Nonlinear Theory: Guided Mode Analysis

This chapter describes the development of slowly varying envelope approximation (SVEA) formulations in the steady-state regime, as well as the application of the analyses to the description of the fundamental physics of the nonlinear saturation mechanism. The SVEA was originally developed for the treatment of continuous wave (CW) traveling wave tubes and has been adapted to the treatment of long-pulse, long-wavelength free-electron lasers or free-electron masers at millimeter wavelengths or longer that are driven by relatively low-energy accelerators such as pulse-line accelerators, modulators, and induction linear accelerators. These accelerators produce relatively low-energy but long-pulse electron beams that are used primarily to generate relatively long wavelengths, which are comparable to the transverse dimensions of the drift tube. As a consequence, the description of these free-electron lasers requires a guided mode analysis. The origins of the SVEA formalism date to the early development of traveling wave tubes for radar applications, where it was often referred to as a “moving window” model because it treats the co-propagation of a “beamlet” of electrons that is one wavelength long with the corresponding electromagnetic field. These tubes employed thermionic cathodes that operated in continuous mode with the corresponding injection of a low-power continuous seed. The structures used in traveling wave tubes produce a sub-luminous wave that can interact directly with the electron beam and induce axial bunching. It was realized early in the development of free-electron lasers that the ponderomotive wave in a free-electron laser resulted in a similar axial bunching mechanism, and a free-electron laser might be thought of as a variant on a traveling wave tube. The steady-state formulations used to model free-electron lasers bear many similarities to these “moving-window” models and were originally developed to treat long-wavelength free-electron lasers that employed long-pulse accelerators. It is important to remark that this formulation is based on the propagation of a single wave frequency; hence, it is applicable for narrow bandwidth interactions. As a result, a single-frequency approximation gives a good approximation to the wave-particle dynamics, and the electromagnetic field is represented using amplitude that varied slowly with position and the fast time scale was removed by averaging Maxwell’s equations; hence, the slowly-varying envelope eapproximation (SVEA). As explained in this chapter, the SVEA formalism is based upon a quasi-static assumption that postulates that particle trajectories in the presence of the radiation field are periodic and match the periodicity of the resonant wave. The resulting formalism integrates the dynamical equations for both the fields and particles in z, the direction of propagation, and represents a Lagrangian formulation of the interaction. This contrasts with the Eulerian formalism used in particle-in-cell simulation models. The distinction between the Lagrangian and Eulerian formalisms is described. The particle dynamics is treated by integration of the full three-dimensional Lorentz force equations, with no average imposed. As such, the step size for integration must resolve the orbital undulations experienced by the electrons as they traverse the wiggler. This requires the integrator to take 20–30 steps through each wiggler period and permits the simulation of the entrance and exit tapers in the wiggler and includes all harmonic components to the motion. As a result, both linear and nonlinear harmonic generations are implicitly included in the formulation (see Chap. 8). Further advantages to this approach accrue when treating short-wavelength free-electron lasers that employ radio frequency linear accelerators and long sequences of relatively short wigglers with quadrupoles located between the wigglers to provide strong focusing. This will be discussed further in Chap. 6. Comparisons with experiments using both helical and planar wigglers that validate the formulation are discussed. In particular, this includes an early SASE experiment at the Massachusetts Institute of Technology which used a helical wiggler at near THz frequencies and a series of MOPA experiments at Lawrence Livermore National Laboratory at millimeter wavelengths using a two-plane focusing planar wiggler that demonstrated an efficiency of 35% using a tapered wiggler.