Investigation of the gain regimes and gain parameters of the free electron laser dispersion equation

We compute the small signal gain curve and various gain parameters by solving numerically the generalized gain-dispersion equation of free electron lasers (FEL), which characterizes the conventional magnetic bremsstrahlung FEL, as well as a large number of other FEL devices. The model includes high gain, collective, and axial velocity spread effects, and some waveguide effects. The FEL gain regimes are investigated and presented in terms of only three universal FEL characteristic parameters. The approximative analytic gain expressions are compared to the numerical computation results, and the approximation error is computed and displayed. In the intermediate regimes (high-low gain, tenuous-collective beam, cold-warm beam), the gain parameters are given in terms of useful curves, and a heuristic approximative formula is suggested for estimating the axial velocity spread gain reduction factor in all gain regimes. We also define and compute gain bandwidth and beam quality acceptance parameters in all gain regimes.