± of the quadratic Hamiltonian are analyzed numerically in both the normal and reversed-field configurations for different values of the ratio
w/
0. The electron trajectories are thus discussed. It is found that pseudo-circular and elliptic trajectories are characterizing both group-I and reversed-field group-II modes while the motion in the normal group-II mode is much more complicated.]]>
Electron beam dynamics in a helical-wiggler free-electron laser (FEL) with a uniform axial guide magnetic field are studied using a three-dimensional Hamiltonian approach. The basic feature of the analysis is the definition of a rotational variable,ĥ, that plays the primordial role in lowering to the half the dimension of the quadratic Hamiltonian as a system of two uncoupled oscillators with definite frequencies and amplitudes. It is through applying this variable in the vicinity of a fixed point that the Heisenberg picture of the dynamics of the particles comes to light, leading thus to the association of the steady-state ideal helical trajectories with arbitrary trajectories. The approach recognized the usual two constants of motion, one being the total energy while the other is the canonical axial angular momentum, P z . If the value of the latter is such that a fixed point exists, the Hamiltonian is expanded about the fixed point up to second order. The so-obtained oscillator characteristic frequencies allowed one to study the different modes of propagation and to identify, and then avoid the problematic operating conditions of the FEL concerned. On the other hand, the amplitudes of the oscillations, which do depend on the frequencies, are fortunately found to be constants of motion and then controlled by the boundary conditions (initial conditions). PACS Nos.: 52.40-w, 52.60+h, 42.55.Tb, 52.75MsRésumé : Utilisant une approche hamiltonienne en 3-D, nous étudions la dynamique d'un faisceau d'électrons dans l'onduleur hélicoïdal d'un laser à électron libres (FEL) avec un guide constitué d'un champ magnétique axial uniforme. L'outil de base de notre analyse est la définition d'une variable rotationnelle qui joue un rôle premier en diminuant de moitié la dimension du Hamiltonien quadratique sous la forme de deux oscillateurs découplés avec fréquences et amplitudes définies. L'utilisation de cette variable dans le voisinage d'un point fixe fait apparaître la dynamique de la particule dans la représentation de Heisenberg, menant à une association entre les trajectoires hélicoïdales idéales et les trajectoires arbitraires. L'approche reconnaît les deux constantes habituelles du mouvement, l'énergie totale et la composante axiale du moment cinétique canonique. Si la valeur de ce dernier permet l'existence de points fixes, nous y centrons une expansion au deuxième ordre pour le Hamiltonien. Les
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