3-D EFFECTS ON SMALL SIGNAL GAIN FOR UNDULATORS WITH VARIABLE PARAMETERS

A new explanation on the quadrature expression for small signal gain(SSG) in a variable parameter undulator is given in terms of Madey's theorem and analogy between the FEL and a phased antenna array and the spontaneous emission spectrum is derived. The generalization of the SSG expression from one to three dimensions is carried out and the effect on SSG due to the collimation error among the axes of the optical,electron beam and undulator is included as well. It shows at three dimensional effects will reduce SSG severely. The comparison with Boeing's FEL experiment is carried out and it is also pointed out that a better operation condition may be obtained if the first uniform section of the Boeing's undulator is reduced a little to make the optimum energy mismatch shifted towards zero.