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 alpha(w)(2)/gamma(0)(2)beta(z0)(2) = 1 with the axial electron velocity Taylor expanded 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.