Conservation of energy and momentum in the classical theory of radiating electrons has been a challenging problem since its inception. We propose a formulation of classical electrodynamics in Hamiltonian form that satisfies the Maxwell equations and the Lorentz force. The radiated field is represented with eigenfunctions using the Gel'fand β-transform. The electron Hamiltonian is the standard one coupling the particles with the propagating fields. The dynamics conserves energy and excludes self-acceleration. A complete Hamiltonian formulation results from adding electrostatic action-at-a-distance coupling between electrons. PACS numbers: 84.40.Fe (microwave tubes) 52.35.Fp (Plasma: electrostatic waves and oscillations) 52.40.Mj (particle beam interaction in plasmas) 52.20.Dq (particle orbits) Keywords : wave-particle interaction, traveling wave tube, β-transform, Floquet boundary conditionTo model consistently the interaction between electrons and waves in devices such as traveling wave tubes, free electron lasers or synchrotrons, we are presently left with two options. The first [6,11,17] is to consider the flow of electrons as a distributed charge and current density coupled with the field through the Maxwell equations. Since it generates diverging singularities, the particle nature of electrons is intentionally overlooked until the question of determining the trajectories of the flow is raised. For the latter, the only possibility is to return to a particle description in which the Lorentz force applies. This change of model for the flow precludes the description of the wave-electron system in Hamiltonian form. One reason is that a procedure is needed to distribute the electron charge and current into a finite volume. This procedure, usually based on meshing space, can only be arbitrary. The second option [8,10,13] is to consistently consider electrons as particles and to determine the field they radiate starting from the Liénard-Wiechert potentials. Dirac's [4] sharp analysis indeed provides an accurate determination of the reaction from the radiated field in the limit of a point electron, yet this approach leads to serious difficulties [4,10,13,16,18]; among these an infinite rest mass of the electron, self-acceleration and acausality, also appear incompatible with the existence of a well-posed Hamiltonian.Hamiltonians are essential to consistently define energy and momentum. They also have great practical usefulness to find approximate solutions in complex systems or to control errors in numerical integration schemes [7]. So their absence in the case of the classical wave-electron interaction is both theoretically and practically unsatisfactory. While the final solution to these problems may involve (upgraded, regular) quantum electrodynamics, one could at least hope for a consistent classical approximation, which would be the classical limit of its quantum counterpart. For instance, in the limit where the electron radiates a large number of photons, each of them having a small energy compared with its kin...

No abstract

No abstract

Mutual phase locking in the system of two limit cycle oscillators with delay coupling is studied. Conditions of phase locking are derived as a result of analysis of a generalized Adler equation. The analytical results are compared with numerical simulation. Depending on the phase shift of the coupling signal propagating between the two oscillators, either in-phase or anti-phase mode of synchronization may arise. The number of possible modes of synchronization increases with the delay time.

No abstract

equipe turbulence plasma, case 322 campus Saint Jérôme, av. esc. Normandie-Niemen, To perform accurate numerical simulations of the wave-particle interaction as it occurs in particular in traveling wave tubes, a new approach using field decomposition with large reduction of degrees of freedom has been proposed : the discrete model. To assess its validity, we compare it with the well-established Pierce equivalent circuit model in small signal regime.Before the comparison, the beam-wave interaction in both models is reformulated using a new pedagogical approach, dealing with associated beam, circuit-beam, and circuit impedances.We show analytically and with a numerical example that the newly developed reduced model is very close to the Pierce model. Interestingly, small deviations do exist at the edges of the amplification band, for which the reduced model offers a physical interpretation. PACS numbers: 84.40.Fe (Microwave tubes), 52.40.Mj (Particle beam interaction in plasmas)

No abstract

The application of chaos in communications and radar offers new and interesting possibilities. This article describes investigations on the generation of chaos in a traveling wave tube ͑TWT͒ amplifier and the experimental parameters responsible for sustaining stable chaos. Chaos is generated in a TWT amplifier when it is made to operate in a highly nonlinear regime by recirculating a fraction of the TWT output power back to the input in a delayed feedback configuration. A driver wave provides a constant external force to the system making it behave like a forced nonlinear oscillator. The effects of the feedback bandwidth, intensity, and phase are described. The study illuminates the different transitions to chaos and the effect of parameters such as the frequency and intensity of the driver wave. The detuning frequency, i.e., difference frequency between the driver wave and the natural oscillation of the system, has been identified as being an important physical parameter for controlling evolution to chaos. Among the observed routes to chaos, besides the more common period doubling, a new route called loss of frequency locking occurs when the driving frequency is adjacent to a natural oscillation mode. The feedback bandwidth controls the nonlinear dynamics of the system, particularly the number of natural oscillation modes. A computational model has been developed to simulate the experiments and reasonably good agreement is obtained between them.Experiments are described that demonstrate the feasibility of chaotic communications using two TWTs, where one is operated as a driven chaotic oscillator and the other as a time-delayed, open-loop amplifier.