11. Electromagnetic Waves in Plasma
11. Electromagnetic Waves in Plasma¶
Learning Objectives¶
- Understand electromagnetic wave propagation in unmagnetized plasmas and the plasma cutoff
- Master the Stix cold plasma dielectric tensor for magnetized plasmas
- Derive dispersion relations for R-wave, L-wave, O-mode, and X-mode
- Analyze whistler wave dispersion and its applications
- Construct and interpret the CMA (Clemmow-Mullaly-Allis) diagram
- Apply Faraday rotation to measure magnetic fields in plasmas
Introduction¶
Unlike electrostatic waves (which involve only $\mathbf{E}$ perturbations), electromagnetic (EM) waves have both electric and magnetic field components:
$$\mathbf{E}_1 \neq -\nabla\phi, \quad \mathbf{B}_1 \neq 0$$
EM waves in plasmas exhibit rich physics: - Cutoffs: frequencies below which waves cannot propagate (evanescent) - Resonances: frequencies where wave properties diverge - Polarization: wave electric field can be linear, circular, or elliptical - Mode conversion: one wave type transforms into another
These waves are crucial for: - Plasma heating: ECRH (electron cyclotron resonance heating), ICRH (ion cyclotron) - Diagnostics: interferometry, reflectometry, polarimetry - Communications: ionospheric propagation, whistler waves - Astrophysics: pulsar emissions, solar radio bursts
We start with the simplest case (unmagnetized plasma) and build up to the full magnetized plasma theory using the Stix formalism.
1. EM Waves in Unmagnetized Plasma¶
1.1 Maxwell Equations and Wave Equation¶
Starting from Maxwell's equations for fields $\propto e^{i(\mathbf{k}\cdot\mathbf{x} - \omega t)}$:
$$\mathbf{k} \times \mathbf{E}_1 = \omega \mathbf{B}_1$$
$$\mathbf{k} \times \mathbf{B}_1 = -\frac{\omega}{c^2}(\mathbf{E}_1 + \mathbf{E}_{\text{plasma}})$$
where $\mathbf{E}_{\text{plasma}}$ is the electric field from plasma currents.
For a cold plasma, the plasma current is: $$\mathbf{j}_1 = -n_0 e \mathbf{v}_1 = -i\frac{n_0 e^2}{m_e \omega}\mathbf{E}_1 = -i\epsilon_0\omega_{pe}^2/\omega \cdot \mathbf{E}_1$$
This gives: $$\mathbf{k} \times \mathbf{B}_1 = -\frac{\omega}{c^2}\left(1 - \frac{\omega_{pe}^2}{\omega^2}\right)\mathbf{E}_1$$
Taking $\mathbf{k} \times$ of the first equation: $$\mathbf{k} \times (\mathbf{k} \times \mathbf{E}_1) = \omega \mathbf{k} \times \mathbf{B}_1$$
Using the vector identity $\mathbf{k} \times (\mathbf{k} \times \mathbf{E}_1) = \mathbf{k}(\mathbf{k}\cdot\mathbf{E}_1) - k^2\mathbf{E}_1$:
For a transverse wave ($\mathbf{k}\cdot\mathbf{E}_1 = 0$):
$$-k^2 \mathbf{E}_1 = -\frac{\omega^2}{c^2}\left(1 - \frac{\omega_{pe}^2}{\omega^2}\right)\mathbf{E}_1$$
This gives the dispersion relation:
$$\boxed{\omega^2 = \omega_{pe}^2 + k^2 c^2}$$
or equivalently:
$$\boxed{k^2 c^2 = \omega^2 - \omega_{pe}^2}$$
1.2 Cutoff and Refractive Index¶
Cutoff: The frequency $\omega = \omega_{pe}$ is a cutoff. For $\omega < \omega_{pe}$: $$k^2 < 0 \Rightarrow k = i\kappa$$
The wave becomes evanescent (exponentially decaying in space): $$\mathbf{E}_1 \propto e^{-\kappa x} e^{-i\omega t}$$
The penetration depth (skin depth) is: $$\delta = \frac{1}{\kappa} = \frac{c}{\sqrt{\omega_{pe}^2 - \omega^2}}$$
For $\omega \ll \omega_{pe}$: $$\delta \approx \frac{c}{\omega_{pe}}$$
This is why low-frequency radio waves cannot penetrate the ionosphere ($\omega_{pe} \sim 2\pi \times 10$ MHz).
Refractive index: Define $n = kc/\omega$:
$$\boxed{n^2 = 1 - \frac{\omega_{pe}^2}{\omega^2}}$$
For $\omega > \omega_{pe}$: $n < 1$ (phase velocity $v_\phi = c/n > c$!)
This does not violate relativity because information travels at the group velocity: $$v_g = \frac{d\omega}{dk} = \frac{k c^2}{\omega} = c \sqrt{1 - \frac{\omega_{pe}^2}{\omega^2}} < c$$
1.3 Physical Picture: Oscillating Electrons¶
EM wave enters plasma:
E-field → accelerates electrons → oscillating current
Current → generates secondary E-field (out of phase)
Net effect: modifies wave propagation
Low ω (ω < ωpe): Electrons respond fast enough to cancel E-field
→ wave cannot propagate (reflected)
High ω (ω > ωpe): Electrons cannot respond fast enough
→ wave propagates (with modified c)
1.4 Ionospheric Applications¶
The ionosphere has density profile $n(h)$ increasing with height:
Height (km) Density (m^-3) f_pe (MHz)
100 10^11 0.3
200 10^12 3
300 10^13 10
An AM radio wave at 1 MHz ($< 10$ MHz) reflects at the height where $f = f_{pe}(h)$. This enables over-the-horizon communication.
FM radio at 100 MHz ($> f_{pe,\max}$) passes through the ionosphere (line-of-sight only).
2. Cold Magnetized Plasma: Stix Formalism¶
2.1 Dielectric Tensor¶
In a magnetized plasma with $\mathbf{B}_0 = B_0\hat{z}$, the plasma response is anisotropic. The displacement is:
$$\mathbf{D} = \epsilon_0 \overleftrightarrow{K} \cdot \mathbf{E}$$
where $\overleftrightarrow{K}$ is the dielectric tensor.
For a cold plasma, the equation of motion for species $s$ is:
$$-i\omega m_s \mathbf{v}_s = e_s(\mathbf{E}_1 + \mathbf{v}_s \times \mathbf{B}_0)$$
Solving for $\mathbf{v}_s$ and substituting into $\mathbf{j} = \sum_s n_0 e_s \mathbf{v}_s$, we get $\overleftrightarrow{K}$.
In the Stix notation, the tensor has the form:
$$\overleftrightarrow{K} = \begin{pmatrix} S & -iD & 0 \\ iD & S & 0 \\ 0 & 0 & P \end{pmatrix}$$
where:
$$\boxed{S = 1 - \sum_s \frac{\omega_{ps}^2}{\omega^2 - \omega_{cs}^2}}$$
$$\boxed{D = \sum_s \frac{\omega_{cs}}{\omega} \frac{\omega_{ps}^2}{\omega^2 - \omega_{cs}^2}}$$
$$\boxed{P = 1 - \sum_s \frac{\omega_{ps}^2}{\omega^2}}$$
Here $\omega_{cs} = e_s B_0/m_s$ is the cyclotron frequency (positive for electrons, negative for ions by our convention).
Convenient combinations: $$R = S + D$$ $$L = S - D$$
$R$ corresponds to right-hand circular polarization, $L$ to left-hand circular.
2.2 Wave Equation¶
The wave equation is:
$$\mathbf{k} \times (\mathbf{k} \times \mathbf{E}_1) + \frac{\omega^2}{c^2}\overleftrightarrow{K}\cdot\mathbf{E}_1 = 0$$
Using $\mathbf{k} \times (\mathbf{k} \times \mathbf{E}_1) = \mathbf{k}(\mathbf{k}\cdot\mathbf{E}_1) - k^2\mathbf{E}_1$ and defining the refractive index $\mathbf{n} = \mathbf{k}c/\omega$:
$$\mathbf{n} \times (\mathbf{n} \times \mathbf{E}_1) + \overleftrightarrow{K}\cdot\mathbf{E}_1 = 0$$
This is the Appleton-Hartree equation in tensor form.
The dispersion relation $D(n, \omega) = 0$ comes from requiring $\det[\mathbf{n} \times (\mathbf{n} \times \overleftrightarrow{I}) + \overleftrightarrow{K}] = 0$.
For propagation at angle $\theta$ to $\mathbf{B}_0$:
$$A n^4 - B n^2 + C = 0$$
where $A, B, C$ are complicated functions of $S, D, P, \theta$. We focus on two special cases.
3. Parallel Propagation ($\mathbf{k} \parallel \mathbf{B}_0$)¶
3.1 Circularly Polarized Modes¶
For $\mathbf{k} = k\hat{z}$ (along $\mathbf{B}_0$), look for solutions with $\mathbf{E}_1 = E_x\hat{x} + E_y\hat{y}$.
The wave equation gives: $$-n^2 E_x + S E_x - iD E_y = 0$$ $$-n^2 E_y + iD E_x + S E_y = 0$$
Combining $E_\pm = E_x \pm iE_y$:
$$(S \mp D - n^2)E_\pm = 0$$
Two modes: 1. R-wave (right circular): $E_+ \propto e^{ikz}$, $n^2 = R = S + D$ 2. L-wave (left circular): $E_- \propto e^{ikz}$, $n^2 = L = S - D$
For electron-ion plasma:
$$R = 1 - \frac{\omega_{pe}^2}{\omega(\omega - \omega_{ce})} - \frac{\omega_{pi}^2}{\omega(\omega + \omega_{ci})}$$
$$L = 1 - \frac{\omega_{pe}^2}{\omega(\omega + \omega_{ce})} - \frac{\omega_{pi}^2}{\omega(\omega - \omega_{ci})}$$
3.2 Electron Cyclotron Resonance¶
The R-wave has a resonance at $\omega = \omega_{ce}$:
$$n^2 = R \to \infty \quad \text{as } \omega \to \omega_{ce}$$
At resonance: - Wave energy is absorbed by electrons gyrating in resonance with the wave - Wavelength $\lambda = 2\pi/k \to 0$ (infinite $k$) - Group velocity $v_g \to 0$ (energy deposited locally)
This is the basis for Electron Cyclotron Resonance Heating (ECRH): - Frequency: $f = n \times f_{ce}$ (typically $n=1$ or $2$) - For ITER: $B \sim 5.3$ T $\Rightarrow f_{ce} \sim 140$ GHz - Launched as X-mode or O-mode, absorbed at cyclotron layer
3.3 Ion Cyclotron Resonance¶
The L-wave has a resonance at $\omega = \omega_{ci}$:
$$n^2 = L \to \infty \quad \text{as } \omega \to \omega_{ci}$$
For Ion Cyclotron Resonance Heating (ICRH): - Frequency: $f \sim 30-100$ MHz for tokamaks - Can selectively heat ion species (minority heating) - Used in combination with mode conversion
3.4 Whistler Waves¶
In the frequency range $\omega_{ci} \ll \omega \ll \omega_{ce}$, the R-wave becomes the whistler mode.
Approximate dispersion (neglecting ions, $\omega_{ce} \gg \omega$):
$$n^2 = R \approx \frac{\omega_{pe}^2}{\omega(\omega_{ce} - \omega)} \approx \frac{\omega_{pe}^2}{\omega \omega_{ce}}$$
Thus: $$\boxed{k = \frac{\omega}{c}\sqrt{\frac{\omega_{pe}^2}{\omega \omega_{ce}}}}$$
or: $$\omega \approx \frac{k^2 c^2 \omega_{ce}}{\omega_{pe}^2}$$
Properties: - Highly dispersive: $\omega \propto k^2$ - Right-hand polarized: $\mathbf{E}_1$ rotates in same sense as electrons - Phase velocity: $v_\phi = \omega/k \propto \omega^{1/2}$ (higher frequency travels faster) - Group velocity: $v_g = d\omega/dk \propto \omega$ (also dispersive)
Discovery: During WWII, audio signals from lightning were detected on radio receivers, exhibiting a characteristic descending tone (falling "whistle"): - Lightning generates broadband EM waves - Higher frequencies travel faster along Earth's magnetic field lines - Arrive earlier → "whistling" sound
Whistlers propagate along magnetic field lines from one hemisphere to the other, providing information about the magnetosphere.
Applications: - Magnetospheric diagnostics - VLF communication with submarines - Wave-particle interactions (radiation belt dynamics)
4. Perpendicular Propagation ($\mathbf{k} \perp \mathbf{B}_0$)¶
4.1 Ordinary and Extraordinary Modes¶
For $\mathbf{k} = k\hat{x}$, $\mathbf{B}_0 = B_0\hat{z}$, there are two independent modes:
Ordinary mode (O-mode): $\mathbf{E}_1 = E_z\hat{z}$ (parallel to $\mathbf{B}_0$) - No $\mathbf{v} \times \mathbf{B}$ force → behaves like unmagnetized plasma - Dispersion: $$\boxed{n^2 = P = 1 - \sum_s \frac{\omega_{ps}^2}{\omega^2}}$$ - Cutoff at $\omega_{pe}$ (or $\omega_R = \sqrt{\omega_{pe}^2 + \omega_{pi}^2}$ for exact treatment)
Extraordinary mode (X-mode): $\mathbf{E}_1$ in $x$-$y$ plane (perpendicular to $\mathbf{B}_0$) - $\mathbf{v} \times \mathbf{B}$ force couples motion - Dispersion: $$\boxed{n^2 = \frac{RL}{S} = \frac{(S+D)(S-D)}{S}}$$
More explicitly: $$n^2 = 1 - \frac{\omega_{pe}^2(\omega^2 - \omega_{UH}^2)}{\omega^2(\omega^2 - \omega_{UH}^2 - \omega_{ce}^2)}$$
where $\omega_{UH}^2 = \omega_{pe}^2 + \omega_{ce}^2$ is the upper hybrid frequency.
4.2 X-mode Cutoffs and Resonances¶
The X-mode has two cutoffs (where $n^2 = 0$):
$$\omega_R = \frac{1}{2}\left(\omega_{ce} + \sqrt{\omega_{ce}^2 + 4\omega_{pe}^2}\right)$$ (right-hand cutoff)
$$\omega_L = \frac{1}{2}\left(-\omega_{ce} + \sqrt{\omega_{ce}^2 + 4\omega_{pe}^2}\right)$$ (left-hand cutoff)
And one resonance (where $n^2 \to \infty$):
$$\omega = \omega_{UH} = \sqrt{\omega_{pe}^2 + \omega_{ce}^2}$$ (upper hybrid resonance)
Near the upper hybrid layer, the wavelength $\lambda \to 0$ and the wave energy is absorbed (converted to electrostatic upper hybrid waves).
4.3 Mode Accessibility¶
The O-mode and X-mode have different accessibility to high-density plasmas:
O-mode: - Cutoff at $\omega = \omega_{pe}$ - Cannot propagate if $n > n_c$ where $\omega_{pe}(n_c) = \omega$ - Limited to densities $n < n_c = \epsilon_0 m_e \omega^2/e^2$
X-mode: - Cutoff at $\omega_R > \omega_{pe}$ (higher than O-mode) - Can access higher densities: $n < n_c^{X-mode} > n_c^{O-mode}$ - Used for overdense plasma heating
For ECRH at $\omega = 2\omega_{ce}$: - O-mode cutoff: $n_c = 4 n_{ce}$ where $n_{ce} = \epsilon_0 m_e \omega_{ce}^2/e^2$ - X-mode cutoff: higher (allows higher density access)
5. CMA Diagram¶
5.1 Construction¶
The Clemmow-Mullaly-Allis (CMA) diagram is a map of wave propagation regimes in parameter space.
Axes: - Horizontal: $X = \omega_{pe}^2/\omega^2$ (density effect) - Vertical: $Y = \omega_{ce}/\omega$ (magnetic field effect)
For each point $(X, Y)$, the diagram shows which modes (R, L, O, X) can propagate or are evanescent.
Boundaries: - Cutoffs: curves where $n^2 = 0$ (boundary between propagation and evanescence) - Resonances: curves where $n^2 \to \infty$ (absorption)
Key curves: - O-mode cutoff: $X = 1$ (vertical line) - R-wave cutoff: $R = 0 \Rightarrow X = 1 - Y$ - L-wave cutoff: $L = 0 \Rightarrow X = 1 + Y$ - Upper hybrid: $X = 1 - Y^2$ (for perpendicular propagation)
5.2 Regions and Mode Characteristics¶
Y (ω_ce/ω)
↑
| R, L, O propagate
| /
| R / O
| cutoff
| /
|/______________→ X (ω_pe²/ω²)
1
Different regions: - Region I ($X < 1 - Y$): All modes propagate - Region II ($1 - Y < X < 1$): R-wave evanescent, L and O propagate - Region III ($X > 1$): O-mode evanescent
The full diagram (including ion effects and perpendicular propagation) has $\sim 10$ distinct regions.
5.3 Applications¶
The CMA diagram is used to: - Design heating and current drive systems (choose appropriate mode) - Plan diagnostic systems (reflectometry, interferometry) - Understand ionospheric propagation - Analyze whistler propagation in magnetosphere
For example, in fusion plasmas: - High density, high $B$ → $(X, Y)$ in region where X-mode needed - Low density edge → O-mode accessible
6. Faraday Rotation¶
6.1 Theory¶
When a linearly polarized wave propagates through a magnetized plasma, the plane of polarization rotates. This is Faraday rotation.
Physical origin: - Linear polarization = superposition of R-wave and L-wave with equal amplitude - R and L have different refractive indices: $n_R \neq n_L$ - Phase difference accumulates: $\Delta\phi = (k_R - k_L) L$ - Polarization plane rotates by angle $\theta = \Delta\phi/2$
For $\omega \gg \omega_{pe}, \omega_{ce}$:
$$n_R - n_L \approx \frac{\omega_{pe}^2 \omega_{ce}}{\omega^3}$$
The rotation angle over distance $L$ is:
$$\boxed{\theta = \frac{\omega_{pe}^2 \omega_{ce}}{2c\omega^2} L = \frac{e^3}{2\epsilon_0 m_e^2 c \omega^2} \int_0^L n_e B_\parallel \, dl}$$
where $B_\parallel$ is the component of $\mathbf{B}$ along the ray path.
The rotation measure is:
$$RM = \frac{e^3}{2\pi \epsilon_0 m_e^2 c} \int n_e B_\parallel \, dl$$
In practical units: $$RM \approx 2.63 \times 10^{-13} \int n_e(\text{cm}^{-3}) B_\parallel(\text{G}) \, dl(\text{pc}) \quad (\text{rad/m}^2)$$
6.2 Astrophysical Applications¶
Faraday rotation is used to measure magnetic fields in:
Pulsars: - Measure $RM$ from multifrequency polarization observations - Infer $\int n_e B_\parallel dl$ along line of sight - Map Galactic magnetic field structure
Active Galactic Nuclei (AGN): - Jets have tangled magnetic fields - Faraday rotation gives field strength and structure
Intracluster Medium: - Galaxy clusters: $n_e \sim 10^{-3}$ cm$^{-3}$, $L \sim$ Mpc - Measure $B \sim \mu$G from rotation measures
Tokamak diagnostics: - Polarimetry measures $\int n_e B_\parallel dl$ - Combined with interferometry ($\int n_e dl$), can infer $B$ profile
6.3 Dispersion and Depolarization¶
At lower frequencies, the rotation angle is larger: $\theta \propto \omega^{-2}$.
For broadband emission with $\Delta\omega$, different frequencies rotate by different amounts, causing depolarization:
$$\Delta\theta \approx 2\theta \frac{\Delta\omega}{\omega}$$
If $\Delta\theta \gtrsim \pi/2$, the net polarization is scrambled.
This sets a depolarization frequency: $$\omega_{\text{depol}} \sim \left(\frac{e^3 n_e B_\parallel L}{\epsilon_0 m_e^2 c}\right)^{1/2}$$
7. Python Implementation¶
7.1 Dispersion in Unmagnetized Plasma¶
import numpy as np
import matplotlib.pyplot as plt
def unmagnetized_dispersion(k, omega_pe):
"""
EM wave dispersion in unmagnetized plasma: ω² = ω_pe² + k²c².
"""
c = 3e8 # m/s
omega = np.sqrt(omega_pe**2 + k**2 * c**2)
return omega
# Parameters
n = 1e19 # m^-3
e = 1.602e-19 # C
epsilon_0 = 8.854e-12 # F/m
m_e = 9.109e-31 # kg
c = 3e8 # m/s
omega_pe = np.sqrt(n * e**2 / (epsilon_0 * m_e))
f_pe = omega_pe / (2 * np.pi)
print(f"Plasma frequency: f_pe = {f_pe / 1e9:.2f} GHz")
print(f"Cutoff wavelength: λ_c = {c / f_pe:.2f} m")
# Wavenumber
k = np.linspace(0, 5, 1000) * omega_pe / c
# Dispersion
omega = unmagnetized_dispersion(k, omega_pe)
# Refractive index
n_refr = k * c / omega
# Group velocity
v_g = c**2 * k / omega
fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, figsize=(14, 10))
# Dispersion relation
ax1.plot(k * c / omega_pe, omega / omega_pe, 'b-', linewidth=2, label='Plasma')
ax1.plot(k * c / omega_pe, k * c / omega_pe, 'k--', linewidth=1.5,
label='Vacuum ($\\omega = kc$)')
ax1.axhline(1, color='r', linestyle=':', linewidth=1.5, label='Cutoff ($\\omega_{pe}$)')
ax1.set_xlabel('$kc/\\omega_{pe}$', fontsize=13)
ax1.set_ylabel('$\\omega/\\omega_{pe}$', fontsize=13)
ax1.set_title('EM Wave Dispersion in Plasma', fontsize=14)
ax1.legend(fontsize=11)
ax1.grid(True, alpha=0.3)
ax1.set_xlim([0, 5])
ax1.set_ylim([0, 6])
# Refractive index
ax2.plot(omega / omega_pe, n_refr, 'b-', linewidth=2)
ax2.axhline(1, color='k', linestyle='--', alpha=0.5, label='Vacuum')
ax2.axvline(1, color='r', linestyle=':', linewidth=1.5, label='Cutoff')
ax2.set_xlabel('$\\omega/\\omega_{pe}$', fontsize=13)
ax2.set_ylabel('Refractive Index $n$', fontsize=13)
ax2.set_title('Refractive Index: $n = \\sqrt{1 - \\omega_{pe}^2/\\omega^2}$', fontsize=14)
ax2.legend(fontsize=11)
ax2.grid(True, alpha=0.3)
ax2.set_xlim([1, 3])
ax2.set_ylim([0, 1.2])
# Phase velocity
v_phase = omega / k
ax3.plot(omega / omega_pe, v_phase / c, 'b-', linewidth=2, label='$v_\\phi$')
ax3.axhline(1, color='k', linestyle='--', alpha=0.5, label='$c$')
ax3.set_xlabel('$\\omega/\\omega_{pe}$', fontsize=13)
ax3.set_ylabel('$v_\\phi / c$', fontsize=13)
ax3.set_title('Phase Velocity (superluminal!)', fontsize=14)
ax3.legend(fontsize=11)
ax3.grid(True, alpha=0.3)
ax3.set_xlim([1, 3])
ax3.set_ylim([0.5, 3])
# Group velocity
ax4.plot(omega / omega_pe, v_g / c, 'r-', linewidth=2, label='$v_g$')
ax4.axhline(1, color='k', linestyle='--', alpha=0.5, label='$c$')
ax4.set_xlabel('$\\omega/\\omega_{pe}$', fontsize=13)
ax4.set_ylabel('$v_g / c$', fontsize=13)
ax4.set_title('Group Velocity (subluminal)', fontsize=14)
ax4.legend(fontsize=11)
ax4.grid(True, alpha=0.3)
ax4.set_xlim([1, 3])
ax4.set_ylim([0, 1.2])
plt.tight_layout()
plt.savefig('unmagnetized_dispersion.png', dpi=150, bbox_inches='tight')
plt.show()
7.2 Stix Parameters and Mode Dispersion¶
def stix_parameters(omega, n, B, Z=1, A=1):
"""
Compute Stix parameters S, D, P for cold magnetized plasma.
Parameters:
-----------
omega : float or array
Wave frequency (rad/s)
n : float
Density (m^-3)
B : float
Magnetic field (T)
Z : int
Ion charge
A : float
Ion mass number
Returns:
--------
dict with S, D, P, R, L
"""
m_i = A * 1.673e-27
omega_pe = np.sqrt(n * e**2 / (epsilon_0 * m_e))
omega_pi = np.sqrt(n * Z**2 * e**2 / (epsilon_0 * m_i))
omega_ce = e * B / m_e
omega_ci = Z * e * B / m_i
# Stix parameters (sum over electrons and ions)
S = 1 - omega_pe**2 / (omega**2 - omega_ce**2) - \
omega_pi**2 / (omega**2 - omega_ci**2)
D = omega_ce * omega_pe**2 / (omega * (omega**2 - omega_ce**2)) + \
omega_ci * omega_pi**2 / (omega * (omega**2 - omega_ci**2))
P = 1 - omega_pe**2 / omega**2 - omega_pi**2 / omega**2
R = S + D
L = S - D
return {
'S': S, 'D': D, 'P': P, 'R': R, 'L': L,
'omega_pe': omega_pe, 'omega_pi': omega_pi,
'omega_ce': omega_ce, 'omega_ci': omega_ci
}
# Parameters
B = 2.0 # T
n = 5e19 # m^-3
params = stix_parameters(1, n, B, Z=1, A=2) # Dummy ω for frequencies
f_ce = params['omega_ce'] / (2 * np.pi)
f_ci = params['omega_ci'] / (2 * np.pi)
f_pe = params['omega_pe'] / (2 * np.pi)
print(f"Electron cyclotron frequency: f_ce = {f_ce / 1e9:.2f} GHz")
print(f"Ion cyclotron frequency: f_ci = {f_ci / 1e6:.2f} MHz")
print(f"Electron plasma frequency: f_pe = {f_pe / 1e9:.2f} GHz")
# Frequency scan
f = np.linspace(0.1, 100, 5000) * 1e9 # Hz
omega = 2 * np.pi * f
params = stix_parameters(omega, n, B, Z=1, A=2)
fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, figsize=(14, 10))
# Stix parameters
ax1.plot(f / 1e9, params['S'], 'b-', linewidth=2, label='S')
ax1.plot(f / 1e9, params['P'], 'r-', linewidth=2, label='P')
ax1.axhline(0, color='k', linestyle='--', alpha=0.3)
ax1.axvline(f_ce / 1e9, color='gray', linestyle=':', alpha=0.5)
ax1.set_xlabel('Frequency (GHz)', fontsize=13)
ax1.set_ylabel('Stix Parameter', fontsize=13)
ax1.set_title('S and P Parameters', fontsize=14)
ax1.legend(fontsize=11)
ax1.grid(True, alpha=0.3)
ax1.set_xlim([0, 100])
ax1.set_ylim([-5, 5])
ax2.plot(f / 1e9, params['D'], 'g-', linewidth=2)
ax2.axhline(0, color='k', linestyle='--', alpha=0.3)
ax2.axvline(f_ce / 1e9, color='gray', linestyle=':', alpha=0.5)
ax2.set_xlabel('Frequency (GHz)', fontsize=13)
ax2.set_ylabel('D Parameter', fontsize=13)
ax2.set_title('D Parameter', fontsize=14)
ax2.grid(True, alpha=0.3)
ax2.set_xlim([0, 100])
# R and L (refractive index squared for parallel propagation)
R_positive = np.where(params['R'] > 0, params['R'], np.nan)
L_positive = np.where(params['L'] > 0, params['L'], np.nan)
ax3.semilogy(f / 1e9, R_positive, 'b-', linewidth=2, label='R (R-wave)')
ax3.semilogy(f / 1e9, L_positive, 'r-', linewidth=2, label='L (L-wave)')
ax3.axvline(f_ce / 1e9, color='b', linestyle=':', alpha=0.5,
label='$f_{ce}$ (R resonance)')
ax3.axvline(f_ci / 1e6 / 1e3, color='r', linestyle=':', alpha=0.5,
label='$f_{ci}$ (L resonance)')
ax3.set_xlabel('Frequency (GHz)', fontsize=13)
ax3.set_ylabel('$n^2$ (R, L modes)', fontsize=13)
ax3.set_title('Parallel Propagation: $n^2 = R, L$', fontsize=14)
ax3.legend(fontsize=10)
ax3.grid(True, which='both', alpha=0.3)
ax3.set_xlim([0, 100])
ax3.set_ylim([0.01, 1000])
# O-mode and X-mode (perpendicular)
n2_O = params['P']
n2_X = params['R'] * params['L'] / params['S']
O_positive = np.where(n2_O > 0, n2_O, np.nan)
X_positive = np.where(n2_X > 0, n2_X, np.nan)
ax4.semilogy(f / 1e9, O_positive, 'b-', linewidth=2, label='O-mode')
ax4.semilogy(f / 1e9, X_positive, 'r-', linewidth=2, label='X-mode')
ax4.axvline(f_pe / 1e9, color='b', linestyle=':', alpha=0.5,
label='$f_{pe}$ (O cutoff)')
ax4.set_xlabel('Frequency (GHz)', fontsize=13)
ax4.set_ylabel('$n^2$ (O, X modes)', fontsize=13)
ax4.set_title('Perpendicular Propagation: O and X modes', fontsize=14)
ax4.legend(fontsize=10)
ax4.grid(True, which='both', alpha=0.3)
ax4.set_xlim([0, 100])
ax4.set_ylim([0.01, 100])
plt.tight_layout()
plt.savefig('stix_dispersion.png', dpi=150, bbox_inches='tight')
plt.show()
7.3 CMA Diagram¶
def cma_diagram():
"""
Generate CMA (Clemmow-Mullaly-Allis) diagram.
"""
X = np.linspace(0, 3, 500)
Y = np.linspace(0, 2, 500)
XX, YY = np.meshgrid(X, Y)
# Cutoff and resonance curves
# O-mode cutoff: X = 1
# R-wave cutoff: R = 0 → X = 1 - Y (for single species, approx)
# L-wave cutoff: L = 0 → X = 1 + Y
# Upper hybrid resonance: X = 1 - Y²
fig, ax = plt.subplots(figsize=(10, 8))
# Define regions based on cutoffs
# (Simplified for electrons only)
# O-mode propagates when X < 1
O_propagate = XX < 1
# R-wave propagates when R > 0 (approx X < 1 - Y for Y << 1)
# More accurately, solve R = 0
R_cutoff_Y = np.linspace(0, 1.5, 100)
R_cutoff_X = 1 - R_cutoff_Y
# L-wave cutoff: X = 1 + Y
L_cutoff_Y = np.linspace(0, 1.5, 100)
L_cutoff_X = 1 + L_cutoff_Y
# Upper hybrid resonance: X = 1 - Y²
UH_res_Y = np.linspace(0, 1.5, 100)
UH_res_X = 1 - UH_res_Y**2
# Plot cutoff/resonance curves
ax.plot([1, 1], [0, 2], 'b-', linewidth=2, label='O-mode cutoff ($X=1$)')
ax.plot(R_cutoff_X, R_cutoff_Y, 'r-', linewidth=2,
label='R-wave cutoff ($X=1-Y$)')
ax.plot(L_cutoff_X, L_cutoff_Y, 'g-', linewidth=2,
label='L-wave cutoff ($X=1+Y$)')
ax.plot(UH_res_X, UH_res_Y, 'm--', linewidth=2,
label='Upper hybrid res ($X=1-Y^2$)')
# Add shaded regions
ax.fill_betweenx([0, 2], 0, 1, alpha=0.1, color='blue', label='O propagates')
ax.fill_between(R_cutoff_X, 0, R_cutoff_Y, alpha=0.1, color='red',
label='R evanescent')
ax.set_xlabel('$X = \\omega_{pe}^2 / \\omega^2$', fontsize=14)
ax.set_ylabel('$Y = \\omega_{ce} / \\omega$', fontsize=14)
ax.set_title('CMA Diagram (Simplified, Electrons Only)', fontsize=15)
ax.legend(fontsize=11, loc='upper right')
ax.grid(True, alpha=0.3)
ax.set_xlim([0, 3])
ax.set_ylim([0, 2])
# Annotate regions
ax.text(0.5, 0.5, 'All modes\npropagate', fontsize=12, ha='center',
bbox=dict(boxstyle='round', facecolor='wheat', alpha=0.5))
ax.text(1.5, 0.3, 'O-mode\nevanescent', fontsize=12, ha='center',
bbox=dict(boxstyle='round', facecolor='lightblue', alpha=0.5))
ax.text(0.3, 1.5, 'R-wave\nevanescent', fontsize=12, ha='center',
bbox=dict(boxstyle='round', facecolor='lightcoral', alpha=0.5))
plt.tight_layout()
plt.savefig('cma_diagram.png', dpi=150, bbox_inches='tight')
plt.show()
cma_diagram()
7.4 Whistler Wave Dispersion¶
def whistler_dispersion(k, omega_pe, omega_ce):
"""
Whistler wave dispersion: ω ≈ k²c²ω_ce/ω_pe².
"""
c = 3e8
# From k²c²ω_ce/ω_pe² = ω, solve for ω
# This is approximate; use quadratic formula
# ω ≈ k²c²ω_ce/ω_pe² for ω << ω_ce
omega = k**2 * c**2 * omega_ce / omega_pe**2
omega = np.minimum(omega, 0.5 * omega_ce) # Limit to whistler range
return omega
# Magnetospheric parameters
n = 1e6 # m^-3 (magnetosphere)
B = 5e-5 # T (Earth's magnetic field at magnetosphere)
omega_pe = np.sqrt(n * e**2 / (epsilon_0 * m_e))
omega_ce = e * B / m_e
f_pe = omega_pe / (2 * np.pi)
f_ce = omega_ce / (2 * np.pi)
print(f"Magnetospheric parameters:")
print(f" f_pe = {f_pe / 1e3:.1f} kHz")
print(f" f_ce = {f_ce / 1e3:.1f} kHz")
# Wavenumber
k = np.linspace(1e-7, 1e-5, 500) # m^-1
omega = whistler_dispersion(k, omega_pe, omega_ce)
f = omega / (2 * np.pi)
# Phase and group velocity
v_phase = omega / k
v_group = 2 * omega / k
fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(16, 5))
# Dispersion
ax1.plot(k * 1e6, f / 1e3, 'b-', linewidth=2)
ax1.axhline(f_ce / (2e3), color='r', linestyle='--',
label='$f_{ce}/2$ (upper limit)')
ax1.set_xlabel('Wavenumber $k$ ($\\mu$m$^{-1}$)', fontsize=13)
ax1.set_ylabel('Frequency (kHz)', fontsize=13)
ax1.set_title('Whistler Dispersion: $\\omega \\propto k^2$', fontsize=14)
ax1.legend(fontsize=11)
ax1.grid(True, alpha=0.3)
# Phase velocity
ax2.plot(f / 1e3, v_phase / c, 'b-', linewidth=2)
ax2.set_xlabel('Frequency (kHz)', fontsize=13)
ax2.set_ylabel('$v_\\phi / c$', fontsize=13)
ax2.set_title('Phase Velocity (higher f travels faster)', fontsize=14)
ax2.grid(True, alpha=0.3)
# Group velocity
ax3.plot(f / 1e3, v_group / c, 'r-', linewidth=2)
ax3.set_xlabel('Frequency (kHz)', fontsize=13)
ax3.set_ylabel('$v_g / c$', fontsize=13)
ax3.set_title('Group Velocity: $v_g = 2v_\\phi$', fontsize=14)
ax3.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig('whistler_dispersion.png', dpi=150, bbox_inches='tight')
plt.show()
# Simulate whistler arrival times
print("\nWhistler arrival times from lightning (L = 10,000 km):")
L = 1e7 # m
for f_khz in [1, 2, 5, 10]:
f_val = f_khz * 1e3
idx = np.argmin(np.abs(f / 1e3 - f_khz))
t_arrival = L / v_group[idx]
print(f" {f_khz} kHz: {t_arrival:.2f} s")
Summary¶
Electromagnetic waves in plasmas exhibit rich behavior governed by the interplay of plasma frequency, cyclotron frequency, and wave frequency:
Unmagnetized plasma: - Dispersion: $\omega^2 = \omega_{pe}^2 + k^2c^2$ - Cutoff at $\omega_{pe}$: waves with $\omega < \omega_{pe}$ cannot propagate - Refractive index $n < 1$: phase velocity $> c$, but group velocity $< c$ - Applications: ionospheric reflection, plasma density diagnostics
Magnetized plasma (Stix formalism): - Dielectric tensor with components $S$, $D$, $P$ - Anisotropic response leads to multiple wave modes - Combinations $R = S + D$, $L = S - D$ for circular polarizations
Parallel propagation: - R-wave (right circular): resonance at $\omega = \omega_{ce}$ → ECRH - L-wave (left circular): resonance at $\omega = \omega_{ci}$ → ICRH - Whistler waves ($\omega_{ci} \ll \omega \ll \omega_{ce}$): highly dispersive, $\omega \propto k^2$ - Applications: magnetospheric physics, VLF communication
Perpendicular propagation: - O-mode: $\mathbf{E} \parallel \mathbf{B}$, cutoff at $\omega_{pe}$ - X-mode: $\mathbf{E} \perp \mathbf{B}$, cutoffs at $\omega_R, \omega_L$, resonance at $\omega_{UH}$ - Mode accessibility: X-mode reaches higher densities than O-mode
CMA diagram: - Maps propagation regions in $(X, Y)$ space where $X = \omega_{pe}^2/\omega^2$, $Y = \omega_{ce}/\omega$ - Shows cutoffs, resonances, and mode boundaries - Essential tool for wave heating and diagnostic design
Faraday rotation: - Rotation angle: $\theta \propto \int n_e B_\parallel dl \cdot \omega^{-2}$ - Measures magnetic fields in astrophysical plasmas - Used in fusion polarimetry for current profile measurements
These wave phenomena enable: - Plasma heating (ECRH, ICRH, LHCD) - Current drive (non-inductive operation) - Diagnostics (interferometry, reflectometry, polarimetry) - Communications (ionospheric propagation) - Astrophysical observations (pulsar RM, AGN jets)
Practice Problems¶
Problem 1: Ionospheric Reflection¶
The ionosphere has a peak density of $n_e = 10^{12}$ m$^{-3}$.
(a) Calculate the plasma frequency $f_{pe}$ at the peak.
(b) An AM radio station broadcasts at 1 MHz. Will the signal reflect off the ionosphere or pass through?
(c) At what minimum frequency will signals pass through the ionosphere?
(d) The ionosphere density varies with time of day. During the day, $n_e$ increases by a factor of 10. How does this affect AM radio propagation?
Problem 2: ECRH System Design¶
A tokamak has $B_0 = 3.5$ T on axis. You are designing an ECRH system for central heating.
(a) Calculate the electron cyclotron frequency $f_{ce}$ and wavelength in vacuum.
(b) The system will use 2nd harmonic ($\omega = 2\omega_{ce}$). What frequency should the gyrotron produce?
(c) For a density $n = 5 \times 10^{19}$ m$^{-3}$, calculate the O-mode cutoff density at $2f_{ce}$. Can the wave reach the center?
(d) If O-mode cannot reach the center, explain how X-mode or electron Bernstein wave (EBW) could be used instead.
Problem 3: Whistler Propagation¶
A lightning stroke at the magnetic equator generates a broadband signal that propagates along field lines to the opposite hemisphere.
(a) For $B = 5 \times 10^{-5}$ T, $n = 10^7$ m$^{-3}$, calculate $f_{ce}$ and $f_{pe}$.
(b) Show that the frequency range $f_{ci} \ll f \ll f_{ce}$ is satisfied for $f \sim 1-10$ kHz (assume hydrogen ions).
(c) Calculate the group velocity at $f = 5$ kHz.
(d) If the path length is $L = 10,000$ km, how long does it take for the 5 kHz component to arrive? How about the 1 kHz component? Explain the "whistling" sound.
Problem 4: CMA Diagram Regions¶
For a plasma with $f_{pe} = 50$ GHz and $f_{ce} = 70$ GHz, determine which modes can propagate at the following frequencies:
(a) $f = 40$ GHz (X-band radar)
(b) $f = 75$ GHz (W-band)
(c) $f = 120$ GHz (ECRH at 2nd harmonic)
(d) For each frequency, calculate $X$ and $Y$, and identify the accessible modes (R, L, O, X).
Problem 5: Faraday Rotation Measurement¶
A polarized wave at $\lambda = 6$ cm (5 GHz) propagates through a plasma slab with $n_e = 10^{18}$ m$^{-3}$, $B_\parallel = 0.1$ T, $L = 1$ m.
(a) Calculate the Faraday rotation angle $\theta$.
(b) If measurements are made at two wavelengths ($\lambda_1 = 6$ cm, $\lambda_2 = 3$ cm), what is the difference in rotation angles $\Delta\theta$?
(c) From $\Delta\theta$, derive the rotation measure $RM = \theta/\lambda^2$.
(d) If only $RM$ is measured (not $n_e$ and $B$ separately), what additional measurement would you need to determine $B_\parallel$?
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