Statistical mechanics and Vlasov equation allow for a simplified Hamiltonian description of Single-Pass Free Electron Laser saturated dynamics

Abstract

A reduced Hamiltonian formulation to reproduce the saturated regime of a Single Pass Free Electron Laser, around perfect tuning, is here discussed. Asymptotically, Nm particles are found to organize in a dense cluster, that evolves as an individual massive unit. The remaining particles fill the surrounding uniform sea, spanning a finite portion of phase space, approximately delimited by the average momenta ω+ and ω−. These quantities enter the model as external parameters, which can be self-consistently determined within the proposed theoretical framework. To this aim, we make use of a statistical mechanics treatment of the Vlasov equation, that governs the initial amplification process. Simulations of the reduced dynamics are shown to successfully capture the oscillating regime observed within the original N -body picture. I. GENERAL BACKGROUNDFree-Electron Lasers (FELs) are coherent and tunable radiation sources, which differ from conventional lasers in using a relativistic electron beam as their lasing medium, hence the term free-electron.The physical mechanism responsible for the light emission and amplification is the interaction between the relativistic electron beam, a magnetostatic periodic field generated in the undulator and an optical wave copropagating with the electrons. Due to the effect of the magnetic field, the electrons are forced to follow sinusoidal trajectories, thus emitting synchrotron radiation. This spontaneous emission is then amplified along the undulator until the laser effect is reached. Among different schemes, single-pass high-gain FELs are currently attracting growing interest, as they are promising sources of powerful and coherent light in the UV and X ranges. Besides the Self Amplified Spontaneous Emission (SASE) setting [1], seeding schemes may be adopted where a small laser signal is injected at the entrance of the undulator and guides the subsequent amplification process [2]. In the following we shall refer to the latter case. Basic features of the system dynamics are successfully captured by a simple one-dimensional Hamiltonian model [15] introduced by Bonifacio and collaborators in [3]. Remarkably, this simplified formulation applies to other physical systems, provided a formal translation of the variables involved is performed. As an example, focus on kinetic plasma turbulence, e.g. the electron beam-plasma instability. When a weak electron beam is injected into a thermal plasma, electrostatic modes at the plasma frequency (Langmuir modes) are destabilized. The interaction of the Langmuir waves and the electrons constituting the beam can be studied in the framework of a self-consistent Hamiltonian picture [4], formally equivalent to the one in [3]. In a recent paper [5] we established a bridge between these two areas of investigation (FEL and plasma), and exploited the connection to derive a reduced Hamiltonian model to characterize the saturated dynamics of the laser. According to this scenario, N m particles are trapped in the resonance, i.e. experience a bouncing motion...

Enhancement of particle trapping in the wave-particle interaction

“…First, we intend to implement the computed control term in the framework of the original N-body self-consistent picture and explore possible beneficial effects for the evolution of the radiation field. Further, it is planned to apply the above technique to the case of the reduced Hamiltonian of [2,3]. We point out that an experimental test of the control method on a modified Travelling Wave Tube has been done [10] in absence of self-consistency.…”

“…Such reduced formulation consists in only four degrees of freedom, namely the wave, the macro-particle and the two boundaries, delimiting the portion of space occupied by the the uniform halo surrounding the inner bulk [2,3].…”

Equilibrium and out of equilibrium phase transitions in systems with long range interactions and in 2D flows

“…In the following subsection, we present recent studies on the equilibrium statistical mechanics of the Vlasov equation (and not of the N particle dynamics) [61,62,63], in the spirit of LyndenBell's work [1] in the context of self-gravitating stars.…”

Enhancement of Particle Trapping in the Free Electron Laser

“…Such reduced formulation consists in only four degrees of freedom, namely the wave, the macro-particle and the two boundaries, delimiting the portion of space occupied by the the uniform halo surrounding the inner bulk, see Antoniazzi et al (2005) .…”

“…It can be therefore hypothesized that a significant reduction in the intensity fluctuations can be gained by implementing a dedicated control strategy, aimed at confining the macro-particle in space. In addition, the size of the macroparticle is directly related to the bunching parameter, a quantity of paramount importance in FEL context, see Antoniazzi and Fanelli (2005). Smith and Pereira (1978) showed that a test-wave, being characterized by a frequency close to the one of the wave, can effectively destroy the macroparticle and activate a consequent detrapping process.…”