Volume 61, number 1 OPTICS COMMUNICATIONS 1 January 1987 HAMILTONIAN MODEL OF A FREE ELECTRON LASER Rodolfo BONIFACIO 1, Federico CASAGRANDE Dipartirnento di Fisica dell'Universitb, Via Celoria 16, 20133 Milan, Italy and Claudio PELLEGRINI Brookhaven National Laboratory, Upton, NY 11973, USA Received 3 September 1986 Both the Compton and the Raman regimes of a free electron laser are described by a relativistic hamiltonian which orig- inates 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 dy- namics of the electrons and the radiation field is such that under proper conditions the radiation emitted by the ac- celerated particles can grow exponentially along the undulator. This high-gain regime is due to a collective instabil- ity 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 ef- fects 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 the radiation field contribution to electron transverse velocity were neglected ;though small in many cases, they can become important for very long or tapered undulators. In this paper we present a set of evolution equations, valid both in the Compton and in the Raman regimes, which is more general than that of refs. [4,5] and preserves the hamiltonian structure of the system, so that energy con- servation and LiouviUe theorem in a (2N+ 2)-dimensional phase space are valid. A suitable scaling of variables al- lows for a description of FEL dynamics only in terms of one parameter. The hamiltonian equations with time as the independent variable can be derived from the modified Hamilton principle [9 ] t2 5 f (PxdX/dt + py dy/dt + Pz dz/dt - H) dt = O. (1) t 1 Since in our problem one follows the evolution of the system along the undulator axis z, we change the indepen- dent variable from t to z, and using H = E we obtain [10] i Also Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Milan, Italy. 0 030-401/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 55Volume 61, number 1 OPTICS COMMUNICATIONS 1 January 1987 z 2 6 f (PxdX/dz +pydy/dz - Edt/dz +Pz) dz = O. (2) 21 In eq. (2) (x, Px), Cv, py) and (t, -E) appear as canonical variables with respect to a new hamiltonian H 1 =- pz. Hence we can write dx/dz = - ~pz/~px, dy/dz = - ~pz/~p),, dPx/dz = ~pz/~X, dP.v/dz = ~pz/O)', (3) dt/dz = Opz/OE, dE/dz = - Opz/Ot. 14) Eqs. (4) are our working equations. Now let H be the relativistic hamiltonian for one electron interacting with elec- tromagnetic fields H=c {[p- (e/e)A]2 +m2c2}1/2 +eV = 7me 2 + eV-K 1St In eq. (5) we assume that the vector potential A = A (z) is transverse and V = V(z) represents space-charge effects due to density fluctuations in an electron beam. Hence eqs. (4) become dt/dz = ( l /mc 2) ~pz/~7, ~ (~ ) dT/dz = - (1/mc2)(Opz/Ot- eEz), E z = - dV/dz, (6') while from eq. (3) it follows that px =py -- const., so that one can setPx =py = 0 thus obtaining from eq. ~5) Pz =mc(72 - l-a2) 1/2, a=-eA/mc2" (7) By assuming that 72 >> 1 + a 2 we have Pz = mc [7- (1 +a2)/27]. (7') Now we specify the dimensionless vector potential a(z, t) as the sum of a magnetostatic, spatially periodic undula- tot potential a0(z ) and a radiation field potential aL(z , t): a =a 0 +aL, a 0 = (ao/X/-2) [0 exp(-ikoz ) + c.c.], a L =- (i/%/'2) {Oa L exp[i(kLZ - COLt)] -- c.c.}, (8) where k 0 = 27r/X 0 = COo/e is the wavenumber associated with the undulator periodicity, k L = 27r/k L = COL/C is the radiation wavenumber, and circular polarization is taken for a helical undulator. Thus eq. (7') becomes Pz = me {7 - (1/2")') [1 +ao 2 + ia 0 (a[ exp(-i0) - c.c.) + laLI2]}, (9) where 0 is the electron-field phase 0 = (k L + ko)z - COLt. (9') As dO/dz = k L + k 0 - coL dt/dz, the evolution equations for the variables 0 and 7 can be obtained at once from (6), (6') and (9); or, alternatively, via a canonical transformation from (t, -E) to (0, 7), as well known in accelera- tor physics [10]. Without any further approximation we get dOf/dz = ko(1 - 72/72) + (kL/2T?) [iao(a L exp(iO/) -- c.c.) -- [aLl2], (10) dTj/dz = - k L ['} a0(a L exp(iOj)/Tj + c.c.) + i(cop/coL) 2 ((exp(--i0)) exp(i0/) -- c.c.)], ( 11 ) where COp = (47reZn/m) 1/2 is the plasma frequency, n =N/V the electron number density, 7R = [COL( 1 + a~)/2 COO] 1/ the resonance energy (at zero initial field) in rest energy units, (exp(i0)) =N -1EN1 exp(-i0/) the electron 56Volume 61, number 1 OPTICS COMMUNICATIONS 1 January 1987 bunching parameter. In eqs. (10), (11) we have added an index4 to distinguish different electrons in a beam. Fur- thermore, the space-charge contribution to dT//dz is obtained from eq. (6') using the result of ref. [5] for the first harmonic contribution to E z , (Ez) / = - (i azren/kL ) [(exp(--i0)) exp(i0j) -- c.c.]. The field equation can be derived as in ref. [4] from Maxwell equations by neglecting slippage effects, and reads daL/dZ = (~2/2C~L) (a 0 (exp(--i0)/7) -- i(1/7)aL), (12) where again any bracket ( ) means an average N -1 GN 1 . The last term in eq. (12), which is usually neglected, comes from the radiation field contribution to electron transverse velocity, just like the terms depending on the field in eq. (10). However, we stress that it is not