Nuclear Instruments and Methods in Physics Research A250 (1986) 369-372 369 North-Holland, Amsterdam NONLINEAR TRAVELING WAVES IN A FREE ELECTRON LASER * R.C. DAVIDSON, G.L. JOHNSTON and A. SEN ** Plasma Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA We investigate nonlinear electromagnetic waves in a system which consists of an intense, cold-fluid relativistic electron beam propagating in a helical magnetic field. By transforming to wiggler coordinates and introducing the traveling wave ansatz, we have obtained a set of three coupled nonlinear ordinary differential equations plus an exact invariant to describe wave motions in the system. In the Compton-approximation limit, the number of differential equations reduces to two. These equations are particularly useful for studying nonlinear saturated states of the free electron laser instability, corresponding to values of the dimensionless phase speed (fl = u/c) that are less than the dimensionless beam speed (fib = Vo/C) or less than unity. This restriction on fl may permit the existence of solitons for discrete values of/3. 1. Introduction Nonlinear saturation of free electron lasers is a subject of considerable interest. We report progress in the development of one model of nonlinear behavior of the free electron laser instability. A relativistic cold-fluid beam of uniform transverse extent propagates axially through an ideal helical wiggler magnetic field. The usual set of four coupled nonlinear partial differential equations for one-di- mensional disturbances in the system is derived. Introduction of the helical transformation and the traveling wave ansatz, in which all dependent variables are assumed to be functions of ~ = z - ut only, plus further algebraic manipulation permit the development of a set of traveling wave equations. This is a set of three coupled nonlinear ordinary differential equations for Pl, 02, and Z (= tiP3- Y) as functions of = kwh. Here, Pl and P2 are the dimensionless components of transverse total mechanical momentum, P3 is the longitudinal component of total mechanical momentum, and ~, [= (1 + p2)1/2] is the dimensionless energy. The Compton-approximation limit is obtained by excluding the longitudinal equation and setting Z = Z 0, the equilibrium (d/d~ = 0) value of Z, in the transverse equations. The full set of traveling wave equations, and the equations in the Compton-approximation limit, both possess exact invariants. Small-signal equations are obtained by linearizing in 8Pl, ~P2, and 8Z, the differences between perturbed and unperturbed quantities. These equations yield traveling wave dispersion relations, which are functions of complex k = k/k w and real ft. They are to be contrasted with the usual normal-mode dispersion relations, which are functions of ~ = ~o/ck w and k, one of which may be complex. Formal correspondence between traveling-wave dispersion relations and familiar normal-mode dispersion relations for FELs is established by the replacement fl ~ &/~:. The Raman approximation to the traveling-wave dispersion relation exhibits a resonant-coupling instability analogous to the normal-mode Raman-regime FEL instability. In particular, fl < fib ( < 1) is required for instability. The Compton-approximation limit of the traveling-wave dispersion relation does not exhibit an instability which corresponds to the usual cube-root instability of the Compton-regime normal mode dispersion relation. But fl < 1 is required for instability in the Compton approximation. Nonlinear wave equations are often associated with saturated states of linear instabilities. Therefore, the results of the small-signal analysis indicate that fl < fib or fl < 1 is required for solutions of the traveling wave equations that correspond to saturated states of the FEL instability. A possible consequence of these * Research supported by the Office of Naval Research. ** Permanent address: Plasma Physics Programme, PRL, Ahmedabad, India. 0168-9002/86/$03.50 © Elsevier Science Publishers B.V. IX. RAMAN/GUIDE FIELD FELS (North-Holland Physics Publishing Division)