Introduction to Plasma Physics and Plasma-based Acceleration Plasma waves

Plasma waves Plasma is a medium that can sustain both electrostatic and electromagnetic waves ES oscillations first studied by Irving Langmuir and Lewi Tonks (Langmuir wave) Later work also performed by e.g. Hannes Alfvén (Alfvén wave)

Non-magnetised plasma Waves in non-magnetised, collision-less plasma most important for wakefield acceleration: Laser propagation Wakefield formation Various laser instabilities Plasma also homogeneous, charge-neutral, has one ion-species

Basic equations Start with continuity equation and momentum balance: Consider small perturbations around static equilibrium

Perturbed equations Consider small perturbations around static equilibrium: Insert perturbed quantities into continuity and momentum equations; Neglect quadratic terms:

Plane waves Insert plane wave: This yields: Thus: Need to add equation of state for p

Equation of state Fast wave: The component α will be “1-D adiabatic”: Slow wave: The component α will be isothermal: Intermediate wave: Strong coupling between wave and plasma, e.g. Landau damping; kinetic model needed

Types of waves Electrostatic waves: Transverse waves:

Ion-acoustic waves Ion-acoustic (ion-sound) wave is a longitudinal electrostatic wave, faster than ions, slower than electrons: Electrons are isothermal, behaviour dictated by their temperature Ions are adiabatic, behaviour dictated by their inertia

Ion-acoustic waves Use plane-wave density equations: Insert into Poisson’s equation, and use kλd<<1:

Ion-acoustic waves Ion sound speed: vs >> vth,i requires: kλd<<1 requires: Otherwise strong interaction with ions: wave is Landau-damped and fluid description no longer valid

Langmuir waves Langmuir wave is a fast, longitudinal, electrostatic wave: Electrons are adiabatic, ions are static. This yields:

Langmuir waves Phase speed: Group speed: Frequency: , but… Strong electron Landau damping for: This determines useful frequency range

Langmuir vs. ion-sound Langmuir: Frequency is roughly plasma frequency, high phase speed, low group speed, may be damped by electrons Ion-sound: Frequency well below plasma frequency (may even be 0), low phase and group speed, may be damped by ions

Transverse waves We consider a fast, high-frequency, transverse plasma wave (transverse electron motion, no ion motion):

Transverse waves The above equations yield a transverse EM wave with dispersion relation: Frequency range: Phase speed: Group speed: Cut-off frequency: Below cut-off, EM waves will be reflected

Basic terminology ωp<ω: Plasma is underdense for light wave ωp>ω : Plasma is overdense for light wave Density/surface where ωp=ω is called critical density/surface; light reflects here Penetration depth: c/ωp

Applications of EM waves Light waves in plasma: laser beams! Laser-wakefield acceleration Laser-driven fusion Plasma heating by laser absorption Plasma density diagnostics Medium-wave radio (radio waves reflected by Earth’s ionosphere)

Landau damping Happens when waves interact with background plasma particles Particles that are just faster than the wave will drive it; Particles that are just slower than the wave will damp it; More slower than faster particles: a net damping results L. Landau, J. Phys. USSR 10, 25 (1946); JETP 16, 574 (translation)

Landau damping Proportional to slope of f(v) at v=vph Landau damping will saturate if slope is (almost) zero Even Landau growth possible for positive slope

Waves in magnetised plasma ion cyclotron wave lower hybrid wave upper hybrid wave O-wave X-wave Too many to deal with here R-wave (Whistler wave) L-wave Alfvén wave magnetosonic wave drift waves Thomas H. Stix, Waves in plasmas (Springer, New York, 1992)

Summary Waves in non-magnetised plasma: Ion-acoustic waves Langmuir waves Electromagnetic waves Each with their own properties, frequency ranges, etc. ES waves may be Landau-damped; EM waves won’t be (vph too large)