Wave-particle interaction and Hamiltonian dynamics investigated in a traveling wavetubea)Fabrice Doveil and Alessandro Macor Citation: Physics of Plasmas (1994-present) 13, 055704 (2006); doi: 10.1063/1.2177201 View online: http://dx.doi.org/10.1063/1.2177201 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/13/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Field theory of a terahertz staggered double-grating arrays waveguide Cerenkov traveling wave amplifier Phys. Plasmas 21, 043103 (2014); 10.1063/1.4870320 A watt-class 1-THz backward-wave oscillator based on sine waveguide Phys. Plasmas 19, 013113 (2012); 10.1063/1.3677889 Observation and Control of Hamiltonian Chaos in Waveparticle Interaction AIP Conf. Proc. 1308, 132 (2010); 10.1063/1.3526149 Absolute instability in a traveling wave tube model Phys. Plasmas 5, 4408 (1998); 10.1063/1.873178 Experimental and theoretical investigations of a rectangular grating structure for low-voltage traveling wave tubeamplifiers Phys. Plasmas 4, 2707 (1997); 10.1063/1.872547

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Wave-particle interaction and Hamiltonian dynamics investigatedin a traveling wave tubea…

Fabrice Doveilb� and Alessandro Macorc�

Physique des Interactions Ioniques et Moléculaires, Unité 6633 CNRS-Université de Provence, EquipeTurbulence Plasma, Case 321, Centre de Saint-Jérôme, F-13397 Marseille cedex 20, France

�Received 21 October 2005; accepted 5 December 2005; published online 11 May 2006�

For wave-particle interaction studies, the one-dimensional �1-D� beam-plasma system can beadvantageously replaced by a Traveling Wave Tube �TWT�. This led us to a detailed experimentalanalysis of the self-consistent interaction between unstable waves and a small either cold or warmbeam. More recently, a test electron beam has been used to observe its non-self-consistentinteraction with externally excited wave�s�. The velocity distribution function of the electron beamis investigated with a trochoidal energy analyzer that records the beam energy distribution at theoutput of the TWT. An arbitrary waveform generator is used to launch a prescribed spectrum ofwaves along the slow wave structure �a 4 m long helix� of the TWT. The nonlinear synchronizationof particles by a single wave responsible for Landau damping is observed. The resonant velocitydomain associated to a single wave is also observed, as well as the transition to large-scale chaoswhen the resonant domains of two waves and their secondary resonances overlap leading to a typical“devil’s staircase” behavior. A new strategy for the control of chaos is tested. © 2006 AmericanInstitute of Physics. �DOI: 10.1063/1.2177201�

I. INTRODUCTION

Wave-particle interaction is central to the operation ofelectron devices1,2 and a long traveling wave tube �TWT� hasbeen extensively used to mimic beam-plasma interaction.3,4

Although chaotic wave-particle interactions have alreadybeen observed in plasma experiments,5 the use of the TWTprovides unique research possibilities. When waves are ex-ternally launched in the tube and the electron beam density isso low that it does not induce any significant growth of thewaves, the beam electrons can be considered as test particlesin the potential of electrostatic waves.

This recently allowed the direct exploration of nonlinearparticle synchronization by a single nonresonant wave.6 Fora single resonant wave, beam trapping can be observed.7 Inthe case of two launched waves, resonance overlap occurs,leading to a large velocity spread of the test beam accordingto the transition to large-scale chaos in phase space by thebreaking of so-called invariant KAM �Kolmogorov-Arnold-Moser� tori.8 Even when a single frequency is launched, thischaotic transition occurs through overlap of the launchedwave and of the beam mode and a “devil’s staircase” behav-ior is recorded. Its existence can be simply related to a para-digm system used to describe the transition to large-scalechaos in Hamiltonian dynamics.9–13 Finally, a new strategyfor the control of chaos by building transport barriers thatprevent electrons to escape from a given velocity region issuccessfully tested.14

The paper is structured as follows: In Sec. II we presentthe principle of the test beam experiment in the TWT and the

trochoidal analyzer used to measure the velocity distributionfunction. In Sec. III, we report the main results on the inter-action of the test beam with one or two waves, and the firsttest of a new method of control of Hamiltonian chaos. Con-clusions and perspectives are drawn in Sec. IV.

II. DESCRIPTION OF THE TWT

Our device consists of three main elements: an electrongun, a slow wave structure �SWS� formed by a helix withaxially movable antennas, and an electron velocity analyzer.It has been described in detail elsewhere.15,16

The electron gun produces a quasimonoenergetic elec-tron beam with a nominal energy spread of about 0.5 eV.16

The electron beam, with radius 1 mm, propagates along theaxis of the SWS and is confined by a strong axial magneticfield of 0.05 T. The SWS consists of a 4 m long wire helix,with a radius of 11.3 mm and a pitch of 0.8 mm, enclosed ina glass vacuum tube evacuated at both ends by two ionpumps at a typical pressure of 2�10−9 Torr. A resistive rftermination serves to reduce reflections at each end of thehelix. The glass vacuum jacket is enclosed by an axiallyslotted cylinder that defines the rf ground and ensures that noother empty waveguide modes than the helix modes canpropagate. These modes have electric field components alongthe axis of the helix and axial phase velocities close to theelectron beam velocity �approximately the velocity of lightmultiplied by the tangent of the helix pitch angle�. They canbe excited by an antenna moving through a cylinder slot andcapacitively coupled to the helix in the frequency range from5 to 95 MHz. The SWS is long enough to allow nonlinearprocesses to develop, such as trapping of the beam in thepotential troughs of a single wave.

a�Paper KI2 6, Bull. Am. Phys. Soc. 50, 183 �2005�.

b�Invited speaker. Electronic mail: doveil@up.univ-mrs.frc�Electronic mail: macor@up.univ-mrs.fr

PHYSICS OF PLASMAS 13, 055704 �2006�

1070-664X/2006/13�5�/055704/5/$23.00 © 2006 American Institute of Physics13, 055704-1

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For a beam intensity of the order of 10 nA, the electronbeam density nb is low enough to ensure that the beam in-duces no wave growth, and the beam electrons can be con-sidered as test particles. Figure 1�a� shows the principle of atest beam experiment. The control parameters are the beamvelocity, the frequency, phase, and amplitude of the variousfrequency components of the signal applied on the antennaand generated by an arbitrary waveform generator, and theinteraction length defined by the position of the emitter alongthe helix. The cumulative changes of the electron beam dis-tribution are measured with a trochoidal velocity analyzer atthe end of the interaction region. Figure 1�b� shows a ren-dering of the analyzer. A small fraction �0.5%� of the elec-trons passes through a hole in the center of the front collec-tor, and is slowed down by three retarding electrodes. Byselecting electrons by means of the drift velocity caused byan electric field perpendicular to the magnetic field, the di-rect measurement of the current collected behind a tiny off-axis hole gives the time-averaged beam axial energy distri-bution with an unprecedented resolution.15

III. TEST PARTICLE EXPERIMENTS

We have first checked that, in the absence of any appliedexternal signal on the antenna, the test electron beam propa-gates without perturbation along the axis of the device.

A. Nonlinear synchronization of the test beam with anonresonant wave

In the reported experiment,6 we apply an oscillating sig-nal at a frequency of 30 MHz on the antenna. According tothe helix dispersion relation, a traveling wave propagatesalong the helix with a phase velocity v�=4.07�106 m/s.Figure 2�a� shows a 2-D contour plot of the velocity distri-bution function for a test beam, with intensity Ib=10 nA and

initial velocity vb=3.82�106 m/s, measured at the outlet ofthe tube after its interaction, over a length of 0.5 m, with thehelix mode. It is the result of linear interpolation by aMATLAB

17 treatment of the recorded measurements obtainedfor different applied signal amplitudes, varying from0 to 1500 mV by steps of 100 mV. When the wave ampli-tude is gradually increased, the beam single peak gives birthto two peaks whose separation increases. The continuous linein Fig. 2�a� gives an estimate of the lower-velocity limit ofthe trapping region of the wave, which scales as the squareroot of the wave amplitude �. As the beam initial velocity isfar out of this trapping domain, the beam electrons merelyexperience a velocity modulation with amplitude �� / �v�

−vb� around their initial velocity vb ��= �q� /m is the electroncharge to mass ratio�. This estimate is obtained by first-orderperturbation theory with respect to the constant helix waveamplitude � around the electron unperturbed free motionwith constant velocity vb. Averaging over the arbitrary initialphase of the electron in the wave yields two peaks at themaximum and minimum electron velocity for the velocitydistribution function, as usual for a sinusoidal motion. These

FIG. 1. �Color online� �a� Principle of the experiment; �b� rendering of thetrochoidal analyzer.

FIG. 2. �Color online� 2-D contour plot of the measured velocity distribu-tion function in a single wave at v�=4.06�106 m/s �a� at fixed interactionlength L=0.5 m=V s increasing wave amplitude with first-order estimatesof modulation �lines� and trapping �half parabola� domains; �b� with fixedwave amplitude �=18 mV versus emitting antenna position z from the gunend with the second-order estimate of mean test beam velocity �continuouscurve�.

055704-2 F. Doveil and A. Macor Phys. Plasmas 13, 055704 �2006�

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two peaks correspond to the two dashed lines of Fig. 2�a�,symmetric around vb. We observe a systematic deviationfrom this simple estimate toward v� as the wave amplitudeincreases. For vb larger than v� we have also observed adeviation toward v�. This deviation is witness to a synchro-nization of the electrons with the wave. It is explained byconsidering the motion of individual test electrons in thepotential of one electrostatic wave described by the equationof the analytically integrable classical nonlinear pendulum.

This is best shown when we keep the wave amplitudeconstant and vary the interaction length z by moving theemitting antenna along the helix. Figure 2�b� is obtained byaccumulating the test beam vdf measured at the output of thehelix for 100 different antenna positions, starting at the gunend of the TWT and spaced every 2.5 cm. We first notice aperiodic velocity bunching of the vdf with a spatial periodLb=vbv� / �f �vb−v���=0.76 m that corresponds to the lengthfor which the electron transit time Lb /vb differs from thewave propagation time Lb /v� by one wave period 1/ f . Wealso note that the amplitude of the beam velocity modulationincreases with z because, when the probe comes closer to theoutput, the wave amplitude is less damped along the SWS. Acloser look at the color contours of Fig. 2�b� shows that theaverage velocity of the test beam oscillates with z. The con-tinuous curve superimposed in the contour plot is the second-order estimate of the phase-averaged test electron velocityfor the measured wave amplitude when the wave damping isalso included. We reported an excellent agreement betweenthis estimate and the measured mean beam velocity obtainedby averaging over the measured velocity distributionfunction.6 We have shown that this effect is at the root of thewell-known Landau damping.6

B. Trapping of the test beam in a resonant wave

Figure 3 is obtained in the same way as Fig. 2�a� butcorresponds to an applied signal at 40 MHz on the antennalocated at L=2.3 m from the device output, and a beam withmean initial velocity equal to the phase velocity v�=3.55

�106 m/s of the helix mode. We observe that the shape ofthe velocity domain in which the test beam electrons arespread is very different from Fig. 2�a�. Its width does notincrease linearly with the wave amplitude but rather like itssquare root, as expected, if the electrons are trapped in thepotential troughs of the wave. We also observe a further ve-locity bunching of the electrons around their initial velocityfor an applied signal amplitude of 900 mV. If we refer to therotating bar model18 to describe the trapped electrons motionwith alternate velocity and spatial bunching in phase space,this phenomenon is also related to beam trapping. Equating Lto a half-trapping length when velocity bunching occurs, wecan calculate the actual helix mode amplitude and check thatit fits with independent determinations from antenna cou-pling measurements19 or from the trapping velocity width.7

C. Resonance overlap for two waves

Three different velocity distribution functions are shownin Fig. 4. The interaction length is equal to 3.6 m. The testbeam with intensity Ib=50 nA is initially centered at v�40

=3.55�106 m/s, the phase velocity for the wave at40 MHz. The blue �respectively, red� curve labeled f30 �re-spectively, f40� shows the normalized beam velocity distribu-tion measured at the output of the TWT after modulation�respectively, trapping� of the test beam in a single wave at30 MHz �respectively, 40 MHz� launched on the fixed an-tenna. The trapping velocity domain for the wave at 30 MHz�respectively, 40 MHz� is indicated by the blue �respectively,red� horizontal bar centered on the respective wave phasevelocity. In both cases, to the experiment resolution, no elec-tron is detected beyond the upper velocity limit of the40 MHz wave trapping domain. As the trapping domains ofthe two waves overlap, we observe that, when both signals atthese two frequencies are applied on the same antenna withan arbitrary waveform generator, the measured electron ve-locity distribution function labeled f30+40 spreads all over theregion above the previous limit. This is a clear indication that

FIG. 3. �Color online� 2-D contour plot of the measured velocity distribu-tion function of a test beam �Ib=120 nA, vb=3.55�106 m/s� trapped in asingle wave at 40 MHz with a trapping domain �continuous curve� for in-creasing amplitude and fixed interaction length L=2.3 m.

FIG. 4. �Color online�. Normalized velocity distribution function for asingle wave at 30 MHz �curve labeled f30�, a single wave at 40 MHz �curvelabeled f40�, and two waves at 30 and 40 MHz �curve labeled f30+ f40�.Circles give wave phase velocities; upper horizontal lines indicate trappingregions.

055704-3 Wave-particle interaction and Hamiltonian dynamics¼ Phys. Plasmas 13, 055704 �2006�

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no barrier to chaotic velocity diffusion persists, as expectedfrom KAM theory to describe the transition to large-scalechaos for the motion of a charged particle in two electrostaticwaves.8

D. Transition to chaos and “devil’s staircase”

Figure 5�a� corresponds to conditions similar to Fig.2�a�, but for larger applied signal on the antenna, longer in-teraction length L=3.6 m, and a test beam with still lowermean initial velocity equal to 2.7�106 m/s. As the signalamplitude is further increased, we see that the measuredbeam distribution spread does not increase linearly, and fol-lows the dashed parabola typical of particle trapping. Thiscorresponds to the existence of a beam mode, associatedwith beam plasma oscillations in the beam frame. The beamelectrons are trapped inside this mode, which propagates atthe same velocity as the beam. It is also to be noticed that werecover the same horizontally wedge-shaped regions as inFig. 3 associated with periodic peaking of the distribution,easily understood as the result of the trapping of the beam inthe beam mode.

When we keep increasing the applied signal amplitude,the trapping domains of the helix mode and of the beammode overlap, and electrons are even accelerated at a veloc-ity larger than the phase velocity of the helix mode, as ex-pected from the transition to large-scale chaos for the motion

of electrons in these two modes with different phase veloci-ties. Moreover, we observe that this transition does not occursmoothly but rather by steps. Figure 5�b� shows some detailsof such steps observed for a slightly different case with a testbeam with initial velocity vb=2.5�106 m/s. Each step canbe related to the presence of secondary resonances, as ex-pected from particle orbit calculations.20 These secondaryresonances exhibit a “devil’s staircase” behavior with infi-nitely nested resonances inside resonances. Only two mainresonances are indicated by the dashed horizontal lines inFig. 5�a�. A comparison with numerical simulation, a moredetailed analysis of this behavior, as well as its explorationwhen two frequencies are excited, have also beenperformed.21

The accurate knowledge of this system allowed us to testa new method of control of Hamiltonian chaos.22 The core ofthis approach is a small apt modification of the system thatchannels chaos by building transport barriers.

E. Control of chaos

Figure 6�a� gives three different velocity distributionfunctions. The sharp peak corresponds to the initial test beamafter traveling along the helix axis in the absence of an ap-plied signal on the antenna. The blue curve with maximumvelocity spread corresponds to the overlap of the beam mode

FIG. 5. �Color online� Transition to large-scale chaos and experimental“devil’s staircase:” �a� 2-D contour plot of the measured velocity distribu-tion function of a test beam �Ib=10 nA, vb=2.7�106 m/s� after interactionover a length L=3.6 m with a single wave at 30 MHz for increasing appliedsignal amplitude; helix mode �respectively, beam mode� trapping domain isindicated by continuous �respectively, dashed� parabola; secondary reso-nances �n ,m� are indicated at velocities m� / �n+m�� with �=v� /vb; �b�velocity distribution functions showing “devil’s staircase” steps.

FIG. 6. �Color online� �a� Normalized beam velocity distribution function atthe output of the TWT: test beam �Ib=50 nA, vb=2.5�106 m/s� �blackpeak�; with helix and beam modes at 30 MHz �phase velocity and trappingdomain of each mode given, respectively, by circles and horizontal bars��blue curve with large spread�; with an additional controlling wave at60 MHz �phase velocity given by middle circle� �red curve with reducedspread�; �b� kinetic coherence versus beam energy.

055704-4 F. Doveil and A. Macor Phys. Plasmas 13, 055704 �2006�

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and of the helix mode described in the previous section. Wethen superpose on the antenna a control signal correspondingto the beating of these two modes, with well-defined ampli-tude and phase. For the proper choice of the phase of Fig.6�a� we observe a peaking of the red velocity distributionwith no electron detected above the phase velocity of thecontrol signal. On the contrary, for a phase increased by �,we would observe an increased spread of the velocity distri-bution function.

We define the kinetic coherence as the ratio of the maxi-mum amplitude of the measured velocity distribution func-tion to the peak amplitude of the controlled test beam in theprevious figure. Figure 6�b� shows that the resonance condi-tion, stating that the beating mode of the helix and beammode at 60 MHz must fit the helix mode at the same fre-quency, is crucial for the control to efficiently work. Thecontrol is realized with an additional cost of energy that cor-responds to less than 1% of the initial energy of the two-modes system. Intuitively it corresponds to balancing thenonlinear secondary resonance associated with the beating ofthe two modes and locally restoring a KAM torus to preventthe diffusion of electrons in phase space. This experimentwas inspired by a general method of control of Hamiltonianchaos22 and was also compared with numericalsimulations.14

IV. CONCLUSION AND PERSPECTIVES

We have observed the interaction of a test electron beamwith modes propagating in the slow wave structure of a trav-eling wave tube when the applied signal consists in one ortwo frequency components with well-defined amplitudes andphases. We have thus exhibited many features of the com-plex Hamiltonian phase space of a charged particle in thepotential of one or two electrostatic waves. This knowledgeallowed us to perform experiments aimed at testing newmethods of control of Hamiltonian chaos.

This experimental proof of the transition to chaos in thepresence of two waves is a step in the direction of the ex-perimental assessment of quasilinear diffusion in a broadspectrum of waves excited with the arbitrary waveformgenerator.23

More details about the test particle dynamics may beobtained by time-resolved measurements of the evolution ofelectron bunches injected with a prescribed phase with re-spect to a wave.24 These experiments are in progress.

Finally the influence of self-consistent effects could bestudied by gradually increasing the beam intensity.

These basic studies open new tracks in the field ofplasma physics and control of complex systems, and of elec-tron devices such as traveling wave tubes and free electronlasers, where the improvement of performances are of crucialimportance.

The authors are grateful to J-C. Chezeaux, D.Guyomarc’h, and B. Squizzaro for their skillful technicalassistance, and to C. Chandre, Y. Elskens and D. F. Escandefor fruitful discussions. This work is supported by Euratom/CEA. A. Macor benefits from a grant by the Ministère de laRecherche Scientifique.

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