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Volume 61 number 1 OPTICS C O M M U N I C A T I O N S 1 January 1987 H A M I L T O N I A N M O D E L OF A FREE ELECTRON LASER Rodolfo B O N I F A C I O 1 Federico C A S A G R A N D E Dipartirnento di Fisica dell Universitb Via Celoria 16 20133 Milan Italy and Claudio P E L L E G R I N I Brookhaven National Laboratory Upton N Y 11973 USA Received 3 September 1986 Both the C o m p t o n and the R a m a n regimes of a free electron laser are described by a relativistic hamiltonian which originates the evolution equations for 2N 2 canonically conjugate electron and field variables with the space coordinate as the independent variable Space charge and field contribution to electron transverse velocity are included Scaled variables are introduced which allow for a description of the behaviour of the system in terms of a single electron beam parameter The free electron laser FEL is potentially an ideal tool for basic and applied research as a powerful source of tunable coherent radiation generated via the injection of relativistic electrons in an undulator 1 The coupled dynamics of the electrons and the radiation field is such that under proper conditions the radiation emitted by the accelerated particles can grow exponentially along the undulator This high gain regime is due to a collective instability of the system 2 5 and has been recently demonstrated both in the single pass amplifier configuration 6 which we shall consider here and even in the oscillator mode of operation 7 The classical hamiltonian approach has been successfully used to describe the FEL process since the early days of single electron small signal treatments 8 We have applied this approach to the investigation of collective effects in the high gain regime In particular a many electron hamiltonian model for FEL amplifiers was discussed in ref 3 in the limit of low density and negligible space charge Then in refs 4 5 FEL dynamics was discussed dropping this restriction In these papers some terms due to

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