The general velocity and current modulation linear transfer matrix of FEL and control over SASE power in the collective regime Egor Dyunin, Avraham Gover à School of Electrical Engineering, Wolfson Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel article info Available online 10 May 2008 Keywords: Shot-noise SASE FEL Transfer matrix abstract We present the general linear transfer matrix of FEL in the collective regime in terms of a single-mode radiation field amplitude and e-beam current and velocity modulation parameters. The formulation is useful to account for composite configurations of the FEL, including non-radiating sections, and is employed to demonstrate the possibility to control the SASE radiation power (including its substantial reduction for facilitating coherent emission with seed radiation amplification) by varying the space- charge parameter in a drift section. & 2008 Elsevier B.V. All rights reserved. 1. Introduction A general description of the FEL as a linear response problem is possible after modal expansion of the coupled Maxwell-plasma equations [1–5]. In the one-dimensional limit and in the case of single radiation and plasma-mode excitation, it is possible to obtain explicit analytical expressions for the transfer matrix in the frequency regime in terms of the small signal variables of the single-mode FEL problem: the radiation-mode amplitude ~ a q ðzÞ, the current density modulation ~ J z ðzÞ, and the beam velocity modulation ~ V ðzÞ. Our approach thus relates to the general FEL start-up problem, studied in numerous other works [6–8]. For useful applications we calculate explicit expressions for all matrix parameters, including the excitation parameters of the beam current and velocity modulation, and also specify operating in the practical cold-beam regime [9]. The velocity modulation noise parameters, which are often ignored, are correlated to the beam current modulation through the Poison equation. We contend that their inclusion in the transfer matrix is important for proper description of the noise amplification process in FEL when collective plasma effects in the wiggler or in other beam transport sections are non-negligible. The derivation of the explicit FEL linear response matrix, presented in this article, would be useful for characterization of coherent and incoherent radiation in various FEL configurations in all gain regimes. In particular, it enables to explore the effect of beam transport sections before and along the wiggler (accelera- tion, drift-free and dispersive sections). 2. General transfer matrix formulation Under the small-signal assumption we can express all para- meters of the electron fluid plasma equations as the sum of a time-averaged part and a time-varying part—whose amplitude is much smaller than the time-averaged part. In conformity with the use of a small signal model, we neglect all cross-products of two time-varying parameters (producing time-independent and sec- ond harmonic quantities). Using such approximations, we can write all quantities as a set of linear equations: nðr; tÞ¼ n 0 þ 1 2 ð ~ NðrÞe iot þ c:cÞ Vðr; tÞ¼ V 0 þ 1 2 ð ~ VðrÞe iot þ c:cÞ jðr; tÞ¼enðr; tÞVðr; tÞ¼ J 0 þ 1 2 ð ~ JðrÞe iot þ c:cÞ Eðr; tÞ¼ E 0 ðrÞþ 1 2 ð ~ EðrÞe iot þ c:cÞ where n(r,t), V(r,t), j(r,t), and E(r ,t) are the beam density, the beam velocity, the beam current, and electric field, respectively. It is convenient to use the frequency domain in order to find the radiative emission from devices employing e-beams. We use a formulation in which the traveling wave spectral radiation fields are expanded in terms of a complete set of transverse modes q (the beam propagation is entirely in the z- direction): E rad ðz; tÞ¼ Re X q ~ a q ðz; oÞ ~ E q? ðr ? Þe iot " # ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/nima Nuclear Instruments and Methods in Physics Research A 0168-9002/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2008.04.062 à Corresponding author. E-mail address: gover@eng.tau.ac.il (A. Gover). Nuclear Instruments and Methods in Physics Research A 593 (2008) 49– 52