Small signal gain based on analytic models of thin intense electron beams

Abstract

We develop the free electron laser theory of the effective energy distribution and the small signal gain for a thin electron beam. The assumption of thinness allows us to treat various transverse locations and electron beam trajectory angles as introducing phase shifts that have the same effect as those introduced by a change in energy of the electron. These ideas extend previous work of Colson et al., Dattoli et al., Scharlemann, and others in five important ways. The first is the ability to treat electron beams with three different classes of matching or symmetry conditions: (i) electron beams with separate betatron matching in each plane. (ii) those with aspect ratio matching, and (iii) crossed matched beams. Manifestations of these symmetries include elliptical cross-sections and electron beams that have modulated spatial profiles. For these we derive analytical expressions for effective energy distributions. Second, two emittance parameters for the electron beam are shown to consolidate into a single parameter that describes most of the energy variation of the effective energy distributions. Thus, the effective energy distribution for a 1:4 ribbon electron beam is nearly equivalent to a distribution for a beam of circular cross-section. Third, these calculations extend to energy distributions, angular distributions, and spatial distributions that all follow Gaussian profiles. Fourth, this model incorporates the description of the incident Gaussian optical beam and the above electron beam dynamics into a single influence function kernel. Emittance, energy spread, diffraction, and gain may be interpreted as limiting the length over which the bunching contributions of the propagating electric fields downstream are important. Fifth, three-dimensional profiles of the optical fields are computed. This work is complementary to the recent work of Yu, Krinsky and Gluckstern in that ours always describes the transition from low gain to high gain for a thin beam and not necessarily the high-gain regime itself. Therefore in this work the parameters of the incident optical beam are included, whereas, their work is not concerned with these parameters. The resulting transverse dependence of the fields may be characterized by an optical beam radius. This optical beam width starts out large compared to the thin electron beam and then, in the example given, contracts to a size that becomes so small that the thin-beam assumption is violated.