Calculation of 3-D free electron laser gain: Comparison with simulation and generalization to elliptical cross section

In the previous paper, we have derived a dispersion relation for the free electron laser (FEL) gain in the exponential regime taking account the diffraction and electron's betatron oscillation. Here, we compare the growth rates obtained by solving the dispersion relation with those obtained by simulation calculation for the waterbag and the Gaussian models for the electron's transverse phase space distribution. The agreement is found to be good except for the limiting case where the Rayleigh length is much longer than the gain length (1-D limit). We also generalize the analysis to the case where the electron beam cross section is elliptical as is usually the case in storage rings, and derive the first-order dispersion relation