A unified analysis of spontaneous and super-radiant emissions in free-electron lasers

A generalized formulation of spontaneous emission and super-radiance effects in a free-electron laser is presented. We consider a stream of electrons of arbitrary temporal duration propagating through the undulator. The sum of the undulator synchrotron radiation emitted by individual wiggling electrons entering the wiggler at random, results in shot-noise in the radiation field. Using the waveguide excitation equations formulated in the frequency domain, an analytical expression for the power spectral density of the electromagnetic radiation is derived. It is shown that for a finite pulse electron beam current, the spectrum of the excited radiation is composed of two terms which are the spontaneous and super-radiant emissions. For an infinitely long e-beam pulse (continuous beam), the shot-noise produces only incoherent spontaneous emission. The power of this radiation is proportional to the DC current I,, of the electron beam. For shorter e-beam pulses, a partially coherent super-radiant emission is also produced with an average power which is proportional to 16. The coherence of this super-radiant emission is enhanced as the pulse duration is reduced. A single formulation describes the coherent features of the super-radiance and the statistical features of the spontaneous emission.