Then this frequency is substituted into the third term, the result is far less than 1 because \(\omega_{UH}>>\omega_{pi}\). In order to obtain the second root, we simplify the equation
\[(\omega^2-\omega_{ce}^2)(\omega^2-\omega_{ci}^2)-\omega_{pi}^2(\omega^2-\omega_{ce}^2)-\omega_{pe}^2(\omega^2-\omega_{ci}^2)=0\]
Expand it to
\[\omega^4-(\omega_{ce}^2+\omega_{ci}^2-\omega_{pi}^2-\omega_{pe}^2)\omega^2+\omega_{ce}^2\omega_{ci}^2+\omega_{pi}^2\omega_{ce}^2+\omega_{pe}^2\omega_{ci}^2=0\]
Let the second root is \(x_2\) and according to the Vieta’s formulas,
\[x_2^2 \omega_{UH}^2=\omega_{ce}^2\omega_{ci}^2+\omega_{pi}^2\omega_{ce}^2+\omega_{pe}^2\omega_{ci}^2\approx\omega_{ce}^2\omega_{ci}^2+\omega_{pi}^2\omega_{ce}^2\]
So, the second root is
\[\omega_{LH}=\sqrt{\frac{\omega_{ce}^2\omega_{ci}^2+\omega_{pi}^2\omega_{ce}^2}{\omega_{pe}^2+\omega_{ce}^2}}\]
Cutoff and Resonance
Snell’s law states that the ratio of the sines of the angles of incidence and refraction is equivalent to the ratio of phase velocities in the two media, or equivalent to the reciprocal of the ratio of the indices of refraction.
\[\frac{\sin\theta_1}{\sin\theta_2}=\frac{v_1}{v_2}=\frac{c/n_1}{c/n_2}=\frac{n_2}{n_1}\]
where, \(\theta_1\) is the incident angle, \(\theta_2\) is the refraction angle. Clear off the fractions,
\[n_1\sin\theta_1=n_2\sin\theta_2\]
In a classic analogy, the area of lower refractive index is replaced by a beach, the area of higher refractive index by the sea. A man runs much faster than he swims in the sea. The fastest way for a rescuer on the beach to get to a drowning person in the sea is to run along a path that follows Snell’s law.
When light travels from a higher refractive index region to a lower refractive index region, \(\theta\) increases because n decreases. It seems to require in some cases (whenever the angle of incidence is large enough) that the sine of the refraction angle be greater than one. The light in such cases is completely reflected by the boundary, a phenomenon known as total total reflection. The largest possible angle of incidence which still results in a refracted ray is called the critical angle; in this case the refracted ray travels along the boundary between the two media. The lower the reflection index is, the smaller the critical angle is. When the reflection index is zero, all the light is reflected. It is said to be cutoff in plasma physics. On the other hand, when light travels from a lower refractive index region to a higher refractive index region, the refractive angle becomes smaller and smaller. When the refractive index tends to infinite, the refractive angle tends to zero. Here the phase velocity \(c/n\) becomes zero. The wave energy will be absorbed by the plasma. It is said to be resonance.
It is very clear that zero is the root of the quadratic equation when C=0. So cutoff occurs when \(C=PRL=0\),
\[P=0\,\,\,\, or \,\,\,\, R=0\,\,\,\, or \,\,\,\, L=0\]
R is given in a plasma which consists of electrons and of one kind of ion by
\[\begin{eqnarray}
R &=& S+D\\
&=& 1-\frac{\omega_{pi}^2}{\omega^2-\omega_{ci}^2}-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2}-\frac{\omega_{pi}^2}{\omega^2-\omega_{ci}^2}\frac{\omega_{ci}}{\omega}-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2}\frac{\omega_{ce}}{\omega} \\
&=& 1-\frac{\omega_{pi}^2}{\omega^2-\omega_{ci}^2}(1+\frac{\omega_{ci}}{\omega})–\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2}(1+\frac{\omega_{ce}}{\omega})\\
&=& 1-\frac{\omega_{pi}^2}{\omega^2-\omega_{ci}^2}\frac{\omega+\omega_{ci}}{\omega}-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2}\frac{\omega+\omega_{ce}}{\omega}\\
&=& 1-\frac{\omega_{pi}^2}{\omega(\omega-\omega_{ci})}-\frac{\omega_{pe}^2}{\omega(\omega-\omega_{ce})}\\
&=& 1-\frac{\omega_{pi}^2(\omega-\omega_{ce})+\omega_{pe}^2(\omega-\omega_{ci})}{\omega(\omega-\omega_{ci})(\omega-\omega_{ce})}\\
&=& 1-\frac{\omega(\omega_{pi}^2+\omega_{pe}^2)-\omega_{pi}^2\omega_{ce}-\omega_{pe}^2\omega_{ci}}{\omega(\omega-\omega_{ci})(\omega-\omega_{ce})}\\
&=& 1-\frac{\omega_{pi}^2+\omega_{pe}^2}{(\omega-\omega_{ci})(\omega-\omega_{ce})}
\end{eqnarray}\]
where \(\omega_{pi}^2\omega_{ce}-\omega_{pe}^2\omega_{ci}=0\), here is the prove
\[\begin{eqnarray}
\omega_{pi}^2\omega_{ce}+\omega_{pe}^2\omega_{ci} &=& \frac{n_i(Ze)^2}{\epsilon_0 m_i}\frac{eB}{m_e}+\frac{n_ee^2}{\epsilon_0 m_e}\frac{-ZeB}{m_i}\\
&=& \frac{n_iZ Ze^2}{\epsilon_0 m_i}\frac{eB}{m_e}-\frac{n_ee^2}{\epsilon_0 m_e}\frac{ZeB}{m_i}\\
&=& \frac{n_e Ze^2}{\epsilon_0 m_i}\frac{eB}{m_e}-\frac{n_ee^2}{\epsilon_0 m_e}\frac{ZeB}{m_i}\\
&=&0
\end{eqnarray}\]
In the electron cyclotron frequency range, \(\omega_{pi}, \omega_{ci}\) are very small, so
\[R\approx1-\frac{\omega_{pe}^2}{\omega(\omega-\omega_{ce})}=0\]
Clear the fraction,
\[\omega^2-\omega\omega_{ce}-\omega_{pe}^2=0\]
We try to find the R cutoff layers in a tokamak with a major radius R and miner radius a. Create a Cartesian coordinate in the plasma centre. The coordinate of the tokamak centre is (-R,0). The toroidal magnetic field at the major radius is \(B_0\).
According to the Ampere’s law, the magnetic field at point(x,y) is
\[RB_0=B(x)(R+x)\]
so,
\[B(x)=\frac{RB_0}{R+x}\]
and the local electron cyclotron frequency is
\[\omega_{ce}=\frac{eB(x)}{m}=\frac{e}{m}\frac{RB_0}{R+x}\]
Considering a parabolic density distribution with a pedestal,
\[n(x,y)=(n_{eo}-n_{ea})(1-\frac{x^2+y^2}{a^2})+n_{ea}\]
where, \(n_{eo}=1.5n_e\) is the density at plasma centre and \(n_{ea}=n_e/5\) is the density at the last closed magnetic flux surface. The square of the local electron plasma frequency is
\[\omega_{pe}^2=\frac{n(x,y)e^2}{\epsilon_0 m}=\frac{n_ee^2}{\epsilon_0 m}(1.5-\frac{x^2+y^2}{a^2}\times 1.3)\]
As a result, the R cutoff condition is
\[\omega^2-\omega\frac{e}{m}\frac{RB_0}{R+x}-\frac{n_ee^2}{\epsilon_0 m}(1.5-\frac{x^2+y^2}{a^2}\times 1.3)=0\]
It is a complicated curve.
By the way, the R cutoff on the CMA diagram is a parabolic curve. In the CMA diagram, the vertical and horizontal ordinates are
\[Y=\frac{\omega_{ce}^2}{\omega^2}\,\,\, and \,\,\, X=\frac{\omega_{pe}^2}{\omega^2}\]
The R cutoff condition is divided by \(\omega^2\), then\
\[1-\frac{\omega_{ce}}{\omega}-\frac{\omega_{pe}^2}{\omega^2}=0\]
\[\frac{\omega_{ce}^2}{\omega^2}=(1-\frac{\omega_{pe}^2}{\omega^2})^2\]
Finally, we get
\[Y=(1-X)^2\]
So, it is a parabolic curve.
The resonance condition is \(n=\infty\), then
\[\tan^2\theta=\lim_{n \to \infty}-\frac{P(n^2-R)(n^2-L)}{(Sn^2-RL)(n^2-P)}=-\frac{P}{S}\]
When \(\theta=0\), the condition \(R \to \infty\) satisfied at \(\omega=\omega_{ce}\). It is called electron cyclotron resonance. The condition \(L\to \infty\) holds when \(\omega=|\omega_{ci}|\). It is called ion cyclotron resonance.
When \(\theta=\pi/2\), \(S=0\) is the resonance condition. It is called hybrid resonance. There are two roots for the hybrid resonance. They are called upper hybrid resonance and lower hybrid resonance.