Wave Propagation 2010 Part 12 doc

Linear dispersion relation frequencyω versus wavenumber k for electromagnetic waves in plasma.. The R-mode and L-modelow-frequency waves are called electromagnetic electron and ion cyclo

12 Electromagnetic Waves

For simplicity, let us assume propagation of plasma waves in the direction parallel to the

ambient magnetic field, i.e., k⊥=0 Then, we have I0[0] =1 and I n [0] =0 These also gives

I±1 [0] =0.5 Thus we obtain K1,1=K2,2, K1,3=03, K2,3=0 and Eq.(26) becomes

The first factor is for transverse waves wherek⊥ E That is, a wave propagates in the z

direction while its electromagnetic fields polarize in the x−y plane The second factor is

for longitudinal waves wherek|| E That is, a wave propagates in the z direction and only

its electric fields polarize in the z direction The longitudinal waves are also referred to as

compressional waves or sound waves Especially in the case ofk || E, waves are called

“electrostatic” waves because these waves arise from electric charge and are expressed bythe Poisson equation (3)

3.1 Transverse electromagnetic waves

The first factor of Eq.(27) becomes the following equation,

Here the argument of the dispersion function x is a complex value.

Let us consider that a phase speed of waves is much faster than velocities of plasma particles.Then, the argument of the plasma dispersion function becomes a larger number Here, the

drift velocity of plasma V dis also neglected Equation (28) is thus rewritten by using Eq.(29)as



=0 if n 0.

Electromagnetic Waves in Plasma 13

(a) Linear scale (b) Logarithmic scale.

Fig 1 Linear dispersion relation (frequencyω versus wavenumber k) for electromagnetic

waves in plasma The quantitiesω and ck are normalized by Πpe.

The solutions to above equation are simplified when we assume that the temperature of

plasma approaches to zero, i.e., V t||→0 and V t⊥→0 Note that this approach is known as the

“cold plasma approximation.” Equation (30) is rewritten by the cold plasma approximationas

The dispersion curves for the high-frequency R-mode and L-mode waves approach to the

following frequencies as k||→0,

323Electromagnetic Waves in Plasma

On the other hand, the low-frequency wave approaches to k||V A as k||→0, and approaches to

Ωc as k||→∞ Note that V A≡cΩ ci/Πpiis called the Alfven velocity The R-mode and L-modelow-frequency waves are called electromagnetic electron and ion cyclotron wave, respectively,

or (electron and ion) whistler mode wave

The temperature of plasma affects the growth/damping rate in the dispersion relation.Assumingω |γ|(where ˜ω≡ω+iγ), the imaginary part of Eq.(30) gives the growth rate γ

Ωce <(ω−Π2Ωcepe )2 (35)Here ion terms are neglected by assuming|ω| Ωciand|ω| Πpi This condition is achieved

when V te⊥>V te||, which is known as electron temperature anisotropy instability

As a special case, electromagnetic electron cyclotron waves are also excited if the ioncontribution (the third line in Eq.(34)) becomes larger than the electron contribution (thesecond line in Eq.(34)) around|ω| ∼Ωci The growth rare becomes positive when

2

pi

(ω−Ωci)2 (36)

Electromagnetic Waves in Plasma 15

(a) Electron temperature anisotropy instability

Here|ω Ωce|and|ω Πpeare assumed This condition is achieved when V ti|| V ti⊥,which is known as firehose instability

For electromagnetic ion cyclotron waves (0<ω<Ωci), the growth rare becomes positivewhen

Ωce > ΩciΠ

2

pi

(ω−Ωci)2 (37)Hereω Ωce|andω Πpe are used This condition is achieved when V ti⊥>V ti||, which isknown as ion temperature anisotropy instability

Examples of these electromagnetic linear instabilities are shown in Figure 2, which areobtained by numerically solving Eq.(28)

325Electromagnetic Waves in Plasma

16 Electromagnetic Waves

3.2 Longitudinal electrostatic waves

The second factor of Eq.(27) becomes

This means that the damping of the Langmuir waves becomes largest at k||∼Πpe/Vte||, which

is known as the Landau damping Note that the second line in Eq.(39) comes from the gradient

in the velocity distribution function, i.e.,∂ f0/∂v|| Thus electrostatic waves are known to bemost unstable where the velocity distribution function has the maximum positive gradient

As an example for the growth of electrostatic waves, let us assume a two-species plasma, andone species drift against the other species at rest Then, Eq.(39) becomes

Electromagnetic Waves in Plasma 17

(a) Electron beam-plasma instability with V te ≡

V t1||=V t2||, V d1=4T te, Πp2=9Πp1.

(b) Electron beam-plasma instability with V te ≡

V t1||=V t2||, V d1=4T te, Πp1=9Πp2.Fig 3 Linear dispersion relation for electrostatic instabilities The frequency is normalized

comes from derivative of the velocity distribution function with respect to velocity with v=

ω/k|| Electrostatic waves are excited when the velocity distribution function has positivegradient This condition is also called the Landau resonance Since the positive gradient inthe velocity distribution function is due to drifting plasma (or beam), the instability is known

as the beam-plasma instability

Examples of the beam-plasma instability are shown in Figure 3, which are obtained bynumerically solving Eq.(38)

327Electromagnetic Waves in Plasma

18 Electromagnetic Waves

3.3 Cyclotron resonance

Since the Newton-Lorentz equation (6) and the Vlasov equation (7) cannot treat the relativism

(such that c V d), plasma particles cannot interact with electromagnetic light mode waves

On the other hand, drifting plasma can interact with electromagnetic cyclotron waves when

a velocity of particles is faster than the Alfven speed V A but is slow enough such thatelectrostatic instabilities do not take place

For isotropic but drifting plasma, Eq.(27) is rewritten as

Electromagnetic Waves in Plasma 19

(a) Ion beam-cyclotron instability with V di=

Here |ω Ωce| and|ω Πpeare assumed The maximum growth rate is obtained at

ω∼k (V di+V A)

These conditions are called the cyclotron resonance Note that electron cyclotron waves areexcited by drifting ions while ion cyclotron waves are excited by drifting electrons Theseinstabilities are known as the beam-cyclotron instability

Examples of the beam-cyclotron instability are shown in Figure 4, which are obtained bynumerically solving Eq.(28)

to an ambient magnetic field is not equal to the parallel temperature Electrostatic wavesare excited when a velocity distribution function in the direction parallel to an ambientmagnetic field has positive gradient Note that the former condition is called the temperatureanisotropy instability The latter condition is achieved when a high-speed charged-particlebeam propagates along the ambient magnetic field, and is called the beam-plasma instability.Charged-particle beams can also interact with electromagnetic cyclotron waves, which iscalled the beam-cyclotron instability These linear instabilities take place by free energysources existing in velocity space

It is noted that plasma is highly nonlinear media, and the linear dispersion relation can beapplied for small-amplitude plasma waves only Large-amplitude plasma waves sometimesresult in nonlinear processes, which are so complex that it is difficult to provide their analyticalexpressions Therefore computer simulations play essential roles in studies of nonlinearprocesses One can refer to textbooks on kinetic plasma simulations (e.g., Birdsall & Langdon,2004; Hockney & Eastwood, 1988; Omura & Matsumoto, 1994; Buneman, 1994) for furtherreading

329Electromagnetic Waves in Plasma

20 Electromagnetic Waves

5 References

Birdsall, C K., & Langdon, A B (2004), Plasma Physics via Computer Simulation, 479pp.,

Institute of Physics, ISBN 9780750310253, Bristol

Buneman, O (1994), TRISTAN, In: Computer Space Plasma Physics: Simulation Techniques and

Software, Matsumoto, H & Omura,Y (Ed.s) pp.67–84, Terra Scientific Publishing

Company, ISBN 9784887041110, Tokyo

Fried, B D & Conte, S D (1961), The Plasma Dispersion Function, 419pp., Academic Press, New

York

Hockney, R W & Eastwood, J W (1988) Computer Simulation Using Particles, 540pp., Institute

of Physics, ISBN 9780852743928, Bristol

Omura, Y & Matsumoto, H (1994), KEMPO1, In: Computer Space Plasma Physics: Simulation

Techniques and Software, Matsumoto, H & Omura,Y (Ed.s) pp.21–65, Terra Scientific

Publishing Company, ISBN 9784887041110, Tokyo

Stix, T H (1992), Waves in Plasmas, 584pp., Springer-Verlag, ISBN 9780883188590, New York Swanson, D G (2003), Plasma Waves, Second Edition, 400pp., Institute of Physics, ISBN

9780750309271, Bristol

Swanson, D G (2008), Plasma Kinetic Theory, 344pp., Crc Press, ISBN 9781420075809, New

York

16

Propagation of Electromagnetic Waves in and around Plasmas

of the electromagnetic wave frequency, the electron plasma frequency, and the electron elastic collision frequency These three parameters determine real and imaginary parts of permittivity In other words, plasmas equivalently behave as conductors or dielectric materials for electromagnetic waves, and these behaviours are controllable by changing complex permittivity, or electron density and gas pressure, which is associated with the electron plasma frequency and the electron elastic collision frequency; this controllability and the time-varying manner for permittivity distinguish plasmas from other electromagnetic media

First of all, in this section, we briefly review the historical perspective of the electromagnetic waves in plasmas, and we point out the reasons why the concept of electromagnetic media composed of plasmas and their discontinuities is focused on in this chapter

Electromagnetic waves in magnetized plasmas have been well investigated for more than half century, aiming at ultimate goals of controlled fusion plasmas for energy production and space plasmas for launching human beings using spacecrafts In a magnetized plasma, various kinds of wave branches are present from low to high frequency ranges, and change

of the external magnetic field induces “walk” on the dispersion curves in a “zoo” of plasma waves Sometimes a branch leads to another totally-different branch; that is called mode conversion (Stix, 1962; Swanson, 1989) These waves can be launched from the outer side of the plasma, but there are also many inherent waves found as magnetohydrodynamic and micro instabilities (Swanson, 1989; Nishikawa & Wakatani, 1990) Other characteristic features of plasma waves are their nonlinearity; shock waves, solitons, and nonlinear mode conversion originate from the aspects of high-energy-state substance (Swanson 1989) The main focus in this chapter is different from such conventional scientific interests

Before we start our description, one more comment about plasma production for industry should be addressed (Lieberman & Lichtenberg, 1994) Plasma production in fabrication

Wave Propagation

332

processes of thin film technology was quite successful, and several different methods have

been developed for semiconductor industry, flat panel display markets, and photovoltaic

cell production Such a technology using plasmas is now somewhat mature, and several

different ideas and schemes for researches on plasma science and engineering are being

tested for other applicable fields

In such a sense, we study new types of plasma-wave interactions, especially arising from

discontinuities in both space and time Progress of techniques to control shape and

parameters of plasmas enables us to make discontinuities in a clear and stable state

1.2 Emerging aspects of plasma wave propagation

Since wave propagation in a magnetized plasma is well described elsewhere (Stix, 1962;

Ginzburg, 1964; Swanson 1989), we here focus on the propagation in and around a

non-magnetized plasma It is not so complicated to describe electromagnetic waves propagating

in and around a bulk non-magnetized plasma (Kalluri, 1998), although the propagation in a

spatially-nonuniform plasma, in which electron-density gradient is significant (Swanson,

1989; Nickel et al., 1963; Sakai et al., 2009) and/or the profiles of electron density is spatially

periodic (Hojo & Mase, 2004; Sakai et al., 2005(1); Sakai et al., 2005(2); Sakai et al., 2007(1);

Sakai et al., 2007(2); Sakaguchi et al., 2007; Sakai & Tachibana, 2007; Naito et al., 2008; Sakai

et al., 2010(1); Sakai et al., 2010(2)), includes novel physical aspects which have not been

described in usual textbooks of plasma physics

Also, complex dielectric constant or permittivity whose imaginary part is significantly large

is observed and easily controlled in a plasma as a macroscopic value (Naito et al., 2008;

Sakai et al., 2010(1)) This imaginary part strongly depends on field profiles of

electromagnetic waves around plasmas when their spatial discontinuities exist, and so

synthesized effects with Bloch modes in periodic structure lead to not only frequency band

gap but also attenuation gap (Naito et al., 2008; Sakai et al., 2009) In another point of view,

power dissipation due to the imaginary part leads to plasma generation (Lieberman &

Lichtenberg, 1994), which is a quite nonlinear phenomenon

In this chapter, considering the spatial discontinuities and the complex permittivity, the

fundamentals of theoretical understandings on electromagnetic waves in and around plasmas

are described In Section 2.1, properties of complex permittivity are generalized using

equations and a newly-developed 3-dimensional (3D) drawing In Section 2.2, starting from

the momentum equation of electrons to treat plasma effects, the complex permittivity in the

Drude model is derived, and the equation is compared with that for metals whose permittivity

is also in the Drude model; such a description will reveal unique features in the case of

discharge plasmas for control of electromagnetic waves In Section 3.1, various methods to

describe effects of periodic spatial discontinuities are demonstrated, including both analytical

and numerical ones, and specific examples of band diagrams of 2D structures are shown In

Section 3.2, another aspect of the spatial discontinuity associated with surface wave

propagation is described, in which propagation of surface waves on the interface with spatial

electron-density gradient is clarified Section 4 summarizes this chapter, showing emerging

aspects of electromagnetic waves in and around plasmas

2 Fundamentals of new aspects for wave propagation

In this Section, we demonstrate features and importance of complex permittivity which is

usual in a low-temperature partially-ionized plasma Section 2.1 includes general

Propagation of Electromagnetic Waves in and around Plasmas 333

description which is also applicable to other lossy materials (Sakai et al., 2010(1)), and

Section 2.2 focuses on the momentum balance equations of electrons in a discharge plasma,

which is the origin of characteristic wave propagation in and around plasmas

2.1 Complex permittivity in a plasma

To describe wave transmission and absorption as well as phase shift and reflection of the

propagating waves, we here introduce a new drawing of dispersion relation in the 3D space

of three coordinates consisting of wave frequency ω/2π, real wavenumber kr, and

imaginary wavenumber ki A propagating wave which is launched at a spatial position

where A (x) is the wave amplitude with the initial boundary condition of A0 =A(x=0), t

is the time, and φ(t,x) is the phase of the wave with the initial condition of φ(0,0)=0 The

dispersion relation in a collisionless plasma is usually expressed in the ω−kr plane, and we

can also obtain a useful information about wave attenuation from ki as a function of ω

when significant loss or wave attenuation takes place

Such a wave propagation in a bulk non-magnetized plasma is characterized by the

permittivity εp in the Drude model in the form

,)/j1(

1

m 2

2 pe

ωε

νm/2 is the electron elastic collision frequency We note that, since we choose a formula

in equation (2.1.1) instead of exp((jkrx−ωt)), the sign in the bracket of equation (2.1.2)

becomes “-”, and the imaginary part of the permittivity becomes negative in general (Pozar,

2005) The detailed derivation of equation (2.1.2) is given in Section 2.2 Figure 1 shows εp

at a fixed wave frequency (4 GHz) as a function of ne with various gas conditions on the

complex plane (Sakai et al., 2010(1)) Here, we assume that the electron energy is 0.5 eV for a

plasma in the afterglow and that the cross section for the electron elastic collisions is

16

10

0

5 × − cm2 for He and1.0×10−16cm2 for Ar from the literature (Raizer, 1991) At 760 Torr

of He, Re(εp) is almost constant at unity for various ne On the other hand, at 5 Torr of Ar,

)

Im(εp is roughly zero while Re(εp) changes significantly in the negative polarity, and this

feature almost corresponds to a collisionless plasma This figure indicates that the change of

gas species and pressure yields εp with Im(εp)/Re(εp) ranging from 0 to infinity for

1

)

Re(εp < on the complex plane

Equation (2.1.2) gives us an understanding of dispersion relation in the 3D space (ω,kr,ki)

Figure 2 displays a dispersion relation in a bulk non-magnetized plasma expressed by

equation (2.1.2) (Sakai et al., 2010(1)) In the case at 5 Torr of Ar, which is nearly collisionless

as mentioned earlier, the trajectory on the (ω,kr) plane is well known in textbooks of

plasma physics The working point is always on the (ω,kr) plane or on the (ω,ki) plane,

Wave Propagation

334

which can be understood easily from equation (2.1.2) However, in the case at 120 Torr of

He, the working point goes far away from the two planes below ωpe and leaves a trajectory

on the (kr,ki) plane; at such a point, the wave suffers attenuation as well as phase shift, as

suggested in equation (2.1.1)

Drawings of dispersion relations in this 3D space reveal significant physical parameters of

electromagnetic media, as shown in the following Knowledge from microwave engineering

(Pozar, 2005) shows that

,1

ωε

σμε

Fig 1 Permittivity in a lossy bulk plasma with various gas condition and various n (Sakai e

et al., 2010(1))

where ε , μ and σ is the permittivity, the permeability, and the conductivity of the media,

respectively From equation (2.1.3), the following equation is derived:

1

s i

Here δs is the skin depth of the wave into the media krki indicates an area on the (kr,ki)

plane, and so a point projected on the (kr,ki) plane expresses conductivity of the media on

the assumption that μ is constant The inverse of the area on the (kr,ki) plane corresponds

to square of δs; as the area is larger, the skin depth is shorter Another physical parameter

which is visible in this 3D drawing is the metallic/dielectric boundary From equation

(2.1.3), we also obtain

2 2 i 2

r − k =ω με

Comprehension of this equation gives us the following result If k >r ki , ε is positive when

μ is positive, leading to the fact that the media is dielectric, and if k <r ki , vice versa, and

Propagation of Electromagnetic Waves in and around Plasmas 335

we can recognize that the media is metallic The line of k =r ki becomes the boundary

between metallic and dielectric media

(a) (b) Fig 2 Dispersion relation of electromagnetic waves in a bulk plasma with 13

e=1×10

in the 3D space (a) In a plasma at 5 Torr of Ar gas (b) In a plasma at 120 Torr of He gas

(Sakai et al., 2010(1))

These characteristics arising from lossy plasmas are distinguishable from other

electromagnetic media; unlike plasmas, any other material never has a variety of parameter

sets such as complex ε(=εr−jεi) and σ Such a characteristic property can be enhanced by

spatial periodicity; a simple periodic εr distribution realized in a solid material makes a

photonic or electromagnetic band material which exhibits band gaps If we introduce the

effects of εi by a plasma array, new features can emerge with the complex-variable effects

(Naito et al., 2008; Sakai et al., 2010(2)) That is, spatial periodic change in εr leads to

formation of frequency band gaps around which propagation-permitted frequency has a

certain gap As a novel feature in plasma cases, wave attenuation due to εi is significantly

different between two bands above and under this band gap; we call it an attenuation gap

This gap arises from change of field profiles of electromagnetic waves; on the upper band,

fields are localized in the lower-εr area, and vice versa on the lower band; such properties

are demonstrated in Section 3.1.1 and 3.1.4

2.2 Electron momentum balance equation and its effects for wave propagation

When we consider wave propagation in a plasma, the rigorous starting point of description

is electron momentum balance equation Here, a non-magnetized plasma is assumed for

simplicity; if an external magnetic field is present, other terms, for instance arising from the

Lorentz force, might be included The electron momentum balance equation deals with a

plasma as a kind of fluids, and it keeps good matching with macroscopic parameters such as

ε and μ Another method containing plasma effects is calculation of direct particle motion,

such as a particle-in-cell simulation in which momentum balance of each single test particle

is treated and collective effects of charged particles are integrated by the Poisson’s

equation Another comment of our treatment based on the fluid model is that ions are

assumed to be immobile due to its huge relative mass compared with electrons Dispersion

relations of some electrostatic waves propagating in a plasma such as ion acoustic waves