PHYSICAL REVIEW A VOLUME 42, NUMBER 12 Nonlinear kinetic theory of the free-electron laser 15 DECEMBER 1990 R. Pratap* and A. Sen Institute for Plasma Research, Bhat, Gandhi nagar 382 424, India (Received 6 July 1990) A kinetic analysis of the nonlinear evolution of the free-electron-laser (FEL) instability is present- ed. The governing equations are the coupled Vlasov-Maxwell equations, which are investigated for a system consisting of a relativistic electron beam propagating through a helical wiggler magnetic field. Assuming that a single cavity mode of the electromagnetic field takes part in the lasing, a gen- eral nonlinear solution of the Vlasov equation is obtained in the resolvent formalism. Use of this solution in the wave equation provides a nonlinear description of the FEL. The saturation proper- ties of the FEL are discussed by numerical and analytical solutions of this equation. I. INTRODUCTION The nonlinear evolution of the free-electron-laser (FEL) instability is a subject of considerable interest, par- ticularly the study of saturation properties. ' ' While linear theory is adequate to describe the exponential gain regime, the saturation regime is difficult to model analyti- cally, and it is customary to rely on the help of computer simulations. In this paper we develop an analytical mod- el for the nonlinear regime, which is based on the resol- vent formalism developed by Prigogine and his co- workers' within the framework of general statistical mechanics of nonequilibrium processes. The principal advantage of the method lies in its being a nonperturba- tive approach and in that it permits a general formulation of the nonlinear evolution problem in the strong-signal regime. For the example studied by us, that of a low- density relativistic electron beam propagating through a helical wiggler magnetic field to amplify a single cavity mode, the nonlinear evolution equation is quite compact and can be solved either numerically or by analytical techniques in some limiting situations. Another unique feature of the method, for the problem at hand, is that it allows an exact solution of the Vlasov equation for an ar- bitrary amplitude of the signal strength. Basically this is achieved by formally writing the solution of the Vlasov equation as an inhomogeneous Volterra equation and em- ploying an iterative method to obtain an infinite-series solution. This series can be closely approximated by a geometric series and summed exactly. This solution is used in the wave equation to obtain a nonlinear evolution equation for the FEL. This equation can be further gen- eralized to include dielectric effects (for Cerenkov radia- tion problems), self-fields, and other wiggler geometries. For the small-signal limit it easily reduces to the standard evolution equation discussed in the literature. We dis- cuss the saturation properties of the FEL by analyzing the evolution equation for a simple Gaussian form of the initial beam distribution function. The paper is organized as follows. Section II describes the basic equations and the physical model. The coupled Vlasov-Maxwell equations are reduced to one- dimensional forms under some standard approximations. In Sec. III we obtain a nonlinear solution to the Vlasov equation in the resolvent formulation. This solution is used in Sec. IV to obtain the final nonlinear evolution equation for the laser wave amplitude, and this equation is solved both numerically and analytically. Saturation properties are discussed. Conclusions and summary dis- cussions are given in Sec. V. II. BASIC EQUATIONS The dynamical system consists of a beam of relativistic electrons traveling through a spatially periodic static magnetic field. We assume that the electron density is sufficiently low that self-fields as well electrostatic efFects (representative of collective effects) can be neglected. The relativistic Vlasov equation governing the dynamics of the system is given by aa af aa af at ap ax ax ap where the Hamiltonian for the single electron is given by 2 1/2 mc+c P— eA (2 C m and e being the mass and charge of the electron, P the canonical momentum, and A the vector potential. A can be written in terms of a sum of plane waves representing the laser field and the static wiggler magnet- ic field. We shall assume that only one mode takes part in the lasing. Further, choosing a helical wiggler field (right circularly polarized), the total field in the cavity can be written as A=&2e„[ A;cos(k, z+co, t)+ A, cos(k, z co, t)]— + &2e [ — A;sin(k;z+co; t)+ A, sin(k, z co, t )], — (3) where e, e are unit polarization vectors. This form is known in the literature as the Williams-Weizsacker ap- proximation, ' in which the static magnetic field is re- 42 7395 1990 The American Physical Society