Strong wiggler field assisted amplification in a second-harmonic waveguide free electron laser

As a technique to reduce the size of compact waveguide free electron lasers (FELs) operated from microwave to the far infrared, a longitudinal interaction mechanism was recently proposed to operate waveguide FELs at the second harmonic. With a gain formulation based on Madey´s theorem in the limit of small wiggler field, it was shown analytically that second harmonic waveguide FELs can reduce significantly the electron energy required for radiation at a given frequency. As it is advantageous to operate second harmonic waveguide FELs with strong wiggler field, Madey´s theorem is used here to reformulate their interaction gain for strong wiggler fields up to a_ω²/γ₀²β _z0²=1with the axial electron velocity Taylor expanded to the eighth order of the wiggler field. Given that Madey´s theorem has not been established for second harmonic waveguide FELs, their interaction gain is also formulated independently by solving their pendulum equation without recourse to Madey´s theorem. These two gain formulas are not analytically identical, but numerically they lead to an excellent agreement over a wide range of system parameters, thus confirming the applicability of Madey´s theorem. The interaction analyses presented form a thorough and detailed description of second harmonic waveguide FELs in the small-signal regime and for wiggler field that is both practical and beneficial