Dispersion characteristics of the electromagnetic waves in a relativistic electron beamguided by the ion channelSaeed Mirzanejhad, Farshad Sohbatzadeh, Maede Ghasemi, Zeinab Sedaghat, and Zeinab Mahdian

Citation: Physics of Plasmas (1994-present) 17, 053106 (2010); doi: 10.1063/1.3425850 View online: http://dx.doi.org/10.1063/1.3425850 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/17/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dispersion relation and growth rate in electromagnetically pumped free-electron lasers with ion-channel guiding Phys. Plasmas 15, 073103 (2008); 10.1063/1.2947278 Erratum: “Wave-mode dispersions in a relativistic electron beam with ion-channel guiding” [Phys. Plasmas13,083101 (2006)] Phys. Plasmas 14, 029902 (2007); 10.1063/1.2458724 Space-charge limiting currents in magnetically focused intense relativistic beams with an ion channel Phys. Plasmas 13, 123505 (2006); 10.1063/1.2402915 Wave-mode dispersions in a relativistic electron beam with ion-channel guiding Phys. Plasmas 13, 083101 (2006); 10.1063/1.2245563 Relativistic Raman backscattering in an electron beam with ion-channel guiding and its application in free-electron laser Phys. Plasmas 12, 113106 (2005); 10.1063/1.2131048

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Dispersion characteristics of the electromagnetic waves in a relativisticelectron beam guided by the ion channel

Saeed Mirzanejhad,a� Farshad Sohbatzadeh, Maede Ghasemi,Zeinab Sedaghat, and Zeinab MahdianDepartment of Physics, Faculty of Science, Mazandaran University, 47416-95447 Babolsar, Iran

�Received 30 September 2009; accepted 15 April 2010; published online 25 May 2010�

In this article, the dispersion characteristics of the paraxial �near axis� electromagnetic �EM� wavesin a relativistic electron beam guided by the ion channel are investigated. Equilibrium fields such asion-channel electrostatic field and self-fields of relativistic electron beam are included in thisformalism. In accordance with the equilibrium field structure, radial and azimuthal waves areselected as base vectors for EM waves. It is shown that the dispersion of the radially polarized EMand space charge waves are influenced by the equilibrium fields, but azimuthally polarized waveremain unaffected. In some wave number domains, the radially polarized EM and fast space chargewaves are coupled. In these regions, instability is analyzed as a function of equilibrium structure. Itis shown that the total equilibrium radial force due to the ion channel and electron beam and alsorelativistic effect play a key role in the coupling of the radially polarized EM wave and space chargewave. Furthermore, some asymptotic behaviors such as weak and strong ion channel, nonrelativisticcase and cutoff frequencies are discussed. This instability could be used as an amplificationmechanism for radially polarized EM waves in a beam-plasma system where a relativistic electronbeam is guided by the ion channel. © 2010 American Institute of Physics. �doi:10.1063/1.3425850�

I. INTRODUCTION

Propagation of electromagnetic �EM� waves in a relativ-istic electron beam-plasma system has many applications incharged particle accelerators, free electron lasers and highpower sources of EM radiation. Therefore, wave propertiesare important to be studied in these cases. Properties of wavepropagation in turn depend on the steady state or the equi-librium conditions. In such systems, solenoidal or quadruplemagnets need to transport intense and relativistic electronbeam. Ion-channel guiding has been proposed as an alterna-tive focusing method of relativistic electron beam.1,2 In thismethod a relativistic electron beam is injected into an under-dense plasma �with density less than or equal to the beamdensity�. The beam front pushes out the plasma electrons,leaving an ion channel. Ion-focusing causes the beam elec-trons to oscillate about the axis and play a similar role as themagnetic guiding field. This new focusing method has beenused in advanced accelerators3,4 and new high power sourcesof EM radiation such as free electron lasers,5–9 ion-channellasers,10,11 ion-channel electron cyclotron masers12 and syn-chrotron x-ray radiation by betatron motion.13–15 Character-istics of the EM wave dispersion in a beam-plasma systemwhere a relativistic electron beam is guided by the ion chan-nel depend on the equilibrium fields and initial conditions ofthe electron beam. In the ion-focused regime �IFR�, the fo-cusing effect of the ion channel overcomes the defocusingeffect of the electron beam and therefore electrons undergobetatron oscillations as they travel along the beam. The sta-bility of a transverse EM wave in IFR was studied byWhittum16 in one dimension and with a frequency muchhigher than the plasma frequency. Dispersion relation of the

EM wave in one dimension was also studied where thesteady state consists of a strong gyration of electrons in thehelical magnetic field of a wiggler.17 Rouhani et al.18 ana-lyzed high frequency eigenmodes of a cylindrical metallicwaveguide partially filled with a relativistic electron beamand guided by the ion channel. In recent works, some prop-erties of space charge waves in a completely filled ion-channel waveguide are studied.19

The EM instability aroused by the interaction betweenEM wave and relativistic electron beam in the ion channelhas been a fascinating research field for over severaldecades.20,21 Recently, EM instability caused by the interac-tion between an EM wave and a plasma wave were describedin a one-dimensional planar structure, while the density ofsteady background ion was assumed to be infinite and uni-form in the y direction.22 Moreover, EM instability in anelectron beam ion-channel system was investigated using ki-netic theory.23 The main goal of the present investigation isto study the instability of radially polarized EM waves in abeam-plasma system where a relativistic electron beam isguided by the ion channel. Since, the ion channel and elec-tron beam density are assumed to be uniform in both longi-tudinal and transverse directions, therefore, our results arevalid at near axis. In paraxial approximation, for narrow rela-tivistic electron beam, the nonrelativistic small amplitude be-tatron motion can be neglected. It is shown that the azimuth-ally polarized EM wave dispersion is obeyed the well-knownEM wave dispersion ��2=�pe

2 +c2k2�. However, radially po-larized EM wave dispersions were coupled to the spacecharge waves and instability is occurred as well. The mecha-nism of this instability is due to the coupling between highfrequency EM and the radial electron oscillation derivedfrom the deflection of longitudinal electron oscillation due toa�Electronic mail: saeed@umz.ac.ir.

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the equilibrium radial forces. In this situation, the plasmawave and the radially polarized EM wave meet the synchro-nization condition and one of them increases while the otherone damps. It is revealed that the radial forces are key pa-rameters, which influence the behavior of radially polarizedEM waves and induce electrons to exchange energy withwaves. Moreover, the effect of equilibrium fields on this in-stability and some asymptotic behaviors are analyzed. It isshown that the ion-channel density can decrease the magni-tude and spread of instability. In addition, instability magni-tude and spread are noticeably affected by the relativisticfactor of the electron beam.

The organization of this paper is as follows: In Sec. II,basic equations are introduced and the dispersion relation forthe radially and azimuthally polarized EM waves and spacecharge waves are derived. In Sec. III, some asymptotic be-haviors are investigated and agreements with known situa-tions are indicated. In Sec. IV, numerical procedure and re-sults are analyzed and some characteristics of the radiallypolarized EM wave instabilities are investigated. In Sec. V,concluding remarks are made.

II. BASIC EQUATIONS AND WAVE DISPERSIONRELATIONS

A relativistic and cold electron beam, with the homoge-neous lab-frame density n0e and constant axial velocity�0=v0 /c, passes through an ion channel with homogeneousdensity n0i. The steady state is in IFR, where the focusingeffect of the ion channel overcomes the defocusing effect ofthe electron beam’s space charge force. The analysis is lim-ited to near-axis waves with negligible betatron motion.Equilibrium fields �due to the ion channel�, the self-electric�due to the electronic charge�, and self-magnetic fields �dueto the axial motion� of the electron beam are

E� 0�r� = E� i�r� + E� s�r� = 2�e�n0i − n0e�rer, �1�

B� 0�r� = B� s�r� = − 2�en0e�0re�. �2�

In which upper index s on the field components denote theself-fields of the electron beam. Now, perturbed amplitudesare added to the steady state quantities,

n = n0 + �n, E� = E� 0 + �E� , B� = B� 0 + �B� , v� = v�0 + �v� .

�3�

Assume plane monochromatic wave dependency for the per-

turbed amplitudes, �A=�A˜ei�kz−�t�. In the cold one fluidmodel, the linearized continuity, momentum, and Maxwellequations in cylindrical coordinates �r ,� ,z� are as follows:

− i��n + ikn0e�vz + n0e�vr

r= 0, �4�

− i��vr +e

m�0�Er −

e�0

m�0�B� + ��pe

2 − �02�pi

2 ��0r

c�vz = 0,

�5�

− i��v� +e

m�0�E� +

e�0

m�0�Br = 0, �6�

− i��vz +e

m�03�Ez − �pi

2 �0r

c�vr = 0, �7�

�Br = −ck

��E�, �8�

�B� =ck

��Er, �9�

− ik�B� = −i�

c�Er −

4�en0e

c�vr, �10�

ik�Br = −i�

c�E� −

4�en0e

c�v�, �11�

1

r�Er + ik�Ez = − 4�e�n , �12�

where �=�−kv0, �pe2 =4�e2n0e /m�0, and �pi

2 =2�e2n0i /m�0 are relativistic plasma frequencies for electron beam andion channel, respectively. Three homogenous equations forthe perturbed electric field �E� components are obtained byeliminating �n, �V� , and �B� from the above equations,

�A + J 0 Cr

0 A 0

Dr +G

r0 B ���Er

�E�

�Ez� = 0, �13�

where constants in the above matrix elements are defined asfollows:

A = �k2c2 − �2 + �pe2 � , �14�

B = ��pe2

�02 − �� − kv0�2� , �15�

C = − i��pe2 − �0

2�pi2 �

��0

c, �16�

D = − i�pi

2 �0

�c�k2c2 − �2� , �17�

J =− �0��pe

2 − �02�pi

2 ��kc − ��0�� − kv0

, �18�

G = ic

��kc − ��0��� − kv0� . �19�

It is seen from the structure of the above matrix elements thatthe radial and axial components of the electric field arecoupled and the azimuthal component is decoupled fromthem. In Sec. IV, it is shown that radially polarized EM waveand longitudinal fast space charge wave meet the synchroni-zation condition in some region and lead to the instability.

053106-2 Mirzanejhad et al. Phys. Plasmas 17, 053106 �2010�

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Then, they can be propagated and amplified through the elec-tron beam. The nontrivial solution of the above homo-geneous equations is obtained as follows:

Det�A + J 0 Cr

0 A 0

Dr +G

r0 B � = A�AB + BJ − CG − CDr2� = 0.

�20�

The first root A=0, corresponds to the azimuthally polarizedwave �0,�E� ,0�. The dispersion of this wave is the well-known dispersion of the transverse EM waves in plasma,�2=�pe

2 +k2c2. This means that the equilibrium fields do notaffect this wave in our one-dimensional linear formalism.The second root �AB+BJ−CG−CDr2�=0, corresponds tothe combined radially polarized and longitudinal waves,��Er ,0 ,�E��. The first term of the second root, AB=0, is thewell-known transverse EM dispersion �2=�pe

2 +k2c2, andlongitudinal fast and slow space charge wave dispersions��−kv0�2=�pe

2 . This term is modified by the equilibriumfields through the other three terms, �AB+BJ−CG−CDr2�=0. Even, on the axis of the ion channel �r=0�, wave disper-sions are modified considerably, �AB+BJ−CG�=0. Byeliminating the effect of the electron beam self-fields fromthe formalism, only two coefficients C and J are changed asfollows:

C� = i�02�pi

2 ��0

c, �21�

J� = �0�02�pi

2 �kc − ��0�� − kv0

. �22�

This condition may be achieved by rare density electronbeam.

III. SOME ASYMPTOTIC BEHAVIORS

In this section, we deduced some asymptotic behaviorsof the prescribed dispersion relation for radially polarizedEM wave such as dense ion channel, rare ion channel, non-relativistic case, and cutoff frequencies.

A. Dense ion channel

For the dense ion channel, the electrostatic force of thechannel overcomes the self-field of the electron beam, hencethe results of this section can be considered as the ion-channel dispersion relation. Ion channel in this case can becreated by the ponderomotive force of intense laserpulse,15,24,25 or by the high density head of the electronbeam. Therefore, the rare electron beam follows the frontdense head and pass through the strong ion channel. For thisstructure, we assume �pe��pi, so the near axis dispersion�r0� is reduced to

�k2c2 − �2 + �pe2 ���pe

2

�02 − �� − kv0�2�

= − v0�pi2 �pe

2 �k − �v0/c2�� − kv0

1 −�pe

2

�02�pi

2 � . �23�

If �pe2 /�0

2�pi2 is neglected in last term, equation is reduced to

the ion-channel dispersion, which can be obtained easily byreplacing C and J by C� and J� from Eqs. �21� and �22�. Theleft-hand side of this equation is the well-known EM andspace charge waves dispersions. We try to calculate instabil-ity effect of the ion channel in weak coupling approximation.In this case, we assume �=�r+�, where �r is the simulta-neous root of the left-hand side dispersions and ���r.In this approximation, one can easily obtain the followingequation:

�2 −kv0�pi

2

4��� − kv0�� +

v0�02�pi

2

4��k − �v0/c2� = 0, �24�

where the instability condition for this case is

�pi2

�pe2

16���v0/c2 − k�k2v0

. �25�

B. Rare ion channel

In this case self-fields of the beam have a key role in thedispersion relation. We assume �pe�pi, therefore disper-sion relation reduces to

�k2c2 − �2 + �pe2 ���pe

2

�02 − �� − kv0�2�

=�pe

4

�02 �v0�k − �v0/c2�

� − kv01 −

�02�pi

2

�pe2 �

−v0

2r2

c4 �k2c2 − �2��0

2�pi2

�pe2 � . �26�

If �02�pi

2 /�pe2 is neglected, dispersion will be independent of

the radial distance. We put �=�r+� in the weak couplingapproximation, therefore � and the instability condition canbe obtained from the following equations, respectively:

�2 +v0�pe

2

4��02�� − kv0�

� −v0�pe

2

4��k − �v0/c2� = 0, �27�

and the instability condition is calculated as

��v0/c2 − k� �k2v0

16��02 . �28�

C. Nonrelativistic case

In nonrelativistic case, we assume, v0 /c�1 ��0→1�. Inthis situation, the magnetic self-field of the electron beam isnegligible and dispersion can be obtained as

053106-3 Dispersion characteristics of the electromagnetic waves… Phys. Plasmas 17, 053106 �2010�

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�k2c2 − �2 + �pe2 ���pe

2 − �2 + 2kv0��

=kv0

��pe

2 ��pe2 − �pi

2 � , �29�

where the left-hand side is the dispersion relations of the EMand space charge waves in nonrelativistic case. Dispersion isindependent of radial distance r, same as rare ion channel.For the weak coupling condition we obtain,

�2 +kv0��pe

2 ��pe2 − �pi

2 �4�4�� − kv0� − kv0�pe

2 ��pe2 − �pi

2 �

�� −kv0�2�pe

2 ��pe2 − �pi

2 �4�4�� − kv0� − kv0�pe

2 ��pe2 − �pi

2 �= 0, �30�

and instability condition can be calculated as

v0 �16

3

�4�� − kv0�k�pe

2 ��pe2 − �pi

2 �. �31�

D. Cutoff frequencies

Using cutoff frequencies for near axes �paraxial� radiallypolarized EM wave r=0 can be obtained from dispersionrelation at the zero wave number �k=0�,

�4 − ��2 − �02��pe

2 ��2 + ��pe2

�02 �0

2��pe2 − �pi

2 �02� +

�pe4

�02 � = 0.

�32�

This is the second order equation for �2 with the followingsolutions:

�2 =�2 − �0

2��pe2 ��0

4�pe4 − 4�0

2�pe2 ��pe

2 − �pi2 �0

2�/�02

2.

�33�

The required condition for the four real solutions of theabove equation is �0

2�pi2 /�pe

2 � �5−�02� /4. Even in these con-

ditions two roots of the equation may be complex. Further-more, the sufficient condition for the four real solutions is�0

2�pi2 /�pe

2 2�02−1 /�0

2−1. The �02�pi

2 /�pe2 parameter in these

inequalities measures the relative strength of the �focusing�ion-channel electric field and self-magnetic field forces to the�defocusing� space charge force. Four root behaviors areshown as a function of relativistic factor �0 in Figs. 1�a� and1�b�, for two arbitrary values of �pi /�pe=0.5 and 2, weakand strong ion channel, respectively. It is shown in thesefigures that for the weak ion channel, �pi /�pe=0.5, fourcomplex roots may occur �Fig. 1�a�� but for the strong ionchannel, �pi /�pe=2, at least two cutoff frequencies are real�Fig. 1�b��.

The cutoff frequencies for case A, dense ion channel, areobtained as

� = ��0�pe�pi�1 − 1 −�0

2

2� �pe

2�0�pi� . �34�

In the self-field free case, last term in above equationis neglected, hence cutoff frequencies are reduced to,�= ��0�pe�pi.

In the case B, rare ion channel, cutoff frequencies areobtained as

� = �pe�p 1

1

2�5p�02 −

4

5�

�pi2

�pe2 � , �35�

where

p = 1 −�0

2

2�

�0

2�5�0

2 −4

5� .

In this case, real cutoff frequencies are obtained only for�0��0.8. If last term in above equation is neglected cutofffrequencies reduced to �= �pe

�p.For the weak relativistic case cutoff frequencies are ob-

tained as

1 2 3 4

2

4

6

8

10

ωpi

/ ωpe

= 0.5

Two real roots

(2γ02−1) / γ

02−1

γ02ω

pi2 / ω

pe2(5−γ

02) / 4

Four complex roots Four real roots

γ0

(a)

Four real roots Two real roots

1 2 3 4

2

4

6

8

10

γ02ω

pi2 / ω

pe2

ωpi

/ ωpe

= 2

(2γ02−1) / γ

02−1

(5−γ02) / 4

γ0

( b )

(a)

(b)

FIG. 1. Behaviors of cutoff frequencies roots as a function of relativisticfactor for two arbitrary values of �a� �pi /�pe=0.5 and �b� �pi /�pe=2.

053106-4 Mirzanejhad et al. Phys. Plasmas 17, 053106 �2010�

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� = �1 −�0

21 �2�pi

2

�pe2 − 1���pe, �36�

real cutoff frequencies are occurred only for �pi /�pe

�1 /�2. The cutoff frequencies of the Eq. �36� reduced tothe well-known cutoff frequencies for transverse EM waveand space charge waves ��� �pe� in nonrelativistic case��0�0�. Moreover, Eq. �33� is related to the following cutofffrequencies in the ultrarelativistic case ��→1�:

�cutoff = �pe�1 �1 + �2�pi/�pe�2

2�1/2

. �37�

It is seen that there are two real and two complex cutofffrequencies. Instabilities are discussed in details in Sec. IVwhen the dispersion relation is solved numerically forvarious parameters.

IV. NUMERICAL PROCEDURE AND RESULTS

To solve dispersion relation numerically, the quantitiesare normalized by the electron rest mass and charge, me ,e,light velocity in vacuum, c and relativistic plasma frequencyin ion channel, �pi as

� =�

�pe, �pi =

�pi

�pe, k =

kc

�pe, r =

r�pe

c. �38�

Figures 2�a� and 2�b� show the coupling of the real part ofdispersion curves for radially polarized EM and fast spacecharge waves which is denoted by R and F, respectively, atarbitrary radial distance r=0.1 with the electron beam axialvelocity �0=0.9 for strong and weak ion channel. In Fig.2�a�, the ion channel is strong �pi=2 and Fig. 2�b� is plottedfor the weak ion channel �pi=0.5. Dashed line shows theinstability region of the waves due to the coupling of thesetwo modes where their real parts are equal. The imaginarypart of the frequency for these two cases is depicted in Fig. 3,in which weak and strong ion channel is indicated by dashedand solid lines, respectively. It is clear that the instabilitymagnitude and spread are smaller for weak ion channel. InFig. 4, the ratio of the perturbed radially electric field to theperturbed axial electric field �Er /�Ez is plotted in terms of

wave number k for radially polarized and fast space chargewaves in strong ion channel �pi=2. It is shown that thesewaves are not pure transverse or longitudinal, but are hybridwaves, especially in the coupling region. The effect of theelectron beam velocity or the relativistic factor �0 on theinstability magnitude and spread for radially polarized andfast space charge waves are shown in Figs. 5�a� and 5�b�. InFig. 5�a�, the imaginary parts of frequency for these twowaves are depicted for �0=1.001, 1.01, 1.1, 1.5, 2.5, 4, and10 at r=0.1 for strong ion channel, �pi=2. By increasing thevalue of �0, the instability curve shifted to a bigger value of

k. In Fig. 5�b�, maximum amplitude of these instabilities isplotted versus log��0−1� in solid line which has a maximumat �01.8. The dashed line shows the instability spread forvarious �0. It is clear that the instability spread increases byincreasing the value of �0 and reaches to �kc /�pe10, as-ymptotically. In Fig. 6, the imaginary part of the dispersion

2 4 6 8 10 12 140

2

4

6

8

10

12

14

kc / ωpe

ωpi

/ ωpe

= 0.5Re ω / ω

pe

R

F

(a)

(b)

2 4 6 8 10 12 140

2

4

6

8

10

12

14

R

F

Re ω / ωpe

kc / ωpe

ωpi

/ ωpe

= 2

FIG. 2. Dispersion curves of radially polarized, R and fast space chargewaves, F, at normalized distance, r=0.1 and electron beam velocity�0=0.9, for �a� strong ion channel �pi /�pe=2 and �b� weak ion channel�pi /�pe=0.5.

2 4 6 8 10 12−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Strong

Weak

Im ω / ωpe

kc / ωpe

FIG. 3. The imaginary part of the normalized frequency as a function ofnormalized wave number at r=0.1 and �0=0.9 for strong ion channel�pi /�pe=2 �solid line� and weak ion channel �pi /�pe=0.5 �dashed line�.

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curve is shown for the case in which the self-fields are ne-glected �coefficients C� and J�, Eqs. �21� and �22� are used indispersion relation� and compared with the general case forweak ion channel �pi=0.5 at r=0.1. It is revealed that in IFRscheme self-fields decrease the instability magnitude andspread, and shift the stability region to a smaller wave num-ber. In Fig. 7, the instability of two modes of radially polar-ized and fast space charge are depicted for strong ion channel�pi=2 by using asymptotic formula of Eq. �23� and generalcase that are coincide perfectly. In Fig. 8, the instability ofthe modes by using asymptotic formula and general case forthe weak ion channel, �pi=0.5, is compared which are in agood agreements with each other. In Fig. 9, the instability ofthe modes for nonrelativistic case of �0=1.001 is depictedusing asymptotic formula �dashed lines� and the general case�solid lines�.

0 5 10 15 20

1

2

3

4

Radially PolarizedFast Space Charge

kc / ωpe

δEr

/ δEz

FIG. 4. The ratio of the perturbed radially electric field to the perturbedaxial electric field vs normalized wave number for radially polarized �solidline� and fast space charge waves �dashed line� in strong ion channel,�pi /�pe=2, �0=0.9 at r=0.1.

3 6 9 12

−0.4

−0.2

0

0.2

0.4

10

4

2.5

1.51.1

1.0011.01Im ω / ω

pe

kc / ωpe(a)

(b)0.001 0.01 0.1 1 10

0.3

0.15

0.09

0.06

0.03

0.018

Log [γ0−1]

0.6

10

5

3

2

1

∆kc/ωpe

[Im ω/ωpe

]max

FIG. 5. �a� The imaginary part of the normalized frequency in terms ofnormalized wave number for �0=1.001, 1.01, 1.1, 1.5, 2.5, 4, and 10 and �b�the maximum amplitude of instability �solid line� and instability regionlength �dashed line� vs log��0−1� for strong ion channel �pi /�pe=2 atr=0.1.

0 1 2 3 4 5 6 7 8

−0.1

−0.05

0

0.05

0.1

0.15

Im ω / ωpe

kc / ωpe

Es, Bs=0

FIG. 6. The imaginary part of the normalized frequency as a function ofnormalized wave number for weak ion channel �pi /�pe=0.5, for the case inwhich the self-fields are neglected �dashed line� and is compared with thegeneral case �solid line� for �0=0.9 at r=0.1.

kc / ωpe

Im ω / ωpe

2 4 6 8 10 12−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

FIG. 7. The imaginary part of the normalized frequency as a function ofnormalized wave number for dense ion channel �pe /�pe=2 by using ofasymptotic formula of Eq. �23� �triangle� and is compared with the generalcase �solid line� that are coincide perfectly.

053106-6 Mirzanejhad et al. Phys. Plasmas 17, 053106 �2010�

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Finally, we are going to introduce the coupling source ofradially polarized EM and fast space charge waves. If we paymore attention to the radial part of momentum equations �Eq.�5��, we will find the term including longitudinal oscillationvelocity �vz �last term�, which could drive the radial compo-nent of the electric field �Er �second term�. These crossterms come from equilibrium fields of ion channel and elec-tron beam and also come from the relativistic effects. Theradial force due to foregoing effects is responsible for thecoupling of space charge and radially polarized waves and isproportional to ��pe

2 −�02�pi

2 �. By omitting this factor, theelectrons do not have any betatron motion through their equi-librium motion and coupling of radially polarized EM, andfast space charge waves are removed. Both analytical andnumerical results confirm this statement. By ignoring thecoupling source just mentioned, C and J factors vanish and

the coupling is removed from the following homogenousequations:

� A 0

Dr +G

rB ��Er

�Ez� = 0. �39�

Furthermore, longitudinal space charge wave and radiallypolarized wave are decoupled from each other and describedby the well-known dispersions i.e., B=0 and A=0, respec-tively. For the radially polarized wave, the ratio of the radialcomponent of the electric field to the axial component isobtained by �Er /�Ez=−B / �Dr+ G

r�. This equation shows that

on the axis, �Er=0, and on the arbitrary radius both radiallyand longitudinal components of the electric fields are presentfor the radially polarized wave. Figure 10 shows the numeri-cal solution of the dispersion relation for the zero betatronoscillation, in the case that ��pe

2 −�02�pi

2 �=0 at an arbitraryradius r=0.1. In this figure, unlike the Figs. 2�a� and 2�b�,fast space charge and radially polarized waves are decoupledand have the well-known dispersion relation. In addition tothese two modes, the slow space charge is also shown whichis indicated by s.

V. CONCLUSIONS

In this article, the dispersion relations of the paraxial EMand space charge waves are derived in the relativistic elec-tron beam with ion-channel guiding. It is shown that thedispersion of the radially polarized EM and space chargewaves are influenced by the equilibrium fields, but azimuth-ally polarized wave are disaffected and have well-knowntransverse EM wave dispersion in plasma ��2=�pe

2 +c2k2�.Moreover, equilibrium fields of ion channel and electronbeam and also relativistic effects have a key role in the cou-pling of radially polarized EM wave and space charge waves.This coupling leads to the wave instability and this effectcould be used as an amplification mechanism for radially

1 2 3 4 5 6

−0.06

−0.04

−0.02

0

0.02

0.04

0.06Im ω / ω

pe

kc / ωpe

FIG. 8. The imaginary part of the normalized frequency as a function ofnormalized wave number for rare ion channel �pi /�pe=0.5 by usingasymptotic formula of Eq. �26� �dashed line� and is compared with thegeneral case �solid line�.

0.2 0.4 0.6 0.8

−0.15

−0.1

−0.05

0

0.05

0.1

kc / ωpe

Im ω / ωpe

FIG. 9. The imaginary part of the normalized frequency as a function ofnormalized wave number for strong ion channel �pi /�pe=2, for relativisticfactor of electron beam �0=1.001 by using asymptotic formula of Eq. �29��dashed line� and is compared with the general case �solid line�.

0 2 4 6 8 10 12

2

4

6

8

10

12

kc / ωpe

R

F

S

ω / ωpe

FIG. 10. Dispersion curves of radially polarized R, fast space charge waveF and slow space charge wave S, at normalized distance, r=0.1 and electronbeam velocity �0=0.9, in the case of ��pe

2 −�02�pi

2 �=0.

053106-7 Dispersion characteristics of the electromagnetic waves… Phys. Plasmas 17, 053106 �2010�

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polarized EM wave in a relativistic electron beam guided bythe ion channel. The effect of inclusion of the self-fieldsleads to decrease in the instability magnitude and spread. Weobtained asymptotic formulas for special cases such as weakand strong ion channel and nonrelativistic case which is in agood agreement with the numerical results of the generalcase.

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