1. Introduction to Plasma Physics and Plasma-based Acceleration
Plasma waves

2. 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)

3. 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

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

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

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

7. 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

8. Types of waves
Electrostatic waves: Transverse waves:

9. 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

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

11. 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1 requires: Otherwise strong interaction with ions: wave is Landau-damped and fluid description no longer valid

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

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

14. 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

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

16. 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

17. 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

18. 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)

19. 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)

20. 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

21. 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)

22. 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)