Effect of beam premodulation on excitation of surface plasma waves in a magnetizedplasmaRuby Gupta, Suresh C. Sharma, and Ved Prakash Citation: Physics of Plasmas (1994-present) 17, 112701 (2010); doi: 10.1063/1.3507293 View online: http://dx.doi.org/10.1063/1.3507293 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/17/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Excitation of surface plasma waves by a density-modulated electron beam in a magnetized plasma cylinder Phys. Plasmas 17, 122105 (2010); 10.1063/1.3528438 Whistler wave excitation and effects of self-focusing on ion beam propagation through a background plasmaalong a solenoidal magnetic field Phys. Plasmas 17, 023103 (2010); 10.1063/1.3280013 Excitation of surface plasma waves by an electron beam in a magnetized dusty plasma Phys. Plasmas 16, 093703 (2009); 10.1063/1.3216918 Excitation of a surface plasma wave over a plasma cylinder by a relativistic electron beam Phys. Plasmas 15, 073504 (2008); 10.1063/1.2955769 Excitation of nonreciprocal electromagnetic surface waves in semibounded magnetized plasmas by an electronbeam Phys. Plasmas 10, 4622 (2003); 10.1063/1.1623765

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Effect of beam premodulation on excitation of surface plasma wavesin a magnetized plasma

Ruby Gupta,1 Suresh C. Sharma,2 and Ved Prakash3

1Department of Physics, Swami Shraddhanand College, University of Delhi, Alipur, Delhi 110036, India2Department of Physics, Maharaja Agrasen Institute of Technology,PSP Area Plot No.-l, Sector-22, Rohini, Delhi 110086, India3India Meteorological Department, Ministry of Earth Science,Lodi Road, New Delhi 110003, India

�Received 19 August 2010; accepted 11 October 2010; published online 4 November 2010�

A density modulated electron beam propagating through a vacuum magnetized plasma interfacedrives electromagnetic surface plasma waves �SPWs� to instability via Cerenkov and fast cyclotroninteraction. Numerical calculations of the growth rate and unstable mode frequencies have beencarried out for the typical parameters of the SPWs. The growth rate � of the unstable waveinstability increases with the modulation index ��� and is maximized for �=1. For �=0, � turns outto be �4.32�1010 rad /s for Cerenkov interaction and �6.81�1010 rad /s for fast cyclotroninteraction. The growth rate of the instability increases with the beam density and scales as one-thirdpower of the beam density in Cerenkov interaction and is proportional to the square root of beamdensity in fast cyclotron interaction. In addition, the real frequency of the unstable wave increaseswith the beam-energy and scales as almost one-half power of the beam-energy. © 2010 AmericanInstitute of Physics. �doi:10.1063/1.3507293�

I. INTRODUCTION

Surface plasma wave �SPW� excitation by laser or elec-tron beam injection has been observed and studied exten-sively in many works during past decades.1–7 These wavesplay a crucial role in laser ablation,8 high sensitivitysensors,9 and are being explored for their potential in sub-wavelength optics, magneto-optic data storage, microscopy,and solar cells.10–13 The problem of transferring energy froma beam of particles into electromagnetic wave energy hasbeen given considerable attention in various fields of phys-ics. Beam-energy extraction requires that phase matching be-tween waves and particles is maintained for as long as pos-sible. It is well known that Cerenkov and cyclotronresonances have this desirable property of maintaining syn-chronization. Denton et al.2 studied the process of SPW ex-citation over a metal surface by charged particles. The exci-tation of surface waves in magnetoactive plasma by amoving charged particle along the spiral line relative to theconstant magnetic field had been studied by Khankina et al.4

Shokri and Jazi5 showed that nonreciprocal electromagneticsurface waves can be excited in semibounded magnetizedplasmas by an electron beam flowing on the plasma surface.Prakash and Sharma7 showed that an electron beam drivessurface plasma waves to instability on a vacuum magnetizeddusty plasma interface and in a magnetized dusty plasmacylinder via Cerenkov and fast cyclotron interaction.

There has been a great deal of interest in studyingelectrostatic14 and electromagnetic15–19 waves by modulatedelectron beams. Sharma et al.14 have examined the effect ofbeam density modulation on excitation of lower hybridwaves in a magnetized plasma cylinder. Volokitin et al.15

considered the theory of the excitation of electromagnetic

waves at Cerenkov resonance by a modulated electron beamin an unbounded space plasma and determined the structuresof potentials and electromagnetic fields inside and outsidethe beam. Lavergnat et al.16 proposed the use of a beam as anantenna for electromagnetic radiation. They showed that anappreciable fraction of the initial kinetic energy of artificiallymodulated injected beams can be spontaneously radiated in acoherent way and found that electromagnetic waves can besignificantly excited by electron beams. Krafft et al.17 havedone the study of the energy transfer between the modulatedspiraling electron beam and the whistler wave. Krafft et al.18

have studied emission of whistler waves by a density modu-lated electron beam in a laboratory plasma and results havebeen compared to the excitation by loop antenna. In this casethe fraction of the power of the modulated electron beamradiated by the whistler wave has been measured and is ofthe order of 10−5. Krafft et al.19 have studied whistler waveexcitation in a magnetized laboratory plasma by a densitymodulated electron beam for frequency modulation belowbut in the range of the electron cyclotron frequency. In thiscase the maximum emission of the whistler waves occurredwhen the phase velocity of the whistler wave was equal tothe beam velocity.

The surface plasma waves have been excited in plasmaby electron beams in the previous works but without beammodulation. In the present paper, we study the effect of pre-modulation of electron beam on excitation of SPWs. In Sec.II we study the excitation of SPWs by a premodulated elec-tron beam moving close to the vacuum plasma interface in amagnetized plasma and obtain the growth rate. The variationof the normalized growth rate of the unstable mode as afunction of the modulation index has been discussed inCerenkov and fast cyclotron interactions. The variation ofthe normalized growth rate with the normalized wave vector

PHYSICS OF PLASMAS 17, 112701 �2010�

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has also been discussed in both the interactions. Results anddiscussions are given in Sec. III. Conclusion part is given inSec. IV.

II. INSTABILITY ANALYSIS

We consider a vacuum plasma interface at x=0, withplasma in region x�0 and the vacuum in region x�0�cf. Fig. 1�. The equilibrium electron and ion densities are ne0

and ni0, respectively, immersed in a static magnetic field Bs

in the z-direction. We assume the t, z variations of fields asE, B�exp�−i��t−kzz�� and consider E field to be polarizedin x-z plane. A density premodulated electron beam withvelocity vb0z�=�O /kOZ�, and density nb0+nbm exp�−i��Ot−kOZz�� �where �O is the modulation frequency, kOZ is thewave number, and nbm=�nb0, Δ being the modulation indexwhose value ranges from 0 to 1� is considered propagatingalong z-axis at a height “h” above the vacuum-plasma inter-face. The response of plasma electrons to the perturbation isgoverned by the equation of motion,

d��v�dt

= −e

mE +

1

cv � Bs +

1

cv � Bw ,

or���v�

�t+ v · ���v� = −

e

mE +

1

cv � Bs +

1

cv � Bw ,

�1�

where v=vb0z+vb1, vb1 refers to perturbed velocity, �=�O

+ ��O3 /c2��vb0 ·vb1� is the relativistic gamma factor, and the

magnetic field of the wave is

BW = �c/���k � E� = �c/���kZEX − kXEZ�y .

Here k22=�2 /c2�� / ��+1�−1� and EX= �ikZ /k2�EZ since

� ·E=0 for surface plasma waves.20

The perturbed electron velocities in x-, y-, andz-directions after linearization are obtained as

vb1x =�e�ikZ/k2� − e�vb0/����ikZ

2 /k2� − kX���� − kZvb0��OEZ

im��� − kZvb0�2�O2 − �ce

2 �,

�2�

vb1y =�e�ikz/k2� − e�vb0/����ikz

2/k2� − kx���ceEz

m��� − kzvb0�2�o2 − �ce

2 �, �3�

and

vb1z =eEZ

im�� − kZvb0���O + �1�, �4�

where �ce=eBs /mc is electron cyclotron frequency and�l= ��O

3 /c2�vb02 .

From equation of continuity �n /�t+� · �nv�=0, wheren=nb0+nb1 exp�−i��t−kZz��+nbm exp�−i��Ot−kOZz��, weobtain the perturbed beam density nb1 as

nb1 =e�nb0 + nbm��ikZ − �vb0/���ikZ

2 − k2kX���OEZ

m��� − kZvb0�2�O2 − �ce

2 �

+e�nb0 + nbm�kZEZ

im�� − kZvb0�2��O + �1�

+enbmkOZEZ

im�� − kZvb0�2��O + �1�. �5�

The perturbed current density will be

J1 = − e�nb0vb1 + nb1vb0z + nbmvb1���x-h� . �6�

By retaining only those terms which go as ��−kZvb0�−2, thez-component of current density is obtained as

J1z = �−e2vb0�ikz − �vb0/���ikz

2 − k2kx���oEz�nb0 + nbm�m��� − kzvb0�2�o

2 − �ce2 �

−e2vb0kzEz�nb0 + nbm�

im�� − kzvb0�2��o + �1�

−e2vb0kozEznbm

im�� − kzvb0�2��o + �1����x-h� �7�

or

J1z = −e2vb0nb0EZ

m � �ikz − �vb0/���ikz2 − k2kx���o�1 + ��

��� − kzvb0�2�o2 − �ce

2 �

+kZ�1 + ��

i�� − kZvb0�2��o + �1�

+kOZ�

i�� − kZvb0�2��o + �1����x-h� , �8�

where �=nbm /nb0.Using Eq. �8� in the z-component of wave equation

�2E − ��� · E� +�2

c2 E = −4i�

c2 J , �9�

we get

VACUUM

Surface Plasma Wavesz

x

PlasmaVacuum-PlasmaInterface

h

Modulated Electron Beam

Bs// z

FIG. 1. �Color online� Schematic of a vacuum-plasma interface with a pre-modulated electron beam propagating above the interface at a distance h.

112701-2 Gupta, Sharma, and Prakash Phys. Plasmas 17, 112701 �2010�

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k22 − kZ

2 +�2

c2 EZ = −�vb0

c2 �pb2 � �kz − �vb0/���kz

2 + ik2kx���o�1 + ����� − kzvb0�2�o

2 − �ce2 �

−kZ + �kZ + kOZ��

�� − kZvb0�2��o + �1��EZ��x-h� , �10�

where �pb2 =4nb0e2 /m.

Multiplying by EZ� and integrating from x=0 to

and taking 0EZ

�EZdx=1, 0EZ

�EZ��x-h�dx=exp�−2k2h� andk2

2=�2 /c2�� / ��+1�−1�, Eq. �10� can be rewritten as

�2 − kZ2c2�+ 1

�

= − �vb0�pb2 �+ 1

�

�� �kz − �vb0/���kz2 + ik2kx���o�1 + ��

��� − kzvb0�2�o2 − �ce

2 �

−kz + �kz + koz��

�� − kZvb0�2��o + �1��e−2k2h �11�

or

�� − ���� + �� = − �vb0�pb2 �+ 1

�

�� �kz − �vb0/���kz2 + ik2kx���o�1 + ��

��� − kzvb0�2�o2 − �ce

2 �

−kZ + �kz + kOZ��

�� − kZvb0�2��o + �1��e−2k2h, �12�

where � is the root of �, given as

� = kZc��+ 1

�. �13�

In Cerenkov interaction, ��−kZvb0�2�0⇒��kZvb0,therefore neglecting the first term on right hand side inEq. �12� and assuming perturbed quantities �=�+� and�=kZvb0+�, where � is the small frequency mismatch.

The growth rate �the imaginary part of �� is obtained as

� =�3

2��pb

2 vb0

2�+ 1

� �kZ + �kZ + kOZ���

��o + �1�e−2k2h�1/3

.

�14�

The real part of frequency �r is obtained from the real part of� as

�r = kZ2 eVb

m1/2

−1

2��pb

2 vb0

2�+ 1

� �kZ + �kZ + kOZ���

��o + �1�e−2k2h�1/3

,

�15�

where Vb is the beam potential.In cyclotron interaction ��−kZvb0�2�o

2��ce2 , therefore

neglecting the second term on right hand side in Eq. �12�, weget

�� − ���� + ��

= − �vb0�pb2 �+ 1

�

�� �kz − �vb0/���kz2 + ik2kx���o�1 + ��

��� − kzvb0�2�o2 − �ce

2 � �e−2k2h,

�16�

where ��−kZvb0��o+�ce corresponds to slow cyclotron in-teraction and ��−kZvb0��o−�ce corresponds to fast cyclo-tron interaction.

Considering slow cyclotron interaction and assumingperturbed quantities �=�+� and �=kZvb0− ��ce /�o�+�, thegrowth rate is obtained as

� = 0. �17�

The phase velocity of the unstable mode is obtained by usingthe real part of � as

�r

kZ= vb0 +

1

kZ��pb

2 vb0

4�ce�+ 1

�

��kz − �vb0/���kz2 + ik2kx���o�1 + ��e−2k2h�1/2

.

�18�

That is, in the case of slow cyclotron interaction, there isno growing mode as the phase velocity exceeds the beamvelocity.

Now, considering fast cyclotron interaction and assum-ing perturbed quantities �=�+� and �=kZvb0+ ��ce /�o�+�, the growth rate is obtained as

� = ��pb2 vb0

4�ce�+ 1

�

��kz − �vb0/���kz2 + ik2kx���o�1 + ��e−2k2h�1/2

. �19�

In the absence of beam, �pb2 =0, therefore Eq. �11� gives

�2 �

�+ 1 = kZ

2c2,

which is the dispersion relation for surface plasma waves.20

III. RESULTS AND DISCUSSIONS

The numerical calculations have been carried outusing the typical parameters of the surface plasma waves:electron plasma density neo=1013 cm−3, mass of electronme=9.1�10−28 g, charge of electron e=4.8�10−10 ergs,magnetic field Bs=1 kG, permittivity �=0.35, beamvelocity vbo=2�1010 cm /s, and modulation frequency

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�o=9.2�1010 rad /s. The modulation index A has beenvaried from 0 to 1 in steps of 0.2 and the beam is assumed totravel at a distance of 2.0 cm from the vacuum-plasma inter-face with beam density nbo=2.8�109 cm−3.

In Fig. 2, we have plotted the variation of the normalizedgrowth rate � /�pb of SPW as a function of normalized wavevector kZ /kOZ using Eq. �14� and taking �=0.9. The normal-ized growth rate of the unstable wave has the value of 13.8and the growth rate is 4.07�1010 rad /s when kZ�kOZ and���o, i.e., when the phase velocity of the wave is compa-rable to the premodulated beam velocity. Krafft et al.18 haveshown in their experimental observations that the maximumemission of the whistler waves occurred when the phase ve-locity was equal to the beam velocity. The normalizedgrowth rate � /�pb is also plotted as a function of modulationindex � in Fig. 3 for the same parameters used for plottingFig. 2 and kZ /kOZ=1.2. The normalized growth rate can beseen increasing with the modulation index � from a value of15.9 to 19.2 as � changes from 0.2 to 0.8. For modulationindex �=0, i.e., without modulation of beam, the value ofgrowth rate turns out to be �4.32�1010 rad /s. The growth

rate of the unstable mode increases with the beam densityand scales as the one-third power of the beam density �cf.Eq. �14��. In Fig. 4, we have plotted the variation of thenormalized growth rate � /�pb of SPW for fast cyclotron in-teraction as a function of normalized wave vector kZ /kOZ

using Eq. �18� for the same values of beam velocity, modu-lation index, and static magnetic field used for plottingFig. 2. The normalized growth rate has the value of 18.07and the growth rate is 5.33�1010 rad /sec when kZ�kOZ

and ���O. Figure 5 shows the variation of the normalizedgrowth rate as a function of modulation index �. It can beseen that the growth rate of the unstable mode increases withthe modulation index �. The normalized growth rate in-creases from a value of 25.07 to 30.63 as � changes from 0.2to 0.8. Without modulation of beam, the value of growth rateturns out to be �6.81�1010 rad /s. The growth rate alsoincreases with the beam density and scales as the square rootof the beam density. The growth rate in slow cyclotron inter-action is found to be zero from Eq. �17�. It can be justifiedfrom the fact that the necessary condition for net energytransfer from beam electrons to the wave is that the beam

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2

pbω

γ

k kz oz

FIG. 2. Normalized growth rate � /�pb of the unstable mode on a vacuum-plasma interface via Cerenkov interaction as a function of normalized wavevector kZ /kOZ for the parameters given in the text.

14

15

16

17

18

19

20

21

0 0.2 0.4 0.6 0.8 1

∆

pbω

γ

FIG. 3. Normalized growth rate � /�pb of the unstable mode on a vacuum-plasma interface via Cerenkov interaction as a function of modulation index�.

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1 1.2

pbω

γ

k kz oz

FIG. 4. Normalized growth rate � /�pb of the unstable mode on a vacuum-plasma interface via fast cyclotron interaction as a function of normalizedwave vector kZ /kOZ.

22

24

26

28

30

32

0 0.2 0.4 0.6 0.8 1

∆

pbω

γ

FIG. 5. Normalized growth rate � /�pb of the unstable mode on a vacuum-plasma interface via fast cyclotron interaction as a function of modulationindex �.

112701-4 Gupta, Sharma, and Prakash Phys. Plasmas 17, 112701 �2010�

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velocity should exceed the phase velocity of the wave. How-ever, in the case of slow cyclotron interaction, it is found thatthe beam velocity is less than the phase velocity of wave �cf.Eq. �18��.

IV. CONCLUSION

A premodulated electron beam can interact with the sur-face plasma waves via Cerenkov interaction and fast cyclo-tron interaction in infinite geometry on a vacuum-magnetizedplasma interface. The interaction between the plasma modeand the beam mode is found to excite the surface plasmawaves, and hence the wave grows. In cyclotron interaction,only fast cyclotron interaction is found to excite the surfaceplasma waves and provide them a growth. The velocity ofthe beam for slow cyclotron interaction is such that it cannotinteract with the surface plasma waves. Mathematically also,the growth rate in slow cyclotron interaction is found to bezero. In conclusion, we may say that the surface plasmawaves are driven to instability in infinite geometry by a pre-modulated electron beam via Cerenkov and fast cyclotroninteractions. The electron beam premodulation offers consid-erable enhancement in growth rate of unstable mode of SPW.The growth rate of the unstable modes decreases with in-crease in the static magnetic field. To the best of authors’knowledge, no experimental or theoretical paper on excita-tion of surface plasma waves by a modulated electron beamin a magnetized plasma has been reported so far, and hencewe cannot compare our theoretical results. A more practicaland realistic problem on excitation of SPWs by premodu-

lated electron beam in a magnetized plasma cylinder is un-derway and will be in a future article.

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