Resonance between betatron and synchrotron oscillations in a free electron laser: A 3-D numerical study

We analyze the effect of betatron and synchrotron resonance on the gain process of the free electron laser (FEL) including betatron oscillations, finite beam emittance, self-consistent electron dynamics and a 3-D radiation field. Betatron oscillations force the electrons to be affected by an oscillatory radiation amplitude and phase in a Gaussian resonator radiation field. We show that not only can the radiation phase variation cause a resonance between approximately twice the betatron wavenumber and the synchrotron wavenumber, but the radiation amplitude variation in an FEL with efficiency enhancement schemes or just a tapered magnetic wiggler field amplitude Bw can also have a similar resonance effect. Numerical results for a 10 micrometer FEL are given. We find that for short wigglers, the effect of the betatron-synchrotron instability on the gain is negligible for Gaussian electron beams with radius less than 0.8 times the radiation beam spot size.