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Electron Bunching in Split-Cavity Monotrons Joaquim J. Barroso
Abstract—The twofold enhancement in the conversion efficiency of a monotron operating in the π mode—instead of the longitudinally uniform TM010 mode for which the efficiency is limited to 20%—arises from a phase synchronization process due to the action of RF electric fields of reversed signs in each partition of the bisected cavity. Index Terms—Electron bunching, microwave generation, monotron, split-cavity oscillator, transit-time tube.
[3]. Assuming 1-D motion and that the space-charge forces are sufficiently weak such that the cavity normal modes are not modified by the presence of the beam, the force equation is dp = qE0 f (z) cos(ωt + φ0 ) dt
in which the electron position is given by the trajectory equation p(t)c dz = dt m2 c2 + p2
I. INTRODUCTION
E
ARLIER conceptual design studies [1]–[4] have demonstrated that in the monotron (the simplest of the microwave tubes), the conversion efficiency is essentially limited to 20%. This efficiency is achieved in circular cavities operating in the TM010 mode for which the axial electric-field distribution is uniform along the interaction space. On the other hand, a cavity bisected by a diaphragm or a conducting foil [5]–[7] to support a π-mode standing wave in the partitioned cavity yields a twofold increase in the conversion efficiency, thus providing a 40% efficiency level [7], [8]. These previous results were obtained through optimization schemes on the basis of particlein cell simulations without much emphasis on the physical mechanisms underpinning the efficiency enhancement process. Instead of resorting to the generic explanation that the splitcavity monotron is twice as much efficient as the single-cavity monotron because a π-mode cavity (with RF electric fields of opposite signs in each partition) corresponds to a pair of single TM010 -mode cavities in tandem, this paper examines the kinematical ballistic bunch in both interaction systems and shows that the efficiency enhancement arises from an extra group of bunched electrons that is formed in the second partition of the bisected cavity. II. EFFICIENCY AND ELECTRON TRAJECTORY The conversion efficiency is calculated by numerically integrating the relativistic equation of motion for an ensemble of electrons injected at given energy W0 , which streams along a cavity of length d, and averaging out the work done on the beam electrons by the RF fields during an oscillation period
Manuscript received April 9, 2009; revised June 5, 2009. First published July 28, 2009; current version published August 21, 2009. This work was supported by the National Council for Scientific and Technological Development (CNPq), Brazil. The review of this paper was arranged by Editor W. Menninger. The author is with the Associated Plasma Laboratory, National Institute for Space Research, 12227-010 São José dos Campos-SP, Brazil (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TED.2009.2026323
(1)
(2)
where p is the electron momentum, q = −|e| and m are the electronic charge and the electron rest mass, ω is the angular frequency of the RF field of amplitude E0 , and φ0 denotes the electron entrance phase. In the force equation, f (z) specifies the electric-field axial distribution. For the TM010 mode, the field is uniform f (z) = 1, and to describe the axial distribution of the π mode, we shall use the analytical function profile f ( z ) = − tanh[100( z − zt )]. In the calculation which follows, we normalize the motion variables according to p = p/mc, z = zω/c, t = ωt whereby the force equation becomes dimensionless d p = −gf (z) cos( t + φ0 ) d t
(3)
where g = qE0 /ωmc. First, using the uniform profile f (z) = 1 and g = 0.045 (to be justified later), the force and trajectories equations are numerically solved subject to the initial conditions z(φ0 ) = 0, p(φ0 ) = p0 , with the initial momentum p0 corresponding to the injection energy W0 = 10 keV, through p(0) = [511 + W0 (keV )]2 − 5112 /511. For 33 electrons with φ0 uniformly distributed over 2π, the resulting distance-time curves are shown in Fig. 1, where the parallel red straight lines are those associated with g = 0 (dc lines). We see that the electron trajectory labeled as 17 runs parallel to the dc lines, thus meaning that its average energy remains close to the 10-keV input energy. By contrast, electron #25 keeps an average velocity of 0.158 (in units of light speed c) noticeably below the dc velocity (v0 = 0.194), while electron #9 streams along the interaction space with an average velocity (0.246) higher than v0 . Such distinct behaviors are explained upon looking at the threshold field amplitude g above which makes an electron to be reflected back to the injection plane (z = 0). Under the weak relativistic approximation, the reflection amplitude in the uniform field (f (z) = 1) is estimated from the dimensionless trajectory equation t − φ0 )(v0 + g sin φ0 ) z( t) = g(cos t − cos φ0 ) + (
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Fig. 1. Distance-time curves for a set of 33 electrons injected at the initial energy of 10 keV into an interaction region with constant-amplitude RF electric field of g = 0.045. The inset shows the distribution of initial phases of the beam electrons, for which the initial change of momentum is proportional to − cos φ0 .
Fig. 2. Distance-time curves for a set of 33 electrons injected at the initial energy of 10 keV into an interaction region with RF electric field of amplitude g = 0.250. Seven electrons are reflected as indicated by the trajectories below (blue) the z = 0 line. (Red) The parallel straight lines are reference dc lines for which g = 0. The inset zooms in on the injection of particles during an RF field period.
which describes a wavy curve that undulates periodically along a straight line of slope v0 + g sin φ0 . Then, the condition for an electron to be reflected requires that the slope be negative, namely v0 + g sin φ0 < 0.
(5)
At fixed v0 and g > 0, we see that the highest slope is for φ0 = π/2 corresponding to electron labeled as #9 and irrespective of field amplitude g > 0 electron #9 will never be reflected. In a complementary fashion, the lowest slope is for φ0 = 3π/2, associated with electron #25. Characterized by the lowest reflection amplitude, such an electron is the first particle to be reflected when g further increases until reaching the minimum threshold value gth = v0 , which accounts for the lowest velocity of electron #25 in Fig. 1. In fact, at g = 0.250 > v0 , we see in Fig. 2 that electron #25 (lowermost trajectory) is reflected together with six accompanying electrons (22–24, 26–28) of initial phases symmetrically distributed around φ0 = 3π/2. Moreover, we note in between the dc and the z = 0 lines in Fig. 2 that some electrons although acquiring negative velocities during their motion, they are reaccelerated and propagate forward in the z-direction. These electrons are identified as reversed particles, for which (derived from the velocity equation t − sin φ0 )) the following relation holds: v( t) = v0 − g(sin v0 + sin φ0 < 1 g
(6)
stating the condition for an electron of initial phase φ0 to attain negative velocity in its journey in the interaction region. Then, a set of parameters {v0 , g, φ0 } satisfying (5) will satisfy (6), but the converse is not necessarily true. For example, let us predict the behavior of electron #20, the third trajectory above the z = 0 line and with entrance phase φ0 /2π = 0.593. At 10-keV initial energy and g = 0.250, the left-hand sides of inequalities (5) and (6) give 0.224 (> 0) and 0.056 (< 1), respectively, thus
Fig. 3. Velocity modulation of electron #20 marked in Fig. 2. During a modulation period, the particle remains for a longer time at positive velocities, thus traveling forward in time.
meaning thus the particle propagates forward in spite of its velocity being periodically reversed, as shown in Fig. 3. In this way (see for instance [9]), the trajectories can be categorized in three types: reflected trajectory (below the z = 0 line), reversed (lying between the lines z = 0 and z = v0 t, corresponding to slow electrons), and transmitted without velocity reversion (above the z = v0 t line, corresponding to fast electrons). Now, considering the stepped profile shown in Fig. 4 with zt = 1.45 (a chosen value to be commented later), the resulting trajectories are shown in Fig. 5, where it is clearly apparent the existence of two bunched groups of electrons. The first group is formed by the same slow electrons 22–33 as in Fig. 1, but now with velocities substantially lower than those in the uniform field. The additional group of slow electrons comprises electrons 1–13, by noting that such otherwise fast electrons after having crossed the sharp transition at z = 1.45 emerge as slow-velocity electrons. This is because during the first part of their journeys, those electrons experience a full
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Fig. 4. Stepped field profile represented by the function f ( z) = z− zt )] with transition position zt = 1.45. − tanh[100)(
Fig. 7. Energy as function of drift distance for electron #9 transiting in a stepped electric-field profile of amplitude g = 0.045 and transition position at z = 1.45 indicated by the dashed vertical line.
Fig. 5. Distance-time curves for electrons crossing a stepped electric-field profile of amplitude g = 0.045. Curves in red are dc lines for vanishing RF fields. The inset clearly shows the development of two groups of bunched particles after the beam has crossed the stepped transition.
Fig. 8. Distance-time curves for a set of 33 electrons injected at the initial energy of 10 keV into an interaction region with stepped electric field of amplitude g = 0.067 for which only electron #22 is reflected. (Red) The parallel straight lines are reference dc lines for which g = 0 and the inset shows electron #22 being reflected after crossing the transition.
Fig. 6. Time-varying velocity of electron #9 and the corresponding synchronous electric field, which reverses in phase at the normalized time 1.21.
cycle of the RF force (of equal durations of retarding and accelerating phases) and just crossing the transition the force still remains in a retarding phase due to the sign reversal of the π-mode oscillation, as shown in Fig. 6 for electron #9. The resulting effect is shown in the energy plot in Fig. 7, where the hitherto fastest electron (#9) upon reaching the field transition at z = 1.45 still remains in a decelerating phase, and thereafter propagates at the average energy of 6.5 keV thus
being maintained in a state of lower energy. Of course, had electron #9 been traveling in the uniform profile, it would have attained a 15.5-keV average energy. In this way, the stepped profile with g = 0.045 converts a group of 13 fast dispersed electrons into a bunch of 13 slow particles, which represents an RF current in the electron beam. In addition, a stepped profile makes the particles more prone to reflection. For instance, increasing the field amplitude to g = 0.067, we see in Fig. 8 that electron #22 gets reflected after crossing the transition. Such an effect is explained by inequality (5). After transiting the region of g > 0, electron #22 arrives at the transition with v0 = 0.065 and φ0 = 0.51π to start its journey through the region where g = −0.067. Using these parameters in the left-hand side of (5) gives a negative value (−0.13), thus indicating that the particle will be reflected somewhere in the second drift space. We see in Fig. 9 that electron #22 after traversing the transition at a (normalized) time 2.23, the particle’s velocity decreases continually and then commutes to a negative-velocity motion.
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klystron, RF fields are sufficiently strong to slow down some electrons to the point where their motion reverses [9], [10]. III. CONCLUSION
Fig. 9. Time-varying velocity of electron #22 in Fig. 8 and the corresponding synchronous electric field acting upon the particle.
Fig. 10. Efficiency dependence on the electric-field amplitude for (dashed line) uniform and (solid line) stepped profiles with optimum interactions lengths of d = 1.40 and d = 2.90, respectively, at initial energy of 10 keV.
At the W0 = 10 keV input energy, therefore, g = 0.067 is the threshold field amplitude that produces electron reflection in the stepped profile, such that by increasing this value more and more electrons will be successively reflected. To see whether this threshold value might have any impact on the monotron’s operation, we plot in Fig. 10 as function of the field amplitude the conversion efficiency for monotrons with uniform and stepped profiles. Since the efficiency is determined by three in this plot the interaction parameters, namely, W0 , g, and, d, has been optimized (averaging over 501 electrons) length, d, with respect to the input energy W0 , yielding the optimal values of 1.40 for the uniform field and 2.90 (related to zt through dopt = 2 zt ) for the stepped profile, such that the cavity is symmetrically partitioned at zt , which justifies the initially chosen value of zt = 1.45). For the uniform profile, the efficiency attains the maximum of 18.20% at g = 0.053 and drops to zero at g = 0.074, a value far below the corresponding threshold amplitude gth v0 = 0.194. By contrast, in addition to a higher efficiency of 43.50% at g = 0.045, for the stepped profile, the efficiency falls to zero at g = 0.069, very close to the threshold value of 0.067. This is consistent with the fact that in highly efficient devices, as for the
In the absence of space-charge forces, a kinematical formulation has been developed to examine the ballistic bunching in a 1-D RF fields for monotrons with uniform and stepped electric-field profiles. Describing the injection of electrons at a prescribed initial energy into the interaction region in terms of their entrance phases relative to a cycle of the driving oscillation, this formalism has given a good insight into the kinematics of electron-velocity reversion and interaction of an electron stream with a standing-wave electric field. The computational problem includes the pertinent set of initial-value quantities {v0 , φ, g}, respectively the injection velocity, entrance phase, and electric-field amplitude, from which a condition has been derived to predict whether a particle is slowed down from its initial velocity and reflected or else it is completely transmitted without velocity reversion. Due to a phase-synchronization mechanism, the steppedfield profile provides a larger number of slow-velocity electrons than those produced by the constant-amplitude electric field. This effect has been clearly shown in distance-time plots of the particle trajectories in stepped bunching fields giving rise to two groups of tightly bunched electrons. In addition, for the input energy of 10 keV, a maximum conversion efficiency of 43.50% is achieved at the optimum values of g = 0.045 and interaction length d = 2.90. Here, we note that the optimum length is uniquely determined from the beam’s input energy. Alternatively, input energies of 20 and 50 keV would yield optimum interaction lengths of 3.96 and 5.74, corresponding to field amplitudes 0.068 and 0.112, respectively, and with conversion efficiencies of about 44.8% A distinguished feature of the monotron is that the optimum RF voltage can be a few times higher than the beam voltage V0 , with no occurrence of reflected electrons. In fact, a voltage ratio 2 /V0 ) of 3.8 corresponds to optimum operation V /V0 = g d(mc of the uniform-profile monotron. This is in contrast with the klystron in which the maximum RF voltage established across the cavity is on the order of the beam voltage, thus posing limitation on output power. For the stepped profile, on the other hand, the corresponding voltage ratio is 3.5 at 43.50% conversion efficiency, thus demonstrating the capability of the splitcavity monotron as an attractive high-power microwave source. R EFERENCES [1] J. J. Müller and E. Rostas, “Un générateur à temps de transit utilisant un seul résonateur de volume,” Helvet. Phys. Acta, vol. 13, no. 3, pp. 435– 450, 1940. [2] Y. K. Yulpatov, “The excitation of oscillations in a cavity resonator by means of a relativistic electron beam,” Radiophys. Quantum Electron., vol. 13, no. 12, pp. 1374–1378, Dec. 1970. [3] J. J. Barroso, “Design facts in the axial monotron,” IEEE Trans. Plasma Sci., vol. 28, no. 3, pp. 652–656, Jun. 2000. [4] V. K. Fedyaev and A. A. Pashkov, “Electron conductance and efficiency of planar microwave gap in the nonlinear mode,” J. Commun. Technol. Electron., vol. 50, no. 3, pp. 361–365, Mar. 2005. [5] R. W. Lemke, “Dispersion analysis of symmetric transverse magnetic modes in a split cavity oscillator,” J. Appl. Phys., vol. 72, no. 9, pp. 4422– 4428, Nov. 1992.
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[6] B. M. Marder, M. C. Clark, L. D. Bacon, J. M. Hoffman, R. W. Lemke, and P. D. Coleman, “The split-cavity oscillator: A high-power e-beam modulator and microwave source,” IEEE Trans. Plasma Sci., vol. 20, no. 3, pp. 312–331, Jun. 1992. [7] J. J. Barroso, “Split-cavity monotrons achieving 40 percent electronic efficiency,” IEEE Trans. Plasma Sci., vol. 32, no. 3, pp. 1205–1211, Jun. 2004. [8] J.-T. He, H.-H. Zhong, B.-L. Qian, and Y.-G. Liu, “A new method for increasing output power of a three-cavity transit-time oscillator,” Chin. Phys. Lett., vol. 21, no. 7, pp. 1302–1305, Jul. 2004. [9] I. A. Chernyavskiy, A. N. Vlasov, T. M. Anderson, Jr., S. J. Cooke, B. Levush, and K. T. Nguyen, “Simulation of klystrons with slow and reflected electrons using large-signal code TESLA,” IEEE Trans. Electron Devices, vol. ED-32, no. 3, pp. 1555–1561, Jun. 2007. [10] W. T. Roybal, B. E. Carlsten, and P. J. Tallerico, “Dynamics of retrograde electrons returning from the output cavity in klystrons,” IEEE Trans. Electron Devices, vol. ED-53, no. 8, pp. 1922–1928, Aug. 2006.
IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 56, NO. 9, SEPTEMBER 2009
Joaquim J. Barroso received the B.S. degree in electrical engineering and the M.S. degree in plasma physics from the Technological Institute of Aeronautics, San José dos Campos-SP, Brazil, in 1976 and 1980, respectively, and the Ph.D. degree in plasma physics from the National Institute for Space Research (INPE), San José dos Campos-SP, in 1988. Since 1982, he has been involved in the design and construction of high-power microwave tubes with INPE. From 1989 to 1990, he was a Visiting Scientist with the Plasma Fusion Center, Massachusetts Institute of Technology, Cambridge. His research interests include microwave electronics and plasma technology.
