Wave Propagation Part 10 potx

Thus the linear dispersion relation ofelectromagnetic waves in plasma is strongly modified from that in vacuum, which is simply ˜ ω=kc where ˜ ω, k, and c represent angular frequency, wav

307 analysis by a method of eigenfunctions and in approximations of GTD is carried out The spherical diffraction antenna array allows: to control amplitude end phase fields distribution on all aperture of HRA; to provide high efficiency because of active radiating units of feeds do not shade of aperture; to realize a combined amplitude/multibase phase method of direction finding of the objects, polarization selection of signals The HRA’s provide: increasing of range of radars operation by 8-10 %; reduce the error of measurement

of coordinates at 6-8 times; reduction of probability of suppression of radar by active interferences by 20-30 %

On the basis of such antennas use of MMIC technology of fabricate integrated feeds millimeter and centimeter waves is perspective Embedding the micromodules into integral feeding-source antennas for HRA’s and spherical diffraction antenna arrays for processing

of the microwave information can be utilized for long-term evolution multifunctional radars Future work includes a more detailed investigation the antennas for solving a problem of miniaturization of feeds for these antennas by means of MMIC technologies

6 References

Bucci, O.M., Elia, G.D & Romito, G (1996) Synthesis Technique for Scanning and/or

Reconfigurable Beam Reflector Antennas With Phase-only Control IEE

Proc.-Microw Antennas Propag., Vol 143, No 5, October, p.p 402-412

Chantalat, R., Menudier, C., Thevenot, M., Monediere, T., Arnaud, E & Dumon, P (2008)

Enhanced EBG Resonator Antenna as Feed of a Reflector Antenna in the Ka Band

IEEE Antennas and Wireless Propag., Vol 7, p.p 349-353

Elsherbeni, A (1989) High Gain Cylingrical Reflector Antennas with Low Sidelobes AEU,

Band 43, Heft 6, p.p 362-369

Eom, S.Y., Son, S.H., Jung, Y.B., Jeon, S.I., ganin, S.A., Shubov, A.G., Tobolev, A.K &

Shishlov, A.V (2007) Design and Test of a Mobile Antenna System With Tri-Band

Operation for Broadband Satellite Communications And DBS Reception IEEE

Trans on Antennas and Propag., Vol 55, No 11, November, p.p 3123-3133

Fourikis, N (1996) Phased Array-Based Systems and Applications, John Willey & Sons., Inc Gradshteyn, I.S & Ryzhik, I.M (2000) Table of Integrals, Series and Products, 930, 8.533,

Academic Press, New York

Grase, O & Goodman, R (1966) Circumferential waves on solid cylinders J Acoust Soc

America, Vol 39, No 1, p.p.173-174

Haupt, R.L (2008) Calibration of Cylindrical Reflector Antennas With Linear Phased Array

Feeds IEEE Trans on Antennas and Propag., Vol 56, No 2, February, p.p 593-596

Janpugdee, P., Pathak, P & Burkholder, R (2005) A new traveling wave expansion for the

UTD analysis of the collective radiation from large finite planar arrays IEEE

AP-S/URSI Int Symp., Washington, DC, July

Jeffs, B & Warnick, K (2008) Bias Corrected PSD Estimation for an Adaptive Array with

Moving Interference IEEE Trans on Antennas and Propag., Vol 56, No 7, July, p.p

3108-3121

Jung, Y.B & Park, S.O (2008) Ka-Band Shaped Reflector Hybrid Antenna Illuminated by

Microstrip-Fed Horn Array IEEE Trans on Antennas and Propag., Vol 56, No 12,

December, p.p 3863-3867

Jung, Y.B., Shishlov, A & Park, S.O (2009) Cassegrain Antenna With Hybrid Beam Steering

Scheme for Mobile Satellite Communications IEEE Trans on Antennas and Propag.,

Vol 57, No 5, May, p.p 1367-1372

Keller, J (1958) A geometrical theory of diffraction Calculus of variations and its

applications Proc Symposia Appl Math., 8, 27-52 Mc Graw-Hill., N.Y

Llombart, N., Neto, A., Gerini, G., Bonnedal, M & Peter De Maagt (2008) Leaky Wave

Enhanced Feed Arrays for the Improvement of the Edge of Coverage Gain in

Multibeam Reflector Antennas IEEE Trans on Antennas and Propag., Vol 56, No 5,

May, p.p 1280-1291

Love, A (1962) Spherical Reflecting Antennas with Corrected Line Sources IRE Trans on

Antennas and Propagation, Vol AP-10, September, No 5-6, p.p.529-537

Miller, M & Talanov, V (1956) Electromagnetic Surface Waves Guided by a Boundary with

Small Curvature Zh Tekh Fiz, Vol 26, No 12, p.p 2755-2765

Ponomarev, O (2008) Diffraction of Electromagnetic Waves by Concave Circumferential

Surfaces: Application for Hybrid Reflector Antennas Bull of the Russian Academy of

Sciences: Physics, Vol 72, No 12, p.p 1666-1670

Rayleigh, J.W.S (1945) The Theory of Sound, 2-nd ed., Vol 2, Sec 287, Dover Publication,

ISBN 0-486-60293-1, New York

Schell, A (1963) The Diffraction Theory of Large-Aperture Sp-herical Reflector Antennas

IRE Trans on Antennas and Propagation, July, p.p 428-432

Shevchenko, V (1971) Radiation losses in bent waveguides for surface waves Institute of

Radioengineering and Electronics, Academy of Sciences of the USSR Translated from

Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol 14, No 5, p.p 768-777, May

Spencer, R., Sletten, C & Walsh, J (1949) Correction of Spherical Aberration by a Phased

Line Source Proc N.E.C., Vol 5, p.p 320-333

Tap, K & Pathak, P.H (2006) A Fast Hybrid Asymptotic and Numerical Physical Optics

Analysis of Very Large Scanning Cylindrical Reflectors With Stacked Linear Array

Feeds IEEE Trans on Antennas and Propag., Vol 54, No 4, April, p.p 1142-1151

Tingye, L (1959) A Study of Spherical Reflectors as Wide-Angle Scanning Antennas IRE

Trans on Antennas and Propagation, July, p.p 223-226

Wave Propagation in Plasmas

1 Introduction

No less than 99.9% of the matter in the visible Universe is in the plasma state The plasma

is a gas in which a certain portion of the particles are ionized, and is considered to be the

“fourth” state of the matter The Universe is filled with plasma particles ejected from theupper atmosphere of stars The stream of plasma is called the stellar wind, which also carriesthe intrinsic magnetic field of the stars Our solar system is filled with solar-wind-plasmaparticles Neutral gases in the upper atmosphere of the Earth are also ionized by aphotoelectric effect due to absorption of energy from sunlight The number density of plasmafar above the Earth’s ionosphere is very low (∼100cm−3or much less) A typical mean-freepath of solar-wind plasma is about 1AU1(Astronomical Unit: the distance from the Sun to theEarth) Thus plasma in Geospace can be regarded as collisionless

Motion of plasma is affected by electromagnetic fields The change in the motion of plasmaresults in an electric current, and the surrounding electromagnetic fields are then modified bythe current The plasma behaves as a dielectric media Thus the linear dispersion relation ofelectromagnetic waves in plasma is strongly modified from that in vacuum, which is simply

˜

ω=kc where ˜ ω, k, and c represent angular frequency, wavenumber, and the speed of light,

respectively This chapter gives an introduction to electromagnetic waves in collisionlessplasma2, because it is important to study electromagnetic waves in plasma for understanding

of electromagnetic environment around the Earth

Section 2 gives basic equations for electromagnetic waves in collisionless plasma Then, thelinear dispersion relation of plasma waves is derived It should be noted that there are manygood textbooks for linear dispersion relation of plasma waves However, detailed derivation

of the linear dispersion relation is presented only in a few textbooks (e.g., Stix, 1992; Swanson,2003; 2008) Thus Section 2 aims to revisit the derivation of the linear dispersion relation.Section 3 discusses excitation of plasma waves, by providing examples on the excitation ofplasma waves based on the linear dispersion analysis

Section 4 gives summary of this chapter It is noted that the linear dispersion relation can beapplied for small-amplitude plasma waves only Large-amplitude plasma waves sometimesresult in nonlinear processes Nonlinear processes are so complex that it is difficult to providetheir analytical expressions, and computer simulations play important roles in studies ofnonlinear processes, which should be left as a future study

2 Linear dispersion relation

The motion of charged particles is described by the Newton-Lorentz equations (5,6)

charge and mass The motion of charged particles is also expressed in terms of microscopicdistribution functions



2.2 Derivation of linear dispersion equation

Let us “linearize” the Vlasov equation That is, we divide physical quantities into an

equilibrium part and a small perturbation part (for the distribution function f=n(f0+f1)

with f0 and f1being the equilibrium and the small perturbation parts normalized to unity,respectively) Then the Vlasov equation (7) becomes

Here, the electric field has only the perturbed component (E0 =0) and the multiplication

of small perturbation parts is neglected ( f1E1→0 and f1B1→0) Let us evaluate the term



v× B0



· ∂ f0

∂ v by taking the spatial coordinate relative to the ambient magnetic field and

writing the velocity in terms of its Cartesian coordinatev= [v⊥cosφ,v⊥sinφ,v||] Here, v||

and v⊥ represent velocity components parallel and perpendicular to the ambient magneticfield, andφ=Ωc t+φ0represents the phase angle of the gyro-motion whereΩc≡ q

m0 represents the plasma angular frequency It follows that

Further taking the wavenumber vector k x=k⊥cosθ,k y=k⊥sinθ,k z=k , we obtain

exp[ik· x−i ˜ ωt] = exp[ik· x−i ˜ ωt]exp[i(ω˜ −v|| ||)(t−t)]

×exp



−i v⊥Ωk c⊥

sin[Ωc(t−t) +φ0−θ] −sin[φ0−θ]

For the time integral in Eq.(15), we use the following relationship,

λ ≡ v⊥Ωk⊥

c ,cosφ = exp[iφ] +exp[−iφ]

d3v=2π

0

∞0

Here Eq.(20) is called the linear dispersion relation and Eq.(21) is called the plasma dielectricequation Note that∑

2.3 Linear dispersion relation for waves in Maxwellian plasma

The Maxwellian (or Maxwell-Boltzmann) distribution is usually regarded as a distribution

of particle velocity at an equilibrium state Plasma or charged particles easily movealong an ambient magnetic field, while they do not move across the ambient magneticfield Distributions of particle velocity often show anisotropy in the direction paralleland perpendicular to the ambient magnetic field That is, the average drift velocity andthe temperature in the direction parallel to the ambient magnetic field differ from those

in the direction perpendicular to the ambient magnetic field Thus the following shiftedbi-Maxwellian distribution is used as a velocity distribution at an initial state or an equilibriumstate,

where V d is the drift velocity in the direction parallel to the ambient magnetic field, and

V t|| ≡ T||/m and V t⊥ ≡√T /m are the thermal velocities in the direction parallel and perpendicular to the ambient magnetic field, respectively, with T being temperature of plasma

particles

When plasma has the Maxwellian velocity distribution (23), we can explicitly perform thevelocity-space integral by using the following properties,

I n



a22

 !

I n



a22



−I n



a22

!

n22a2I n

 !

n22a2I n

I n



t22

,

!

n22t2I n

and Z p[x]is the plasma dispersion function (Fried & Conte, 1961)

t||

ξ n Z0[ζ n]

,

3 Excitation of electromagnetic waves

Eq.(26) tells us what kind of plasma waves grows and damps in arbitrary Maxwellian plasma.This section gives examples on the excitation of plasma waves based on the linear dispersionanalysis

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

(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 Π pe

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,

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≡cΩci/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−Π2Ωpe

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

ce < ΩciΠ

2

pi

(ω−Ωci)2 (36)

(a) Electron temperature anisotropy instability

Fig 2 Linear dispersion relation for electromagnetic instabilities

Here|ω Ωce|and|ω Πpe are 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)

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 /V te||, 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