Self-consistent undulator radiation via Lienard-Wiechert fields

Abstract Using Lienard-Wiechert fields and the Lorentz Force relation we present self consistent three-dimensional radiationstudies of electron beams moving through periodic electromagnetic structures such as those present in synchrotronsand free-electron laser undulators. Besides providing an economical means of calculating three dimensional vectorradiation fields, our approach yields new insights into individual electron motion as it is driven by both, the velocityfields (Coulomb Fields) and the radiation fields generated by other electrons. We present results ofelectron beam com-pression resulting from longitudinal radiation forces competing in opposition with repulsive velocity field forces. Wediscuss results ofnoiseless three dimensional SelfAmplified Spontaneous Emission (SASE) in the X-Ray region re-sulting from the interaction of a filamentary electron beam with a circularly polarized magnetic undulator. Introduction The usual technique used to study the free-electron laser amplification process entails solving a paraxial version ofthe wave equation at one or more signal frequencies [1,2,3,4]. Although this approach has been quite successful in ex-plaining most features of the free-electron laser (FEL) stimulated emission process, by the nature of its approxima-tions, it has some limitations. For example one disadvantage of working with a paraxial wave equation is that thedescription of the radiation field has incomplete three-dimensional features and consequently only waves having sim-pie spatial structure can be used to solve the FEL problem. Also, working in the frequency domain yields satisfactoryresults only when a few longitudinal modes are needed to deal with the problem. Thus it is impractical to use the fre-quency domain approach when very short electron beam, and consequently short optical pulses, are considered. Per-haps the greatest limitation of present FEL theories is that a solution of the homogeneous wave equation is alwayspresent at the beginning ofthe FEL interaction process. That is, an input signal is introduced so that the so called elec-tron ponderomotive phase is a well defmed quantity. Although this assumption is needed to study the FEL wave am-plification process (stimulated emission) it does not provide sufficient flexibility to study the FEL start-up process.The limitation ofthese theories is even more evident when applied to study coherent radiation effects that occur in self-amplified spontaneous emission (SASE) processes.In this paper we present the derivation of equations of motion for electrons and fields based on Lienard-Wiechertfields, which are exact, three-dimensional solutions of the wave equation in free-space and, as it is done in all othertheories, the electron's motion is governed by the relativistic Newton's Second Law in which the driving force is theLorentz force. The Lorentz force includes fields generated by all electrons in the beam as well as any arbitrary exter-nally prescribed fields. To be more specific we deal with the self-consistent radiation problem of an electron beambeing accelerated by periodic electromagnetic structures (undulators and wigglers), such as those used in synchrotronsand free-electron lasers (FELs). Unlike the restricted solutions of Maxwell's equations obtained by others, theLienard-Wiechert fields are exact, time and space domain free-space solutions of Maxwell's equations for a pointcharge. The attributes ofthese particular solutions allows us to incorporate in a natural way the three dimensional ef-fects of internal forces existing within the electron beam and as a result we can explore easily the three dimensionalnature ofthe radiated fields. This approach becomes particularly useful in dealing with the FEL start-up problem aswell as when studying non-stimulated coherent radiation effects. Unlike the approach used by other, our scheme re-quires no initial artificial electromagnetic seed to start the numerical solution of the problem. Furthermore becauseLienard-Wiechert fields are time-domain solutions of the wave equation, we can study non-periodic electron beam sys-tems. In particular we can deal satisfactorily with three-dimensional effects of very long and very short electron bunch-es. As an example of the power of our approach we present results of three dimensional radiation features of self-490/SPIE Vol. 2522