Dispersion Relation ω(k)
Phase & Group Velocity
Pulse Propagation & Broadening
Interactive visualization of electromagnetic wave propagation in plasma — dispersion relation, cutoff frequency, phase and group velocity, and pulse broadening
In an unmagnetized cold plasma, electromagnetic waves satisfy ω² = ωp² + k²c², where ωp = √(ne²/(ε₀mₑ)) is the electron plasma frequency determined by electron density n. For ω < ωp, k becomes imaginary — the wave is evanescent and decays exponentially. This is the plasma cutoff. Above ωp, waves propagate with phase velocity vφ = ω/k = c/√(1 - ωp²/ω²) > c and group velocity vg = dω/dk = c√(1 - ωp²/ω²) < c. The product vφ·vg = c², ensuring relativistic causality is never violated despite superluminal phase velocity.
When ω < ωp, the refractive index n = √(1 - ωp²/ω²) becomes imaginary, and the wave decays as exp(-κx) with κ = (ωp/c)√(1 - ω²/ωp²). This skin depth δ = 1/κ determines how far the wave penetrates. Metals reflect visible light because their plasma frequency (~10¹⁵ Hz) is in the UV — visible light (ω < ωp) cannot propagate. The ionosphere reflects HF radio (3-30 MHz) because its plasma frequency is ~10 MHz, enabling long-range radio communication. Signals at ω > ωp (VHF and above) pass through for satellite communication.
A pulse is a superposition of many frequencies, each propagating at a different phase velocity. Near the cutoff, the dispersion is strong and the pulse broadens significantly. The group velocity vg = c√(1 - ωp²/ω²) approaches zero as ω→ωp, causing severe delay and distortion. This dispersive broadening is characterized by the group velocity dispersion (GVD) parameter β₂ = d²k/dω², which determines how different spectral components drift apart. In interstellar communication, plasma dispersion from the interstellar medium causes pulse arrival time to depend on frequency — higher frequencies arrive first (dm⁻¹·Δt ∝ Δf/f²), which is how pulsars were discovered.
The simplest case: no external magnetic field B₀. The dispersion relation is ω² = ωp² + k²c². Transverse EM waves have a single cutoff at ω = ωp. Below this frequency, no electromagnetic wave can propagate through the plasma. The wave electric field is perpendicular to the propagation direction (transverse), and both left and right circular polarizations propagate identically.
With an external magnetic field B₀ along the propagation direction, left and right circularly polarized waves experience different dispersion: n²(L,R) = 1 - ωp²/[ω(ω ∓ ωc)], where ωc = eB₀/mₑ is the electron cyclotron frequency. The R-wave has a resonance at ω = ωc (electron cyclotron resonance) and a cutoff at ωR = (ωc + √(ωc² + 4ωp²))/2. The L-wave has no cyclotron resonance (for electrons) and a cutoff at ωL = (-ωc + √(ωc² + 4ωp²))/2. This birefringence causes Faraday rotation — the polarization plane rotates as the wave propagates.
When the wave propagates perpendicular to B₀ with E ∥ B₀, the dispersion relation is the same as the unmagnetized case: n² = 1 - ωp²/ω². The magnetic field has no effect because the electron oscillation is along B₀. This is the "ordinary" mode with a single cutoff at ω = ωp. The extraordinary mode (X-mode, E ⊥ B₀) has a more complex dispersion with two cutoffs and resonances, involving both ωp and ωc.
The ionosphere (50-1000 km altitude) has plasma frequencies of 3-10 MHz. HF radio (3-30 MHz) reflects off the ionosphere, enabling over-the-horizon communication. Higher frequencies (VHF/UHF) penetrate for satellite links. The maximum usable frequency (MUF) depends on the electron density profile and path geometry. Ionospheric sounding (ionosondes) measure the plasma frequency vs altitude by sweeping frequencies and detecting echoes — directly measuring the dispersion relation.
Pulsar radio pulses propagate through the interstellar medium (ISM), a tenuous plasma with ~1 electron/cm³. Different frequency components travel at different group velocities, causing the pulse to arrive later at lower frequencies: Δt ∝ DM/f², where DM = ∫ne·dl is the dispersion measure along the line of sight. Jocelyn Bell Burnell discovered pulsars in 1967 partly because the dispersed pulses had a distinctive frequency-swept signature. Measuring DM provides the integrated electron density and thus distance estimates to pulsars.
In tokamak fusion reactors, plasma density is measured by launching microwave beams and detecting the cutoff. A reflectometer sweeps frequency and measures the time delay of reflected signals — the cutoff location moves as ωp changes with density, providing a density profile. Interferometry measures the phase shift of a probing beam through the plasma: Δφ = (ω/c)∫(1-n)dl, giving line-integrated density. These diagnostics are essential for controlling fusion plasmas at 10²⁰ m⁻³ densities and keV temperatures.
The solar corona has ωp ~ 300 MHz at the base, decreasing with altitude. Radio observations at different frequencies probe different coronal heights. Type III solar radio bursts show rapid frequency drift (high to low) as energetic electrons stream outward through decreasing density — the frequency drift directly maps the electron density profile. CME-driven shock waves produce Type II bursts with slower drift. Space weather prediction relies on understanding these plasma wave phenomena.