M C De Sousa

F Doveil

Y Elskens
email: yves.elskens@univ-amu.fr

I L Caldas

An Upgraded 4 Meters Long Traveling Wave Tube for Plasma Physics Research

Keywords: traveling wave tube (TWT), wave-particle interactions, beam-plasma systems

We consider a 4 meters long traveling wave tube (TWT) that presents an upgraded slow wave structure. We analyze linear and nonlinear phenomena due to the beam-wave interaction in the device, e.g. beam modulation, and wave growth and saturation.

I. INTRODUCTION

Traveling wave tubes (TWTs) [START_REF] Pierce | Traveling Wave Tubes[END_REF], [START_REF] Gilmour | Principles of traveling wave tubes[END_REF] are vacuum electron devices mainly used for wireless communications [START_REF] Minenna | The traveling-wave tube in the history of telecommunication[END_REF], e.g. space telecommunication. Helix TWTs are very useful for this kind of application because they present broad bandwidths with a rather simple design. Whereas industrial devices are 2 to 30 cm long, TWTs with some meters in length are used for research in plasma physics [START_REF] Dimonte | Destruction of trapping oscillations[END_REF]- [START_REF] Doveil | Two regimes of self-consistent heating of charged particles[END_REF].

In this paper, we work with a 4 meters long TWT [START_REF] Doveil | Experimental observation of nonlinear synchronization due to a single wave[END_REF]- [START_REF] De Sousa | Wave-particle interactions in a long traveling wave tube with upgraded helix[END_REF] specially designed to simulate beam-plasma systems with low noise. Such device is presented in Fig. 1, and is currently the only one of its kind in operation. The slow wave structure (SWS) of the TWT was recently upgraded. Its helix is now rigidly wrapped in a dielectric polyimide tape to guarantee a nearly constant helix pitch all along the TWT. The upgraded SWS enables [START_REF] De Sousa | Wave-particle interactions in a long traveling wave tube with upgraded helix[END_REF] much more precise measurements, the execution of new experiments, and the benchmarking of numerical models [START_REF] Minenna | Electromagnetic power and momentum in N-body Hamiltonian approach to wave-particle dynamics in a periodic structure[END_REF], [START_REF] Kh | Timedomain non-linear simulations of a 3 meter long traveling wave tube[END_REF].

II. WAVE PROPAGATION

In [START_REF] De Sousa | Wave-particle interactions in a long traveling wave tube with upgraded helix[END_REF], we introduced a theoretical model describing wave propagation through the different components of the SWS. The parameters estimated with this model (dispersion relation, phase velocity, and impedance) present excellent agreement with the experimental data. The model also estimates the electromagnetic field in the SWS and the group velocity of waves propagating through it [START_REF] De Sousa | Wave-particle interactions in a long traveling wave tube with upgraded helix[END_REF].

III. BEAM-WAVE INTERACTION

When the electromagnetic wave interacts with an electron beam, several linear and nonlinear phenomena take place in the TWT. Among these phenomena we may cite the beam modulation. An initially monokinetic beam gets modulated by the wave and presents two peaks with distinct voltages at the end of the TWT, as can be seen in Fig. which corresponds to a beam initial velocity equal to the wave phase velocity for the 30 MHz wave [START_REF] De Sousa | Wave-particle interactions in a long traveling wave tube with upgraded helix[END_REF]. After interaction with the wave, the beam distribution function displays two peaks with V b-= 16.4 V and V b+ = 21 V, as well as a local minimum around 18 V. It means some electrons lost energy to the wave and were decelerated to 16.4 V, whereas other electrons were accelerated by the wave and reached 21 V.

In Fig. 3(b), the beam was emitted with I b = 10.3 µA and V b0 = 30 V, and it interacts with a 15 MHz wave. In this case, electrons with velocity equal to the wave phase velocity have V bϕ = 39.7 V of voltage. The beam distribution function at the end of the TWT presents peaks with V b-= 25.6 V and V b+ = 40.4 V, and a local minimum with 32.8 V. Thus, some electrons were decelerated to 25.6 V, and other electrons were accelerated towards a velocity close to the wave phase velocity, reaching 40.4 V.

When the beam initial velocity is slightly higher than the wave phase velocity, electrons and wave interact resonantly. Electrons transfer energy to the wave and its amplitude increases until the wave reaches saturation [START_REF] De Sousa | Wave-particle interactions in a long traveling wave tube with upgraded helix[END_REF]. After saturation, electrons are trapped by the wave, and move back and forth in its potential. As a consequence, the wave amplitude oscillates along the TWT [START_REF] De Sousa | Wave-particle interactions in a long traveling wave tube with upgraded helix[END_REF].

Fig. 4 depicts the growth coefficient and saturation amplitude as a function of beam current for a wave emitted with 40 MHz. When the wave saturates due to trapping of electrons in its potential, the theoretical predictions [START_REF] Pierce | Traveling Wave Tubes[END_REF], [START_REF] Gilmour | Principles of traveling wave tubes[END_REF], [START_REF] Dimonte | Destruction of trapping oscillations[END_REF] estimate an increase in the growth coefficient (saturation amplitude) proportional to

I 1/3 b (I 2/3
b ). This is the case in Fig. 4, showing that waves propagating in the TWT saturate because of the development and trapping of electron bunches.

For beam currents below 100 µA, approximately, the growth coefficient and saturation amplitude in Fig. 4 follow the linear theory predictions. When I b 100 µA, nonlinear space charge effects become important [START_REF] De Sousa | Wave-particle interactions in a long traveling wave tube with upgraded helix[END_REF] and these parameters deviate from the linear prediction and tend to a constant value.

IV. CONCLUSIONS

We carried out experiments in a 4 meters long TWT specially conceived to simulate beam-plasma systems. This TWT presents an upgraded SWS that increases the precision of experimental data and makes new experiments possible. The coupling impedance defines the interaction between electromagnetic waves and electron beam, and it also agrees with the model.

We analyzed the beam modulation that occurs after its interaction with the wave. Another effect investigated is the wave growth and saturation process. We showed that waves in the TWT saturate due to trapping of beam electrons that move back and forth in the wave potential. For low values of beam current, the wave growth and saturation amplitude agree with the linear theory. When the beam current is high, these parameters deviate from the linear theory because of space charge forces.

Fig. 1 .

 1 Fig. 1. 4 meters long TWT at PIIM Laboratory.

Fig. 2 .

 2 Fig. 2. Dispersion relation. Most error bars are within the marker size.

Fig. 2

 2 shows the dispersion relation as a function of the wave frequency. Most error bars in this figure are within the marker size, showing the great precision of the upgraded TWT.

3 .

 3 In panel (a), the electron beam interacts with a 30 MHz wave. The beam was emitted with current I b = 5.3 µA and voltage V b0 = 19 V, 978-0-7381-4516-7/21/$31.00 ©2021 IEEE

Fig. 3 .

 3 Fig. 3. Beam distribution function displaying the electrons voltage at the end of the TWT.

Fig. 4 .

 4 Fig. 4. (a) Growth coefficient and (b) saturation amplitude as a function of beam current for a 40 MHz wave.

ACKNOWLEDGMENT D. Guyomarc'h, J.-P. Busso, J.-B. Faure, V. Long and J.-F. Pioche are acknowledged for technical support with the experimental device, and Thales for contributing to the TWT upgrade.