Hamiltonian description of self-consistent wave-particle dynamics in a periodic structure

Abstract

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...

Recent discrete model for small-signal analysis of traveling-wave tubes

“…While Pierce's development leads rapidly [13] to eq. (18), it bypasses the circuit-beam impedance (13) and (16) needed in our comparison.…”

“…This decomposition was then adapted to simulate helix traveling wave tubes by a collaboration [27,28] between Thales Electron Devices at Vélizy, and Aix-Marseille University, as an alternative to equivalent circuit approaches [29]. Finally renamed discrete model, Kuznetsov's decomposition has been extended [16,18,19,30] to express a self-consistent hamiltonian in a periodic structure with conjugate variables representing electric and magnetic fields.…”

“…In this section, we review the basics of the (nonlinear) Kuznetsov discrete theory [14][15][16]. Electromagnetic fields in three-dimensional Euclidean space are governed by the Maxwell equations with sources, and generate the Lorentz force on particles.…”

“…In [16], this normalisation is chosen equal to the eigenfield pulsation Ω s β so that the canonical variables of the Hamiltonian (not discussed here) are the field coefficients V s n and I s n in (29)-(30) ; their dimension is then the square root of an action. In [15], this normalisation has the dimension of an energy, and V s n and I s n become dimensionless.…”

“…The more recent discrete model (a.k.a. Kuznetsov discrete model) [14][15][16] is a promising tool to analyse such regimes beyond the capability of Pierce's model.…”