PProceedings of the 2019 CERN–Accelerator–School course on
High Gradient Wakefield Accelerators, Sesimbra, (Portugal)
Introduction to Plasma Physics
Paul Gibbon
Forschungszentrum Jülich GmbH, Institute for Advanced Simulation, Jülich Supercomputing Centre,D-52425 Jülich, GermanyCentre for Mathematical Plasma Astrophysics, Katholieke Universiteit Leuven, 3000 Leuven, Belgium
Abstract
The following notes are intended to provide a brief primer in plasma physics,introducing common definitions, basic properties and processes typicallyfound in plasmas. These concepts are inherent in contemporary plasma-basedaccelerator schemes, and thus build foundation for the more advanced lectureswhich follow in this volume. No prior knowledge of plasma physics is re-quired, but the reader is assumed to be familiar with basic electrodynamicsand fluid mechanics.
Keywords
Plasma definitions; wave propagation; electron motion; ponderomotive force. Plasma types and definitions
Plasmas are often described as the fourth state of matter, alongside gases, liquids and solids; a defini-tion which does little to illuminate their main physical attributes. In fact, plasmas can exhibit behaviourcharacteristic of all three of the more familiar states depending on its density and temperature, so weobviously need to look for other distinguishing features. A simple textbook definition [1, 2] would be:a quasi-neutral gas of charged particles showing collective behaviour , which sounds a bit more authori-tative, but demands further explanation of the rather fuzzy-sounding ‘quasi-neutrality’ and ‘collectivity’.The first of these is actually just a mathematical way of saying that even though the plasma particles con-sist of freely moving electrons and ions, their overall charge densities cancel each other in equilibrium.So, if n e and n i are the number densities of electrons and ions with charge state Z respectively, thenthese are locally balanced : n e (cid:39) Zn i . (1)The second property, collective behaviour, arises because of the long range nature of the /r Coulomb potential, which means that local disturbances in equilibrium can have a strong influence onremote regions of the plasma. In other words, macroscopic fields usually dominate over short-livedmicroscopic fluctuations, and a net charge imbalance ρ = e ( Zn i − n e ) will immediately give rise toan electrostatic field according to Gauss’ law: ∇ . E = ρε . Likewise, the same set of charges moving with velocities v e and v i respectively, will give rise to a current density J = e ( Zn i v i − n e v e ) . This in turn induces a magnetic field according to Ampères law: ∇ × B = µ J . It is these internally driven electric and magnetic fields which largely determine the dynamics of the plasma,including its response to externally applied fields through particle or laser beams – like, for example, inthe case of plasma-based accelerator schemes.Now that we have established what plasmas are, it is natural to ask where we can find them. In factthey are rather ubiquitous: in the cosmos, 99% of the visible universe is in a plasma state: stars, the in-terstellar medium and jets of material emanating from various astrophysical objects. Closer to home,Available online at https://cas.web.cern.ch/previous-schools a r X i v : . [ phy s i c s . acc - ph ] J u l he ionosphere extending from around 50 km = 10 Earth-radii to 1000 km provides vital protection tolife on Earth from solar radiation. Terrestrial plasmas can be found in fusion devices, machines designedto confine, ignite and ultimately extract useful energy from deuterium-tritium fuel; street lighting; in-dustrial plasma torches and etching processes; and lightning discharges. Needless to say, plasmas playa central role in the present school, providing the medium to support very large, travelling-wave fieldstructures for the purposes of accelerating particles to high energies. Table 1 provides a brief overviewof these various plasma types and their properties. Table 1:
Densities and temperatures of various plasma types.
Type Electron density Temperature n e n e n e ( cm − ) T e T e T e (eV ∗ ) Stars × Laser fusion × Magnetic fusion Laser-produced − − Discharges − ∗ ≡ In most types of plasma, quasi-neutrality is not just an ideal equilibrium state, it is something thatthe plasma actively tries to achieve by readjusting the local charge distribution in response to a dis-turbance. Consider a hypothetical experiment in which an ion or positively charged ball is immersed intoa plasma – see Fig. 1. After some time, the ions in the ball’s vicinity will be repelled and the electronsattracted, leading to an altered average charge density in this region. In turns out that we can calculatethe potential φ ( r ) of this sphere after such a readjustment has taken place. Fig. 1:
Debye shielding of charged spheres immersed in a plasma
First of all, we need to know how fast the electrons and ions actually move. For equal electron andion temperatures ( T e = T i ), we have by definition: m e v e = 12 m i v i = 32 k B T e , (2)where v e and v i represent the respective average election and ion velocities. Therefore, for a hydrogen2lasma, where the ion charge and atomic number are both unity, Z = A = 1 , we find: v i v e = (cid:18) m e m i (cid:19) / = (cid:18) m e Am p (cid:19) / (cid:39) . In other words, ions are almost stationary on the electron timescale. To a good approximation, we canoften write: n i (cid:39) n , (3)where the material (e.g. gas) number density, n = N A ρ m /A and ρ m is the usual mass density; N A the Avogadro number. In thermal equilibrium, the electron density follows a Boltzmann distribution [1]: n e = n i exp( eφ/k B T e ) , (4)where n i is the ion density, k B is the Boltzmann constant, and φ ( r ) is the potential created by the externaldisturbance. From Gauss’ law (Poisson’s equation), we can also write down: ∇ φ = − ρε = − eε ( n i − n e ) . (5)So now we can combine Eq. (5) with Eqs. (4) and (3) in spherical geometry to eliminate n e and arriveat a physically meaningful solution: φ D = 14 πε e − r/λ D r . (6)This latter condition supposes that φ → at r = ∞ . The characteristic length scale λ D inside the expo-nential factor is known as the Debye length , given by: λ D = (cid:18) ε k B T e e n e (cid:19) / = 743 (cid:18) T e eV (cid:19) / (cid:18) n e cm − (cid:19) − / cm . (7)The Debye length is a fundamental property of nearly all plasmas of interest and depends equally on itstemperature and density. An ideal plasma has many particles per Debye sphere: N D ≡ n e π λ D (cid:29) , (8)a prerequisite for the collective behaviour encountered earlier. An alternative way of expressing this isvia the so-called plasma parameter : g ≡ n e λ D , (9)which is basically the reciprocal of N D . Classical plasma theory is based on assumption that g (cid:28) ,which also implies dominance of collective effects over collisions between particles. Before we returnto refine our plasma classification therefore, it is worth having a quick look at the nature of collisionsbetween plasma particles. Where N D ≤ , screening effects are reduced and collisions will dominate the particle dynamics. Inintermediate regimes, collisionality is usually measured via the electron-ion collision rate , given by: ν ei = π n e Ze ln Λ2 (4 πε ) m e v te s − , (10) ∇ → r ddr ( r dφdr ) v te ≡ (cid:112) k B T e /m e is the electron thermal velocity and ln Λ is a slowly varying term, the Coulomblogarithm, which typically takes on a numerical value O (10 − . The numerical coefficient in expres-sion (10) may vary in textbooks depending on the definition taken. This one is consistent with Refs. [7]and [3], which define the collision rate according to the average time taken for a thermal electron to bedeflected by o via multiple scatterings from fixed ions. The collision frequency can also be written as ν ei ω p (cid:39) Z ln Λ10 N D ; with ln Λ (cid:39) N D /Z, where ω p is the electron plasma frequency to be defined shortly in Eq. (11). Armed with our definition of plasma ideality (Eq. (8)), we can proceed to make a classification of plasmatypes in density-temperature space. This is illustrated for a few examples in Fig. 2: the ‘accelerator’plasmas of interest to the present school are found right in the middle of this chart, having densitiescorresponding to roughly atmospheric pressure and temperatures of a few eV ( Kelvin) as a result offield ionization – see Section 1.5.
Fig. 2:
Examples of plasma types in the density-temperature plane
So far we have considered characteristics, density and temperature, of a plasma in equilibrium. We canalso ask how fast the plasma responds to some external disturbance, which can be due to electromagneticwaves (eg laser pulse), or particle beams. Consider a quasi-neutral plasma slab in which an electron layeris displaced from its initial position by a distance δ – Fig. 3. This creates two ’capacitor’ plates withsurface charge σ = ± en e δ , resulting in an electric field: E = σε = en e δε . The electron layer is accelerated back towards the slab by this restoring force according to: m e dvdt = − m e d δdt = − eE = e n e δε . ig. 3: Slab or capacitor model of oscillating electron layer.
Or: d δdt + ω p δ = 0 , where ω p ≡ (cid:32) e n e ε m e (cid:33) / (cid:39) . × (cid:18) n e cm − (cid:19) / s − . (11)is the electron plasma frequency .This quantity can be obtained via another route by returning to the Debye sheath problem ofSection 1.1 and asking how quickly it takes the plasma to adjust to the insertion of the foreign charge.For a plasma with temperature T e , the reponse time to recover quasi-neutrality is just the ratio of Fig. 4:
Response time to form a Debye sheath the Debye length to the thermal velocity v te ≡ (cid:112) k B T e /m e : t D (cid:39) λ D v te = (cid:18) ε k B T e e n e · mk B T e (cid:19) / = ω − p . If the plasma response time is shorter than the period of a external electromagnetic field (such asa laser), then this radiation will be shielded out . To make this more quantitative, consider ratio: ω p ω = e n e ε m e · λ π c . Setting this to unity defines the wavelength for which n e = n c , or n c (cid:39) λ − µ cm − , (12)5bove which radiation with wavelengths λ > λ µ will be reflected. In the pre-satellite/cable era of the 20thcentury, this property was exploited to good effect in the transmission of long-wave radio signals, whichutilises reflection from ionosphere to extend its reception range. Fig. 5:
Left: overdense plasma, ω < ω p , showing mirror-like behaviour; right: underdense case, ω > ω p , whereplasma behaves like a nonlinear refractive medium. In both cases the incoming laser pulse is focussed from the leftonto the target. Typical gas jets have P ∼ bar; n e = 10 − cm − and from Eq. (5), the critical densityfor a glass laser is n c (1 µ ) = 10 cm − . Gas-jet plasmas are therefore underdense , since ω /ω p = n e /n c (cid:28) . In this context, collective effects are important if ω p τ int > , where τ int is some characteristicinteraction time - for example the duration of a laser pulse or particle beam entering the plasma. Forexample, if τ int = 100 fs, and n e = 10 cm − then we have → ω p τ int = 1 . , and we will need toconsider the plasma response on the interaction timescale. Generally this is the situation we seek toexploit in all kinds of plasma applications: short-wavelength radiation; nonlinear refractive properties;generating high electric or magnetic fields; and of course, for particle acceleration. Plasmas are created via ionization, which can occur in a number of ways: through collisions of fastparticles with atoms; photoionization via electromagnetic radiation, or via electrical breakdown in strongelectric fields. The latter two are examples of field ionization , which is the mechanism most relevant tothe plasma accelerator context. To get some idea of when this occurs, we need to know the typical fieldstrength required to strip electrons away from an atom. At the Bohr radius a B = 4 πε (cid:126) me = 5 . × − cm , the electric field strength is: E a = e πε a B (cid:39) . × Vm − . (13)This threshold can be expressed as the so-called atomic intensity : I a = ε cE a (cid:39) . × Wcm − . (14)A laser intensity of I L > I a will therefore guarantee ionization for any target material, though in fact thiscan occur well below this threshold value (eg: ∼ Wcm − for hydrogen) via multiphoton effects.Simultaneous field ionization of many atoms produces a plasma with electron density n e , temperature T e ∼ − eV. 6 .6 Relativistic threshold Before we tackle the topic of wave propagation in plasmas, it is useful to have some idea of the strengthof the external fields used to excite them. To do this we resort to the classical equation of motion foran electron exposed to a linearly polarized laser field E = ˆ yE sin ωt : dvdt (cid:39) − eE m e sin ωt, which implies that the electron will acquire a velocity → v = eE m e ω cos ωt = v os cos ωt. (15)This is usually expressed in terms of a dimensionless oscillation amplitude: a ≡ v os c ≡ p os m e c ≡ eE m e ωc . (16) In articles and books this is often referred to as the ‘quiver’ velocity or momentum and can exceed unity.In this case, normalised momentum (3rd term) is more appropriate, since the real particle velocity is justpinned to the speed of light. The laser intensity I L and wavelength λ L are related to E and ω by: I L = 12 ε cE ; λ L = 2 πcω . Substituting these into Eq. (16) one obtains: I L = 2 π ε m c e a λ L (cid:39) . × a λ µ Wcm − (17)Conversely, a (cid:39) . I λ µ ) / , (18)where I = I L Wcm − ; λ µ = λ L µm . From this expression we see that we will already have relativistic electron velocities, or v os ∼ c , forintensities I L ≥ Wcm − , at wavelengths λ L (cid:39) µ m. Comparing this to thermal velocities v te /c = (cid:113) k B T e /m e c = 0 . for T e = 50 eV, we see that at relativistic laser intensities, the laser field willnormally completely dominate the electron motion over the ambient plasma temperature. Electron dynamics in electromagnetic waves
As we have already seen, real plasma dynamics involves a collective response of the constituent particlesto the influence of external fields. However, it is still helpful to first examine how single electrons respondto electromagnetic laser fields before tackling the more complex ‘many particle’ problem.
The simplest model of a laser field starts with a plane-wave geometry as in Fig. 6, with the transverseelectromagnetic fields E L = (0 , E y , E z ); B L = (0 , B y , B z ) . These wave can be described equivalentlyby a general, elliptically polarized vector potential A ( ω, k ) travelling in the positive x -direction : A = A (0 , δ cos φ, (1 − δ ) sin φ ) , (19)7 ig. 6: Geometry for treating motion in one-dimensional plane wave where φ = ωt − kx is the phase of the wave; A its amplitude ( v os /c = eA /mc ) and δ the polarizationparameter. For linear polarization (LP), δ = ± , , we have A = ± ˆ y A cos φ ; A = ˆ z A sin φ, whereas for circularly polarized (CP) light, δ = ± √ , the vector potential becomes: A = A √ ± ˆ y cos φ + ˆ z sin φ ) . The electron momentum in electromagnetic wave with fields E and B given by Lorentz equation: d p dt = − e ( E + v × B ) , (20)with p = γm v , and relativistic factor γ = (1 + p /m c ) . This has an associated energy equation,after taking dot product of v with Eq. (20): ddt (cid:16) γmc (cid:17) = − e ( v · E ) . (21)The solution to this problem is treated in many texts [4, 5, 8], so we give just a simple recipe here:1. Compute laser fields from vector potential with given polarization: E = − ∂ t A , B = ∇ × A
2. Use dimensionless variables such that ω = k = c = e = m = 1 (eg: p → p /mc, E → e E /mωc , etc.)3. Evaluate first integrals to yield conservation relations: p ⊥ = A , γ − p x = α , where γ − p x − p ⊥ =1 ; α = const.4. Change of variable to wave phase φ = t − x
5. Solve for p ( φ ) and r ( φ ) .In the normal laboratory frame, the electron is initially at rest before the EM wave arrives, so thatat t = 0 , p x = p y = 0 and γ = α = 1 . Then we can write p x = a (cid:104) δ −
1) cos 2 φ (cid:105) ,p y = δa cos φ, (22) p z = (1 − δ ) / a sin φ. These can be integrated again to get the particle trajectories: x = 14 a (cid:34) φ + 2 δ −
12 sin 2 φ (cid:35) , ig. 7: Electron motion in LP plane wave ( δ = 1 , left); CP plane wave ( δ = ± / √ , right) y = δa sin φ, (23) z = − (1 − δ ) / a cos φ. Note that the solution is self-similar in the variables ( x/a , y/a , z/a ) . The trajectories are illustratedgraphically in Fig. 7. In both cases the electron drifts with average momentum p D ≡ p x = a , orvelocity v D c = v x = p x γ = a a . In a CP wave the oscillating p x component at φ vanishes, but drift p D remains. The orbit is a helix withradius kr ⊥ = a / √ , momentum p ⊥ /mc = a / √ and pitch angle θ p = p ⊥ /p D = √ a − . A laser pulse normally has a finite duration, but we can still apply some of the above solutions substitutinga temporal envelope A ( x, t ) = f ( φ ) a cos φ, for the constant amplitude assumed before. In generalthough, it is straightforward (and more useful) to integrate the momentum equations numerically, asshown below in Fig. 8. Note that in both cases there is no net energy gain, in agreement with the Lawson-Woodward theorem. In the CP case the oscillations in p x are suppressed, but the drift is still there.Moreover, the v × B oscillations also nearly vanish, but the ’DC’ part, a manifestation of the longitudinalponderomotive force , is retained. The simplest way to break the symmetry of the plane wave solutions illustrated above is to introducea finite transverse dimension into the wave. This is the normal case for a short-pulse laser, and althoughwe can no longer find exact solutions, we can still make a number of useful deductions about the electronmotion. Consider first a single electron oscillating slightly off-centre of focused laser beam. After 1stquarter-cycle, the electron sees a lower field, and doesn’t quite return to its initial position. Therefore,it is gradually accelerated away from the laser axis. Mathematically, this process can be analyzed witha simple perturbative calculation. In the limit v/c (cid:28) , the equation of motion Eq. (46) for the electronbecomes: ∂v y ∂t = − em E y ( r ) . (24)9 ig. 8: Electron motion in laser pulse with finite number of cycles: LP pulse (left), CP pulse( right). The fourinsets show respectively (clockwise from top left): transverse momentum, longitudinal momentum, v × B forceand longitudinal displacement. Taylor expanding electric field about the current electron position: E y ( r ) (cid:39) E ( y ) cos φ + y ∂E ( y ) ∂y cos φ + ..., where φ = ωt − kx as before.To lowest order, we therefore have v (1) y = − v os sin φ ; y (1) = v os ω cos φ, where v os = eE L /mω . Substituting back into Eq. (24) gives: ∂v (2) y ∂t = − e m ω E ∂E ( y ) ∂y cos φ. Multiplying by m and taking the laser cycle-average, f = (cid:90) π f dφ, p f E (r) y e−
I(r) lasery x
Fig. 9:
Schematic view of the radial ponderomotive force due to a focused beam. transverse ponderomotive force on the electron: f py ≡ m ∂v (2) y ∂t = − e mω ∂E ∂y . (25)A generalized relativistic version of Eq. (25) can be found by rewriting the Lorentz Eq. (20) in terms ofthe vector potential A : ∂ p ∂t + ( v. ∇ ) p = ec ∂ A ∂t − ec v × ∇ × A . (26)Make use of identity: v × ( ∇ × p ) = 1 mγ p × ∇ × p = 12 mγ ∇ | p | − mγ ( p. ∇ ) p , separate the timescales of the electron motion into slow and fast components p = p s + p f and averageover a laser cycle, get p f = A and f p = d p s dt = − mc ∇ γ, (27)where γ = (cid:16) p s m c + a y (cid:17) / , a y = eA y /mc . Wave propagation in plasmas
The theory of wave propagation is an important subject in its own right, and has inspired a vast body ofliterature and a number of textbooks [6, 7, 10]. There are a great many possible ways in which plasmacan support waves, depending on the local conditions, presence of external electric and magnetic fields,and so on. Here we will concentrate on two main wave forms: longitudinal oscillations of the kind metalready, and electromagnetic waves. To derive and analyse wave phenomena, there are also a number ofpossible theoretical approaches depending on the length and time scales of interest, which in laboratoryplasmas can range from nanometres to metres, and femtoseconds to seconds, respectively:1. First principles N-body molecular dynamics2. Phase-space methods – Vlasov-Boltzmann equation3. 2-fluid equations4. Magnetohydrodynamics (single, magnetised fluid).The first of these approaches is rather costly and limited to much smaller regions of plasma than usuallyneeded to describe most types of wave which supported by plasmas. Indeed, the number of particlesneeded for first-principles modelling of a tokamak would be around ; a laser-heated gas requires – still way out of reach of even the most powerful computers available. Clearly a more tractablemodel is needed and in fact, many plasma phenomena can be analysed by assuming that each chargedparticle component with density n s and velocity u s behaves in a fluid-like manner, interacting with otherspecies ( s ) via the electric and magnetic fields (method 3). The rigorous way to derive the governingequations in this approximation is via kinetic theory , starting from method 2 [2, 7], which is beyondthe scope of this lecture. Finally, slow wave phenomena on more macroscopic, ion timescales can behandled with the 4th approach above [2].For the present purposes we therefore begin with the 2-fluid equations for a plasma with a finitetemperature ( T e > ), and assumed to be collisionless ( ν ie (cid:39) ) and non-relativistic, so that the fluidvelocities u (cid:28) c . The equations governing the plasma dynamics under these conditions are: ∂n s ∂t + ∇ · ( n s u s ) = 0 (28)11 s m s d u s dt = n s q s ( E + u s × B ) − ∇ P s (29) ddt ( P s n − γ s s ) = 0 , (30)where P s is the thermal pressure of species s ; γ s the specific heat ratio, or (2 + N ) /N ) , where N isthe number of degrees of freedom.The continuity equation (Eq. (28)) tells us that (in the absence of ionization or recombination) the numberof particles of each species is conserved. Noting that the charge and current densities can be written ρ s = q s n s and J s = q s n s u s respectively, Eq. (28) can also be written: ∂ρ s ∂t + ∇ · J s = 0 , (31)which expresses the conservation of charge .Equation (29) governs the motion of a fluid element of species s in the presence of electric andmagnetic fields E and B . In the absence of fields, and assuming strict quasineutrality ( n e = Zn i = n ; u e = u i = u ), we recover the more familiar Navier-Stokes equations: ∂ρ∂t + ∇ · ( ρ u ) = 0 ,∂ u ∂t + ( u · ∇ ) u = 1 ρ ∇ P. (32)By contrast, in the plasma accelerator context we will usually deal timescales on which the ions can beassumed to be motionless u i = 0 , and with un magnetised plasmas, so that the momentum equation thenreads: n e m e d u e dt = − en e E − ∇ P e (33)Note that E can include both external and internal field components (via charge-separation). A characteristic property of plasmas is their ability to transfer momentum and energy via collectivemotion. One of the most important examples of this is the oscillation of the electrons against a stationaryion background, or
Langmuir wave . Returning to the 2-fluid model, we can simplify Eqs. (28)–(30) bysetting u i = 0 , restricting the electron motion to one dimension ( x ) and taking ∂∂y = ∂∂z = 0 : ∂n e ∂t + ∂∂x ( n e u e ) = 0 n e (cid:18) ∂u e ∂t + u e ∂u e ∂x (cid:19) = − em n e E − m ∂P e ∂x (34) ddt (cid:18) P e n γ e e (cid:19) = 0 The above system (34) has 3 equations and 4 unknowns. To close it we need an expression for the electricfield, which, since B = 0 , can be found from Gauss’ law (Poisson’s equation) with Zn i = n : ∂E∂x = eε ( n − n e ) (35)12he system of equations (34) and (35) is nonlinear, and apart from a few special cases, cannot be solvedexactly. A common technique for analyzing waves in plasmas therefore is to linearize the equations,assuming the perturbed amplitudes are small compared to the equilibrium values: n e = n + n ,u e = u ,P e = P + P ,E = E , where n (cid:28) n , P (cid:28) P . These expressions are substituted into (34,35) and all products n ∂ t u , u ∂ x u etc. are neglected to get a set of linear equations for the perturbed quantities: ∂n ∂t + n ∂u ∂x = 0 ,n ∂u ∂t = − em n E − m ∂P ∂x , (36) ∂E ∂x = − eε n ,P = 3 k B T e n . The expression for P results from specific heat ratio γ e = 3 and assuming isothermal backgroundelectrons, P = k B T e n (ideal gas) – see Kruer (1988). We can now eliminate E , P and u fromEq. (36) to get: (cid:32) ∂ ∂t − v te ∂ ∂x + ω p (cid:33) n = 0 , (37)with v te = k B T e /m e and ω p given by Eq. (11) as before. Finally, we look for plane wave solutions ofthe form A = A e i ( ωt − kx ) , so that our derivative operators become: ∂∂t → iω ; ∂∂x → − ik . Substitutioninto Eq. (37) yields the Bohm-Gross dispersion relation: ω = ω p + 3 k v te . (38)This and other dispersion relations are often depicted graphically on a chart such as that in Fig. 10, whichgives an overview of which propagation modes are permitted for low- and high-wavelength limits. Fig. 10:
Schematic illustration of dispersion relations for Langmuir, electromagnetic and ion-acoustic waves. .2 Transverse waves To describe transverse electromagnetic (EM) waves, we need two additional Maxwells equations; Fara-day’s law (48) and Ampère’s law (49), which we will introduce properly shortly (see Eqs. ( 48) and(49)). For the time-being it is helpful to simplify things by making use of our previous analysis of small-amplitude, longitudinal waves. Therefore, we linearize and again apply the harmonic approximation ∂∂t → iω : ∇ × E = − iω B , (39) ∇ × B = µ J + iε µ ω E , (40)where the transverse current density is given by: J = − n e u . (41)This time we look for pure EM plane-wave solutions with E ⊥ k – see Fig. 6, and also note that thegroup and phase velocities are assumed to be large enough v p , v g (cid:29) v te , so that we can assume a cold plasma with P e = n k B T e (cid:39) . The linearized electron fluid velocity and corresponding current arethen: u = − eiωm e E , J = n e iωm e E ≡ σ E , (42)where σ is the AC electrical conductivity . By analogy with dielectric media (see e.g., Ref. [9]), in whichAmpere’s law is usually written ∇ × B = µ ∂ t D , by substituting Eq. (42) into Eq. (49), can showthat: D = ε ε E , with ε = 1 + σiωε = 1 − ω p ω . (43)From Eq. (43) it follows immediately that: η ≡ √ ε = ckω = (cid:32) − ω p ω (cid:33) / , (44)with ω = ω p + c k (45)The above expression can also be found directly by elimination of J and B from Eqs. (39)–(42). Fromthe dispersion relation Eq. (45), also depicted in Fig. 10, a number of important features of EM wavepropagation in plasmas can be deduced. For underdense plasmas ( n e (cid:28) n c ):Phase velocity: v p = ωk (cid:39) c (cid:32) ω p ω (cid:33) > c Group velocity: v g = ∂ω∂k (cid:39) c (cid:32) − ω p ω (cid:33) < c In the opposite case, n e > n c , the refractive index η becomes imaginary, and the wave can no longerpropagate, becoming evanescent instead, with a decay length determined by the collisionless skin depth c/ω p – Fig. 11 14 F i e l d i n t en s i t y -4 -3 -2 -1 0 1 2 kx B y2 E z2 E (0) = 2E n c /n n /n c c/ p Fig. 11:
Electromagnetic fields resulting from reflection of an incoming wave by an overdense plasma slab.
So far we have considered purely longitudinal or transverse waves: linearising the wave equations en-sures that any nonlinearities or coupling between these two modes is excluded; a reasonable approxima-tion for low amplitude waves, but inadequate to describe strongly driven waves in the relativistic regimeof interest for plasma accelerator schemes. The starting point for most analyses of nonlinear wave prop-agation phenomena is the Lorentz equation of motion for the electrons in a cold ( T e = 0 ), unmagnetizedplasma, together with Maxwell’s equations [7, 8]. We also make two assumptions i) that the ions areinitially assumed to be singly charged ( Z = 1 ) and are treated as a immobile ( v i = 0 ), homogeneousbackground with n = Zn i , and ii) that thermal motion can be neglected, since the temperature remainssmall compared to the typical oscillation energy in the laser field ( v os (cid:29) v te ). The starting equations (SIunits) are then as follows: ∂ p ∂t + ( v · ∇ ) p = − e ( E + v × B ) , (46) ∇ · E = eε ( n − n e ) , (47) ∇ × E = − ∂ B ∂t , (48) c ∇ × B = − eε n e v + ∂ E ∂t , (49) ∇ · B = 0 , (50)where p = γm e v and γ = (1+ p /m e c ) / . To simplify matters we first assume a plane-wave geometrylike that in Fig. 6, with the transverse electromagnetic fields given by E L = (0 , E y , B L = (0 , , B z ) .From Eq. (46) the transverse electron momentum is then simply given by: p y = eA y , (51)where E y = ∂A y /∂t . This relation expresses conservation of canonical momentum. Substituting E = −∇ φ − ∂ A /∂t ; B = ∇ × A into Ampère Eq. (49) yields: c ∇ × ( ∇ × A ) + ∂ A ∂t = J ε − ∇ ∂φ∂t , where the current J = − en e v . Now we use a bit of vectorial wizardry, splitting the current into rotational(solenoidal) and irrotational (longitudinal) parts: J = J ⊥ + J || = ∇ × Π + ∇ Ψ J || − c ∇ ∂φ∂t = 0 . Applying the Coulomb gauge ∇ · A = 0 and v y = eA y /γ from (51), to finally get: ∂ A y ∂t − c ∇ A y = µ J y = − e n e ε m e γ A y . (52)The nonlinear source term on the RHS contains two important bits of physics: n e = n + δn , whichcouples the EM wave to plasma waves; γ = (cid:113) p /m e c which accounts for the relativisticallyenhanced electron inertia. The above wave equation thus already describes a host of effects which high-intensity laser pulses will be subjected to in a plasma:– diffraction due to finite focal spot σ L : Z R = 2 πσ L /λ – ionization effects dn e /dt ⇒ refraction due to radial density gradients– relativistic self-focusing and self-modulation ⇒ η ( r ) = (cid:115)(cid:18) − ω p ( r ) γ ( r ) ω (cid:19) – ponderomotive channelling ⇒ ∇ r n e – scattering by plasma waves ⇒ k → k + k p .All of these nonlinear effects are important for laser powers P L > T W . One of the most important ofthese is relativistic self-focussing, for which there is a power threshold [11–13]. The laser power can bewritten: P L = (cid:16) mωce (cid:17) (cid:18) cω p (cid:19) c(cid:15) (cid:90) ∞ πra ( r ) dr = 12 (cid:16) me (cid:17) c (cid:15) (cid:18) ωω p (cid:19) ˜ P , (cid:39) . (cid:18) ωω p (cid:19) ˜ P GW , where ˜ P ≡ πa ( ω p σ L /c ) . The normalized critical power often quoted in early texts ˜ P c = 16 π thus corresponds to: P c (cid:39) . (cid:18) ωω p (cid:19) GW . (53)For example, a laser with wavelength λ L = 0 . µm propagating in an electron gas with n e = 1 . × cm − sees a normalized density of n e n c = (cid:0) ω p ω (cid:1) = 0 . and will start to focus at the threshold power ⇒ P c = 0 . TW. An example of this behaviour for a long pulse in an underdense plasma is shown inFig. 12.Taking the longitudinal component of the momentum Eq. (46) gives: ddt ( γm e v x ) = − eE x − e m e γ ∂A y ∂x . We can eliminate v x using Ampère’s law (49) x : − eε n e v x + ∂E x ∂t , Fig. 12:
Laser propagation in underdense plasma for P L /P c (cid:28) (left) and P L = 2 P c (right) while the electron density can be determined via Poisson’s Eq. (47): n e = n − ε e ∂E x ∂x . The above (closed) set of equations can in principle be solved numerically for arbitrary pumpstrengths. For the moment, we simplify things by linearizing the plasma fluid quantities: n e (cid:39) n + n + ...v x (cid:39) v + v + ... and neglect products like n v etc. This finally leads to: (cid:32) ∂ ∂t + ω p γ (cid:33) E x = − ω p e m e γ ∂∂x A y . (54)The driving term on the RHS is the relativistic ponderomotive force , with γ = (1 + a / / . Somesolutions of Eq. (54) are shown in Fig. 13 below, for low- and high-intensity laser pulses respectively. Theproperties of these wakes will be discussed in detail in subsequent lectures, but we can already see someobvious qualitative differences in the linear and nonlinear waveforms; the latter typically characterised bya spiked density profile, saw-tooth electric field and longer wavelength. The coupled fluid Eqs. (52) and(54) and their fully non-linear counterparts describe a vast range of nonlinear laser-plasma interactionphenomena, many of which are treated in the later lectures of the school: plasma wake generation; blow-out regime laser self-focussing and channelling; parametric instabilities; harmonic generation, and so on.Plasma-accelerated particle beams , on the other hand, cannot be treated with fluid theory and demanda more sophisticated kinetic approach, usually requiring the assistance of numerical models solved withthe help of powerful supercomputers. 17 k p -0.020.00.020.040.06 w a k e fi e l d Enpump k p -10123 w a k e fi e l d pumpEn Fig. 13:
Wakefield excitation by a short-pulse laser propagating in the positive x-direction in the linear regime(left) and nonlinear regime (right).
References [1] F. F. Chen,
Plasma Physics and Controlled Fusion , 2nd Ed. (Springer, 2006)[2] R. O. Dendy (ed.),
Plasma Physics, An Introductory Course , (Cambridge University Press, 1993)[3] J. D. Huba,
NRL Plasma Formulary , (NRL, Washington DC, 2007) [4] J. Bardsley et al.,
Relativistic dynamics of electrons in intense laser fields , Phys. Rev. A , 3823(1989)[5] F. V. Hartemann et al., Nonlinear ponderomotive scattering of relativistic electrons by an intenselaser field at focus , Phys. Rev. E , 4833â ˘A ¸S4843 (1995).[6] J. Boyd and J. J. Sanderson, The Physics of Plasmas [7] W. Kruer,
The Physics of Laser Plasma Interactions , Addison-Wesley, 1988[8] P. Gibbon,
Short Pulse Laser Interactions with Matter: An Introduction , IC Press, London, 2005[9] J. D. Jackson,
Classical Electrodynamics , Wiley 1975/1998[10] J. P. Dougherty in Chapter 3 of R. Dendy
Plasma Physics , 1993[11] A. G. Litvak,
Finite-amplitude wave beams in a magnetoactive plasma , Sov. Phys. JETP ,344â ˘A ¸S347 (1970).[12] C.E. Max, J. Arons, A. B. Langdon, Self-Modulation and Self-Focusing of Electromagnetic Wavesin Plasmas , Phys. Rev. Lett. , 209â ˘A ¸S212 (1974).[13] P. Sprangle, E. Esarey, E., A. Ting, Nonlinear theory of intense laser-plasma interactions , Phys.Rev. Lett. 64, 2011â ˘A ¸S2014 (1990). 18
Useful constants and formulae
Table 2:
Commonly used physical constants
Name Symbol Value (SI) Value (cgs)
Boltzmann constant k B . × − JK − . × − erg K − Electron charge e . × − C . × − statcoulElectron mass m e . × − kg . × − gProton mass m p . × − kg . × − gPlanck constant h . × − Js . × − erg-sSpeed of light c × ms − × cms − Dielectric constant ε . × − Fm − —Permeability constant µ π × − —Proton/electron mass ratio m p /m e e/k B N A . × mol − . × mol − Table 3:
Formulae in SI and cgs units
Name Symbol Formula (SI) Formula (cgs)