FREE ELECTRON LASER OSCILLATOR STARTUP ANALYSIS

Abstract
An analysis of the free electron laser (FEL) oscillator startup problem is presented. The model is spatially one-dimensional though many important three-dimensional effects are heuristically included. The electron beam consists of pulses of arbitrary shape and are mono-energetic with no spread in either the longitudinal or transverse velocities. The wiggler parameters are taken to be fixed. Finally the analysis is carried out in the low gain, small signal regime. The radiation field within the resonator is represented by a superposition of spatial modes which are such that the tangential electric field vanishes on the mirrors. The vector potential of the linearly polarized radiation field is written as [MATH] where kn = ωn/c = π_n/L, L is the separation between the mirrors, a_n (t) is the Fourier coefficient of the nth mode and c.c. denotes the complex conjugate. For the oscillator startup, the quantity of real interest is the energy density matrix [MATH] defined by the elements where ε_nm (t) = k_n² < a_n (t) a_m* (t) >, where the bracket < > denotes the ensemble average of the enclosed quantity. The energy density matrix [MATH] for the Nth pass is found to be [MATH] where ω_L is the characteristic laser frequency, the Q factor represents the resonator losses, [MATH] is the source matrix representing the spontaneous (incoherent) radiation, [MATH] is the gain matrix representing the stimulated (coherent) radiation, [MATH] is the Hermitian conjugate of the gain matrix, [MATH] ≤ L_W/v_zo, L_W is the length of the wiggler, v_zo is the axial velocity within the wiggler, and t_N is the time the Nth electron pulse entered the wiggler. Numerical results are presented using the parameters of the FEL oscillator experiment at Stanford University and LASL.