Numerical Experiments on Free-Electron Lasers Without Inversion Yuri Rostovtsev, 1 Simeon Trendaﬁlov, 1 Alexander Artemiev, 1,3,5 Kishore Kapale, 1 Gershon Kurizki, 5 and Marlan O. Scully 1,2,4 1 Department of Physics and Institute for Quantum Studies, Texas A&M University, College Station, Texas 77843-4242, USA 2 Department of Electric Engineering, Texas A&M University, College Station, Texas 77843-4242, USA 3 General Physics Institute Russian Academy of Sciences, 38 Vavilov Street, Moscow, 117942, Russia 4 Max-Planck-Institut fu ¨r Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748, Garching, Germany 5 Chemical Physics Department, Weizmann Institute of Science, Rehovot 76100, Israel (Received 23 February 2003; published 30 May 2003) The inversionless free-electron laser having a drift region consisting of two magnets is analyzed. Performing numerical simulations of electron motion inside wigglers and the drift region, we have shown that this system has a positive mean gain over the entire energy distribution of the electron beam. We study the inﬂuence of emittance and the spread of electron energies on the gain. DOI: 10.1103/PhysRevLett.90.214802 PACS numbers: 41.60.Cr In a free-electron laser (FEL) [1,2], the accelerated motion of electrons in the ‘‘pondermotive potential’’ of the combined ﬁeld of the wiggler and the laser produces coherent stimulated radiation. Under the inﬂuence of the pondermotive potential, a grating in the spatial density of electrons (‘‘bunching’’) on the scale of the laser wave- length is produced. As a result net emission is enhanced. FELs are able to produce radiation in widely different frequency domains: from microwaves [3] to x rays [4]. They are used for spectroscopy, laser surgery, material research, and the production intense x-ray beams. A main limitation on the gain, both for small-gain and large-gain regimes, is set by the spread in the longitudinal momentum of the electrons in the beam. For this reason, much effort has been devoted to producing highly mono- energetic electron beams [5]. Recently, new approaches to increasing the gain in atomic lasers, based on coherence and interference, have been the subject of investigation [6]. This concept has interesting implications for the free-electron laser with- out inversion (FELWI) as well, even though it is a purely classical rather than quantum device [7]. The FELWI is conceptually implemented via interfer- ence of the radiation from two wigglers and an appropri- ate phasing of the electrons in the drift region between the wigglers [8–11] (shown in Fig. 1). Let us note here that an optical klystron also utilizes a two-stage setup [12], which allows one to increase the maximum gain, but the average gain is zero. The important difference between these two concepts is that the drift region of the FELWI and the dispersion region of the optical klystron have signiﬁcantly different phase-shift functions. It was shown that a FELWI requires a correlation between the electron energy change in the ﬁrst wiggler and the transverse electron velocity [13,14]. A noncol- linear FEL geometry provides this correlation and allows for the essentially two-dimensional electron dynamics in the drift region, which is needed for the FELWI phase control [15,16]. It was shown that the FELWI is consistent with Liouville’s theorem as well as the generalized Madey’s theorem [15]. However, it has not been clear from Refs. [13–15] that a realistic drift region can pro- duce the phase shifts to implement FELWI. In this Letter we report the results of numerical experi- ments which have been performed by using a drift region consisting of two bending magnets (defocusing TM 1 and focusing TM 2 ) as shown in Fig. 1. Simulating the motion of electrons in the wigglers and in the drift region, we ﬁnd the gain dependence on the energy of the electrons which is depicted in Fig. 2(a) and represents the main result of the paper. It is clear that the average FELWI gain is positive in contrast with an ordinary FEL [Fig. 2(c)] or an optical klystron (OK) which is obtained by removing the bending magnets from the drift region and adjusting the wigglers in a linear conﬁguration [Fig. 2(b)]. The gain for FEL and OK is positive for some energies and negative for other energies, and the average gain over energy is zero. Let us note here that for the ordinary FEL the electron beam should have higher energy than the reso- nant energy to have gain, and this may be viewed as the inversion condition for the electron beam; in the same sense, the FELWI concept overcomes this condition. We note also that we consider a small-gain small- signal regime, not a high-gain one. Thus, our numerical simulation allows us to conﬁrm the results of previously published work on this subject [14,15]. In particular, we TM TM electron beam Wiggler II drift region Wiggler I 1 2 FIG. 1. A FELWI setup. PHYSICAL REVIEW LETTERS week ending 30 MAY 2003 VOLUME 90, NUMBER 21 214802-1 0031-9007= 03=90(21)=214802(4)$20.00  2003 The American Physical Society 214802-1