A multimode nonlinear particle simulation code is used to find the saturated efficiency for power transfer into modes of a cylindrical waveguide carrying a spatiotemporally modulated gyrating electron beam. For a TEsj-mode fifth harmonic 94-GHz harmonic converter using a 150-kV, 6.7-A cold beam, this code predicts a conversion efficiency of 57% when a linearly tapered guide magnetic field is used, and 70% when a nonlinear taper is used. Efficiency in the linear taper case is shown to be insensitive to beam axial velocity spread, while both cases show negligible power flow into competing modes.PACS numbers: 41.60.Ap, 52.75.Ms Efficient, high-power rf sources are in demand for many applications, such as drivers for next generation electron-positron colliders, as sources for fusion plasma heating and control, and as amplifiers for advanced mmwave radar systems. Extensions of proven technologies are being undertaken to meet these demands. Thus advanced klystrons [l] and gyroklystrons [2] show promise as 50-MW sources above 10 GHz for driving next generation colliders, cavity gyrotrons [3] have generated 940 kW at 140 GHz for plasma heating, and 35-GHz gyrotron traveling-wave amplifiers have shown 32% bandwidth and 30% efficiency at output levels of 25 kW [4]. This Letter reports first results on the nonlinear theoretical properties of a recently proposed [5] alternative mechanism to satisfy these demands, namely, gyroharmonic radiation from a spatiotemporally modulated gyrating electron beam. Prior to the work reported here, it was speculated that harmonic conversion based on this process could be highly efficient, that radiation into competing modes could be small, and that a moderate axial velocity spread on the electron beam could be tolerated [5,6]. Re-
Hirshfield and Park Reply:We are grateful for the opportunity Larson [1] provides for us to clarify our published Letter. As shall be shown, there is no violation of the second law of thermodynamics implied by our results, since the total phase space for the system does not decrease.Equation (1) of our Letter, from which shrinkage in the energy spread of an electron beam was calculated, was derived from the Vlasov equation, where the electrons were subjected to assigned cavity fields. For noninteracting particles, it is well known [2] that the quantity f(p,r,t)d 3 pd 3 r is conserved, where /(p,r,f) is the timedependent distribution function in the sixfold momentum (p) and configuration (r) space. For an electron beam, this phase-space volume is related to the emittance. The entropy -A:J"^/ 3 /7^/ 3 ry*(p,r,^ )ln[y(p,r,/)] can be shown to be constant in this case as well, although it is customary in beam physics applications to deal with the phase-space volume directly.Larson has suggested that somehow, in discussing the phase-averaged energizing or deenergizing of groups of electrons with energies lower or higher than a mean value, we have obscured what is in fact a broadening in the energy distribution. It is indeed true that a particle's initial phase with respect to the external field will determine whether or not it initially experiences an increase or decrease in its energy. A detailed examination of the phase-dependent dynamical equations reveals that most particles with energy higher than the mean experience a decrease in energy, while most with energy lower experience an increase [3]. However, when the particle energies are distributed narrowly about the mean energy, the energies will eventually all coalesce to the mean. Near the discrete energies identified in our Letter, the rate of coalescence was predicted to be more rapid than for other energies. We (perhaps intemperately) labeled this coalescence process "cooling" in our Letter, but the quotes were prominently displayed.So how is the phase space conserved? Clearly, while the momentum part is shrinking, the configuration part is swelling. This phenomenon has been exhibited for single-particle orbits by Humphries [4], and shown in parti-
A formulation to characterize stimulated emission of synchrotron radiation is presented, based on simplification of a theory developed by Sokolov and Ternov. It is shown that optical gain may be obtained up to gyration harmonics of the order of the critical harmonic number for spontaneous radiation, 3y\ where / is the relativistic energy factor. Synchrotron-radiation lasers (SRLs) in the infrared and visible portions of the spectrum are shown to be feasible, provided beams with high energy definition are used. SRL gain can exceed free-electron-laser gain for similar beams.PACS numbers: 42.55. Tb, 41.70.+t, 41.80.Ee, 52.75.Ms Recently, convincing experimental evidence [1] showed that strong cooperative effects in the spontaneous synchrotron radiation of gyrating electrons will prevail when the electrons are localized within a spatial bunch that is smaller than the radiation wavelength. As a result, the intensity of the radiation observed was nearly N times that for a single electron, where N is the number of electrons in a bunch. Cooperative radiation effects among electrons which are not localized are described by the phenomena of stimulated emission and absorption, wherein an imposed monochromatic radiation field can excite randomly and widely spaced electrons to either absorb or emit radiation in phase coherence with the imposed radiation. Considerable interest exists in stimulated emission processes involving free electrons, as in gyrotrons [2] and free-electron lasers (FELs) [3] for the generation and amplification of radiation at wavelengths from millimeters to the vacuum ultraviolet.This Letter describes the conditions for obtaining optical gain through stimulated emission of synchrotron radiation from a beam of gyrating electrons in a strong magnetic field. Of course, at the fundamental and first few cyclotron harmonics, such a mechanism has been long
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