Startup Scenarios in High-Power Gyrotrons

IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004 841 Startup Scenarios in High-Power Gyrotrons Gregory S. Nusinovich, Fellow, IEEE, Oleksandr V. Sinitsyn, Leonid Velikovich, Muralidhar Yeddulla, Thomas M. Antonsen, Jr., Senior Member, IEEE, Alexander N. Vlasov, Senior Member, IEEE, Stephen R. Cauffman, Member, IEEE, and Kevin Felch, Member, IEEE Invited Paper Abstract—To realize continuous-wave (CW) operation of millimeter-wave gyrotrons at megawatt (MW)-power levels, these devices must operate in very high-order modes. To excite such an operating mode and to drive it into the regime of MW-level operation with high efficiency requires careful consideration of the startup scenario through which the operating parameters of the device are brought to their nominal values. In the present paper, several common startup scenarios and the most important physical effects associated with them are discussed. Then, the paper presents the results of startup simulations for a 140-GHz, MW-class gyrotron developed by Communications and Power Industries (CPI) for electron-cyclotron plasma heating and current drive experiments on the “Wendelstein 7-X” stellarator. The simulations were done with MAGY, a multifrequency, self-consistent code developed at the University of Maryland. Simulations tracking six competing modes show that, with a proper choice of operating parameters, stable excitation of the desired TE28 7 -mode at 1 MW level can be realized, despite the presence of dangerous parasites in the resonator spectrum. These results are in approximate agreement with experimental tests, in which the gyrotron demonstrated reliable operation at power levels up to 900 kW. PACS numbers: 84.40.Ik, 52.75 Ms. Index Terms—Electron-cyclotron current drive (ECCD), electron-cyclotron resonance heating (ECRH), multimode simulations. I. INTRODUCTION S A RULE, any microwave oscillator progresses through a time-dependent variation of operating conditions before it reaches its nominal steady-state operating point. For proper operation, this startup scenario should first fulfill the conditions for self-excitation, and then evolve to the nominal operating point in a way that ensures that the desired mode is excited with maximum efficiency at the desired power level, while neighboring modes are suppressed. In microwave sources driven by electron beams, the self-excitation conditions are usually characterized by the starting cur- A Manuscript received August 20, 2003; revised November 14, 2003. This work was supported in part by the Office of Fusion Energy, U.S. Department of Energy and in part by the Multidisciplinary University Research Initiative on Vacuum Electronics sponsored by the Air Force Office of Scientific Research. G. S. Nusinovich, O. V. Sinitsyn, L. Velikovich, M. Yeddulla, and T. M. Antonsen, Jr. are with the Institute for Research in Electronics and Applied Physics (IREAP), University of Maryland, College Park, MD 20742-3511 USA (e-mail: [email protected]). A. N. Vlasov is with SAIC, McLean, VA 22102 USA. S. R. Cauffman and K. Felch are with Communications and Power Industries (CPI), Palo Alto, CA 94304-1031 USA. Digital Object Identifier 10.1109/TPS.2004.828854 which is a function of various operating parameters rent (such as the beam voltage, beam position, etc.) So, when the the self-excitation conditions are beam current exceeds fulfilled, and this gives rise to oscillations, which at reasonably chosen parameters reach, in a certain transient time, the is steady-state regime. The parameter region, where known as the region of soft self-excitation. In this region, oscillations can start to grow from the noise level, which is typically determined by the presence of the beam. In many oscillators, in addition to the region of soft self-excitation, there is also a region of hard self-excitation, where the self-excitation conditions starting from low noise level are not ), but where the oscillations can neverthefulfilled (i.e., less be sustained, once their initial amplitude exceeds a certain threshold level. This classification was first introduced by Appleton and van der Pol for radio oscillators [1] and then used in consideration of various sources of coherent electromagnetic radiation. In the region of hard self-excitation, devices exhibit hysteresis. That is, depending on the history of parameter values, one can observe for a given final set of parameters either the presence or the absence of oscillations. Typical dependencies of the device efficiency on the beam current in the regimes of soft and hard self-excitation are shown qualitatively in Fig. 1. Very often the maximum efficiency can be obtained only in the region of hard self-excitation. In such cases, the device parameters should, first, pass through the region of soft self-excitation before reaching the point of the most efficient operation in the hard self-excitation region. Of course, in the process of this transition, the oscillations should remain stable. When a device is designed to operate in a high-order mode, the problem of the startup scenario is even more complicated, because the mode spectrum is very dense and, therefore, the self-excitation conditions can simultaneously be fulfilled for several modes. In this case, it is desirable to excite the operating mode prior to the others and then to maintain the conditions under which this mode will suppress all competitors. Suppression occurs due to nonlinear competition among the modes [2]. The mode competition results in the fact that effectively the starting current of a parasitic mode is increased by the presence of the desired operating mode. Denoting the starting current of and the amplitude of the operating the parasitic mode by mode by , one can formulate the requirement for suppression as 0093-3813/04$20.00 © 2004 IEEE (1) 842 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004 interaction between competing modes. These comments lay the groundwork for Section III, which discusses recent simulations of the startup scenario for CPI’s 140-GHz gyrotron. Then, Section IV presents the conclusions from the study and Section V summarizes this work. II. GENERAL COMMENTS A. Controlling Parameters Fig. 1. (a) Temporal dependence of the beam voltage and current during the voltage rise. (b) Dependence of the efficiency on the beam current for the instants of time, corresponding to soft (t = t ) and hard (t = t > t ) self-excitation. In the hard self-excitation region, the hysteresis exists in the range of currents where the dependence  (I ) shows three equilibrium states, among which one in the middle is unstable, while two others are stable. So, relying on the effect of mode competition, it becomes possible to drive the desired mode to the point of the most efficient operation even in the presence of many competing modes. (Note that the mode competition is not the only effect occurring in mode interaction in microwave oscillators, see, e.g., [3].) The problems described above in general terms are typical in high-power gyrotrons. It was shown by Moiseev and Yulpatov [4] a long time ago that the maximum gyrotron efficiency often occurs in the regime of hard self-excitation. Then, the desire to develop gyrotrons with higher operating frequency, higher power levels, and longer pulse lengths has led to operation at higher order modes. The problem of the startup scenario for gyrotrons operating in high-order modes was first formulated in [5]. During the past three decades, practically all researchers involved in the development of high-power long-pulse [or continuous-wave (CW)] millimeter-wave gyrotrons have had to contend with various aspects this problem (see [6]–[12] and references therein). This paper is written with an attempt to overview this problem and to present some of the recent results of simulations of startup scenarios in a megawatt (MW)-class, millimeter-wave gyrotrons developed at Communications and Power Industries (CPI). This paper is organized as follows. Section II contains general comments regarding: 1) the operating parameters that must be considered when modeling realistic startup scenarios; 2) the nature of mode competition in gyrotrons exhibiting both soft and hard self-excitation; and 3) the effects of parametric In general, gyrotron operation can be controlled by the beam voltage , the current , and the magnetic field in the interaction cavity . When a triode-type electron gun is used, there is . also the opportunity to vary the modulating anode voltage Also, in many cases, there is an additional coil in the electron gun region, which is used to adjust the magnetic field in this . This coil allows for better adjustment of the elecregion tron orbital-to-axial velocity ratio and the radius of the annular electron beam in the interaction space. So, in general, it is possible to manipulate up to five independent parameters to devise an optimal startup scenario. (We assume the beam current to be independent of the beam voltage because thermionic cathodes used in long-pulse/CW gyrotrons typically operate in the regime of temperature-limited emission.) In the regimes of pulsed operation, not all of these parameters can be varied during one pulse. For instance, the coils used in millimeter-wave gyrotrons have a very large inductance. Therefore, it is infeasible to vary the magnetic field ( , as well as ) during the time of the voltage rise . The time it requires to vary the cathode temperature (in order to modify the operating current at a given voltage) is also much larger than . This means that the beam current cannot be treated as an independent parameter during the voltage rise, because it depends solely on the beam voltage when the cathode temperature is held constant. These limitations limit the available means to control the startup scenario in pulsed gyrotrons. For instance, in the case of gyrotrons with triode-type electron guns, once the desired final operating parameters are chosen, the only free parameters are and ) the time-dependencies of the two voltages ( as they evolve to their target values. So, in this case we have only two independent parameters. The situation is even more . stringent in gyrotrons with diode-type guns, where Here, we have only one free parameter, which varies from zero to its final value. is many orders In realistic scenarios, the voltage rise time . Typof magnitude larger than the cavity fill time ically, in long-pulsed gyrotrons, ranges from hundreds of microseconds to milliseconds, while is on the order of nanoseconds. Therefore, we can treat the variation of oscillation amplitudes with voltage during the voltage rise as an adiabatically slow process. That is, we may assume that at any given voltage (in the process of the voltage rise), the gyrotron oscillations will have time to achieve a steady-state equilibrium. The fact is extremely important, because in the opposite that case , which is often called “an instant turn-on,” the voltage changes too rapidly for the oscillations to reach a steady state, making controlled startup scenario impossible. Both instant turn-on and slow turn-on scenarios were studied in [9], NUSINOVICH et al.: STARTUP SCENARIOS IN HIGH-POWER GYROTRONS Fig. 2. Example of starting currents of the operating TE 843 -mode and parasites as functions of the beam voltage. where it was shown that in some cases the time required to reach the equilibrium is much longer than the cavity fill time . (As shown below, our results confirm this conclusion.) Another physical effect, which is extremely important and favorable for gyrotron operation is the inverse dependence of the relativistic electron-cyclotron frequency on electron energy. As shown in numerous studies (see, e.g., [4], [5], [13], [14]), the region of soft self-excitation corresponds to smaller values , (here, is the of the cyclotron resonance mismatch gyrotron operating frequency, and is the cyclotron resonance harmonic number) than the region of hard self-excitation. In the hard-excitation region, in the presence of large amplitude microwave oscillations, this mismatch can be larger, first, because a large-amplitude wave can trap particles gyrating with a more detuned frequency and, second, because the particles decelerated by the microwave field increase their cyclotron frequency and, thereby, stay in resonance with the microwave oscillations. This favorable phenomenon makes it possible, even in the absence of other variable parameters, to raise the voltage from zero to the final operating voltage in a manner that first excites oscillations from noise in the soft-excitation regime, and then drives the device into the higher power and more efficient hardexcitation regime as shown schematically in Fig. 1. A typical example illustrating this scenario is shown in Fig. 2, reproduced from [15]. Of course, this figure is not sufficient to predict which of the many possible modes will be excited. This conclusion can be drawn only after numerical analysis of excitation of all potentially competing modes, as will be shown below. B. Mode Competition in Gyrotrons With Soft and Hard Self-Excitation In general, in high-power gyrotrons with a dense mode spectrum, the excitation of many modes during the voltage rise can result in various complicated nonlinear phenomena. To give the readers some insight into the physics of mode interaction, we start by considering the simple case of interaction between only two modes. If the gyrotron interaction space is cylindrically symmetric, i.e., consists of an annular beam of gyrating electrons and a cylindrical cavity, then the electron beam may excite -modes, which in the case of nonzero azimuthal index rotate in the azimuthal direction. If the two modes under consideration are in resonance with electrons at the same cyclotron harmonics, and these modes have different azimuthal , then, in the equations for mode excitation, indices averaging the source term over the cross section of the interaction space and over entrance phases of electrons results in averaging over the phase difference of these rotating modes, i.e., (here, is the azimuthal coordinate) [16]. After averaging, the equations for the mode amplitudes become independent of the mode phases. This is the same situation that occurs in radio oscillators with two degrees-of-freedom [2], as well as in other microwave [17] and optical [18] oscillators, where the mode frequency separation is much larger than the mode resonance width, which is inversely proportional to the cavity fill time . That is: . To qualitatively analyze the temporal evolution of the amplitudes of two modes, it is convenient to represent the nonlinear dependence of the source terms on mode amplitudes as a polynomial in mode amplitude and to keep in this representation only the lowest-order terms needed to describe the effects. In the regime of soft self-excitation, the saturation effects are described by the lowest-order nonlinear terms. A typical state space for such a two-mode gyrotron with a soft self-excitation is shown in Fig. 3(a). Here, and are normalized amplitudes of modes and arrows show a motion of a system to one of the two stable equilibrium states on axes. In the regime of hard self-excitation, the next-order terms in the polynomial representation of the source term should be taken into account. This makes the state space for a device more complicated. A typical example of such a space is shown in Fig. 3(b). As one can see, in addition to stable single-mode equilibria on axes shown in Fig. 3(a) for the case of soft self-excitation, there is now also a stable equilibrium state with nonzero amplitudes of two modes. Fig. 3(c) shows the state space for the case when one mode is in the region of soft excitation, while another is in the hard excitation regime. In this case, only single-mode oscillations of either first or second 844 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004 Fig. 4. Regions with different regimes of gyrotron operation in the plane “‘interaction length’ versus ‘resonator diameter.’” Fig. 3. Possible state spaces of two-mode oscillators in the cases. (a) Both modes are in the soft self-excitation region. (b) Both modes are in the hard selfexcitation region. (c) One mode is in the soft, while another is in the hard self-excitation regions (F are normalized amplitudes of the modes). modes are stable as in the case of soft excitation of both modes shown in Fig. 3(a). Systems having more than one stable equilibrium state can exhibit various hysteresis phenomena [1], [2]. (These phenomena in gyrotrons were studied not only theoretically in numerous papers, but also experimentally [19], [20]; see, also, references [20].) For instance, for the case shown in Fig. 3(a), an answer to the question “Which of the two modes will oscillate in such a device?” is: “The mode, for which the self-excitation conditions are fulfilled first.” With regard to the state space shown in Fig. 3(a), this means that the final equilibrium state, which will be reached by an oscillator, is determined by the initial conditions in this state space. In the hard self-excitation regimes, where additional stable equilibria exist, the situation with hysteresis phenomena is even more complicated. It has been shown [4], [13], [14] that, as the cavity length increases, the maximum efficiency point moves further into the hard self-excitation region, i.e., the ratio of the optimum current to its starting value gets smaller. Therefore, qualitatively, one can surmise the existence of the following regions of mode excitation and interaction in the plane of parameters “resonator length versus resonator diameter.” (Note that the enlargement of the resonator diameter corresponds to the mode spectrum densification.) First, at very short lengths, when the resonator , the oscillalength is shorter than the starting length tions do not appear because the starting conditions are not fulfilled: the cyclotron absorption inherent in nonmodulated electron flows dominates over coherent radiation processes [21]. Then, at , the self-excitation conditions can be fulfilled for a device in the region of soft excitation. At these lengths, when the resonator diameter is relatively small and, therefore, the device operates at relatively low-order modes, whose frequency separation is larger than the cyclotron resonance band, the modes can be excited separately. In resonators of a larger diameter, the modes can be excited simultaneously. Correspondingly, the state space for such a device with the soft self-excitation and dense spectrum of modes looks like the one shown in Fig. 3(a). In the case of even longer resonators, where the maximum efficiency point is in the hard self-excitation region, the situation is more complicated and, depending on the mode spectrum density, the following scenarios should be expected. First, when the gyrotron operates at relatively low-order modes, such modes will be excited separately in the process of voltage rise. Then, in the case of operation at higher order modes, the modes can be excited simultaneously. Possible startup scenarios for these cases are shown in Fig. 4. We show in this figure the dependencies of the efficiencies on the beam voltage for a gyrotron with a diode-type electron gun, because while the current varies as the voltage is increased, as discussed above, we cannot treat the beam current as an independent parameter during the voltage rise. The case A shown in Fig. 4 corresponds to the situation when the starting conditions for the second mode start to be fulfilled when the first mode is in the regime of hard self-excitation. The state space for such a gyrotron is shown in Fig. 3(c). Here, the first mode, being excited first, can remain stable until the voltage reaches the border of single-mode oscillations of this mode or will be in its vicinity. At this instant of time, the second mode is still in the region of soft self-excitation. So, one should expect here the mode hopping effect. The case B corresponds to the situation when the mode spectrum is denser and, therefore, the starting conditions for the second mode start to be fulfilled when the first mode is still in the region of soft self-excitation. So, for a certain interval NUSINOVICH et al.: STARTUP SCENARIOS IN HIGH-POWER GYROTRONS 845 of time, the state space of such a gyrotron with both modes in the region of soft self-excitation can look like the one shown in Fig. 3(a). Then, at higher voltages, the first mode moves in the region of hard self-excitation and the situation becomes similar to the case A described above. The cases C and D correspond to the gyrotron with a harder self-excitation of two modes. Here, in the case C, the region of the soft self-excitation of the second mode lies within the region of the hard excitation of the first mode. Therefore, when the voltage reaches the border of single-mode oscillations of the first mode, the first mode will disappear, but the self-excitation conditions for the second mode will not be fulfilled. So, the second mode cannot be excited in such a case. The case D differs from the case C only by the fact that the mode frequencies are closer and, therefore, in a certain range of voltages, both modes are in the soft excitation region. Nevertheless, the first mode is excited first, thus, the second mode cannot be excited here. Before closing this subsection, let us note that the approach based on a simple polynomial approximation of the source terms was recently used for a qualitative description of the effect of the radial thickness of electron beams on the mode competition in gyrotrons [22]. C. Parametric Interaction Above, we have considered the simplest case of interaction between two modes, the amplitudes of which evolve in a manner independent on the phase relation between the modes. This kind of interaction is often called a nonsynchronous one [3]. The case when the evolution of mode amplitudes depends on the phase relations is known as the synchronous [3] or parametric interaction. When all modes are in resonance with electrons at the same cyclotron harmonic, the synchronous or parametric interaction may occur between three modes, whose frequencies and azimuthal indices obey the following conditions [3]: (2) The first and second conditions can be treated, respectively, as the energy and angular momentum conservation laws in the four-photon decay process. The first condition is an approximate one because the modes have a finite width of the resonance curves. So, it is permissible to have the mode frequencies slightly detuned from an exactly equidistant spectrum. The corresponding condition can be written more precisely as (3) Then, the phase difference will vary with the same temporal scale as the mode amplitudes. So a self-consistent set of equations describing the parametric interaction between such modes should consist of the equations for mode amplitudes and the equation for this phase difference. Neighboring whispering gallery modes with the same radial index generally satisfy the conditions given by (2) and (3). So between such modes the parametric interaction can take place. Of course, this interaction becomes significant only when the mode frequency separation is smaller than the cyclotron resonance band, i.e., when each of the modes can interact with electrons resonantly. Note that this parametric interaction is typical not only for gyrotrons operating in whispering gallery modes, but also for quasi-optical gyrotrons, where many equidistant modes of the Fabry–Perrot resonator can be excited simultaneously [9]. Coming back to the gyrotrons operating in high-order azimuthally rotating modes let us consider an example. -mode, Consider the spectrum in the vicinity of the which is often used in MW-class gyrotrons operating at 100–110-GHz frequencies. The nonequidistance of cutoff , , and , which frequencies for modes form a triplet of quasi-equidistant modes, is very small: 10 . (Here, is the eigen-number, which determines the cutoff frequency , of a -mode in a cylindrical cavity of a radius ). At the same time, cavity -factors for high-power gyrotron oscillators are typically about 10 , so the conditions for synchronous interaction between these modes are fulfilled. However, the frequency separation of these modes is about 2.7%, while the cyclotron resonance band typically does not exceed 1%. Therefore, when one of such modes is excited by the beam, its low- and high-frequency satellites can be present due to the parametric interaction, but the amplitudes of these satellites should be small, because they do not interact resonantly with the beam. There are numerous factors that can influence the parametric interaction between almost equidistant modes. One such factor is the azimuthal nonuniformity of the electron emission. This effect was recently analyzed qualitatively [23] and simulated using a modified version of the self-consistent, multifrequency code MAGY to confirm the results of the qualitative analysis [24]. In a more general case, the parametric interaction can also occur between modes resonant with different cyclotron harmonics. Corresponding resonance conditions and effects are discussed in [25]. Also, reference [26] examines the parametric interaction between three modes that are resonant with an electron beam at three different cyclotron harmonics. III. ANALYSIS AND SIMULATIONS A. Starting Current and the Growth Rate of Oscillations Above, we have stated that 1) the oscillations of any mode start to grow when the beam current is larger than the starting current and 2) the temporal growth rate of oscillations is on the order of . Both of these statements need some clarification. First, it should be noted that during the voltage rise, both the beam current and the starting current vary. The evolution of the beam current and the electron orbital-to-axial velocity ratio with the voltage can be determined with the use of either the adiabatic theory of magnetron-type electron guns [27]–[29], or numerical codes (such as the widely used E-gun code [30]). Determination of the starting current is somewhat more complicated. In principle, a procedure for deriving an expression for the starting current is very straightforward, viz. the device operation is considered in the framework of small-signal theory, in which the effect of the radio frequency (RF) field on electron motion is treated as perturbation, so only the first-order terms in this perturbation are taken into account. Then, the linearized expression for the RF component of the electron current density is substituted 846 Fig. 5. Dependence of the starting currents of two competing modes on the -mode. assumed exit coordinate for a 140-GHz gyrotron operating at the TE into the equation describing the RF field excitation. The resultant equation, which describes the balance between the power of microwave losses in the cavity and the microwave power withdrawn from the beam, does not contain the RF field amplitude, but determines the current at which the growth rate of the mode equals zero (and above which the growth rate is positive). The specific procedure and equations are described elsewhere [31], [32]. This procedure and corresponding results, however, are simple and transparent only when one can use a cold-cavity approximation and assume that there is no interaction between the electrons and the RF field after the resonator output cross section. The cold-cavity approximation implies that the axial structure of the RF field in the resonator does not depend on the presence of the electron beam. This approximation works well, when the diffractive -factor is much larger than its minimal value, which can be estimated as [33]. The resonators presently used in MW-class CW gyrotrons often have a smooth transition between a cylindrical portion, which plays the role of the resonator, and an output uptaper. The diffractive -factor of such resonators is close to its minimum value. Even more important, however, is the fact that the interaction between an electron beam and the outgoing radiation continues into the output uptaper. Indeed the angle of tapering is rather small (it typically ranges from 2 to 5 ), and the profile of the externally applied magnetic field is often somewhat wider than the cylindrical cavity region. Therefore, the interaction can extend into this uptaper for many cyclotron orbits and it is very difficult, if not impossible, to precisely determine the cross section, in which this interaction stops. Recent analysis of starting currents in gyrotrons [34] has shown that, depending on the “output” cross section, i.e., the axial position at which we assume the interaction to have stopped, the starting current can vary significantly, and moreover, the starting currents of the operating and parasitic modes can vary in such a way that the mode with the higher starting current depends upon the choice of interaction length. An example of this effect is shown in Fig. 5 reproduced from [34]. (Results shown in Fig. 5 correspond to the case when IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004 the straight part of a waveguide, which just forms a cavity, is located between 4.14 and 5.52 cm.) These results indicate that reliance on linearized starting current calculations is insufficient for determining which of two neighboring modes will be excited first during a voltage rise. Now, let us discuss our estimate for the growth rate of oscillations. Since we are interested in the initial stage of the growth of oscillations, we can neglect the nonlinear terms in equations describing the field excitation. Then, we immediately find that the oscillation intensity grows in time as . So, in addition to our estimate , we get a factor, which characterizes the excess of the beam current over the starting current. Now, we should estimate the time it takes for our mode to grow from the noise level to the large-signal level, where this amplitude becomes large enough to affect the starting current of a second mode. The noise level is a level of spontaneous radiation providing a so-called white noise. According to [9], the amplitude of this , where is the number noise is inversely proportional to of electrons passing the resonator during the cavity decay time. This number can be estimated as 10 . So, for the typical , 10 , 100 GHz, parameters such as this number is on the order of 10 and, therefore, the initial noise-level amplitude is on the order of 10 . At the same time, numerous simulations show that large-signal effects begin to occur when the amplitude reaches the level of 0.1–1. Correis on the spondingly, the time it takes to reach saturation order of . Near the saturation, it is necessary to take into account nonlinear terms in equations for the field excitation that increases the saturation time. Now, we can come back to the classification of the cases of the instant turn-on and the adiabatically slow voltage rise, which we discussed in Section II-A, and determine our classification more accurately by considering the following example. Assume that we are dealing with two competing modes and that the voltages at which self-excitation conditions are fulfilled for and , respectively. Then, during the these modes are voltage rise, which we assume to be linear the time interval between these voltages can be estimated as . Correspondingly, we can identify as an “instant turn-on” the case when during this time interval the intensity of . the first mode will grow insignificantly, i.e., In such a case, the self-excitation conditions for the second mode will not be affected by the first mode, which has a small amplitude. In the opposite case (4) the first mode has sufficient time to grow large enough to suppress the excitation of the second mode. Let us note that the coefficient , which characterizes the voltage rise time, in short pulses can be very different from that in long pulses, depending on the characteristics of the power supply employed. For instance, in short-pulse gyrotron experiments at MIT, it takes about 1 s to reach a nominal voltage of about 80 kV [35], while in long pulse gyrotron tests at CPI, this time is larger than 100 s. NUSINOVICH et al.: STARTUP SCENARIOS IN HIGH-POWER GYROTRONS Fig. 6. Coupling impedances of six competing modes in the vicinity of the beam position as functions of the electron guiding center radius. B. Results of Simulations Detailed simulations have been performed for the 140-GHz gyrotron developed at CPI for electron-cyclotron resonance heating and current drive in the German stellarator “Wendelstein , and the 7-X” [36]. The operating mode in this tube is device operates at the fundamental cyclotron resonance. Our preliminary analysis has shown that for a given beam radius the most important competing modes are and , which form with the operating mode a triplet of modes corotating with gyrating electrons. Also, important is the triplet of counterrotating modes, having the radial index : , and , which form a counterrotating triplet. Since the latter triplet is formed by modes having a larger radial index, the inner peak of the radial profile of their coupling impedance corresponds to a smaller radius of electron guiding centers . [The coupling impedance of a thin annular electron beam to rotating modes in cylindrical waveguides/cavities is equal to . Here, “minus” and “plus” stand for the corotating and counterrotating modes, respectively.] The radial profile of the coupling impedance for the six modes under consideration in the vicinity of the beam position is shown in Fig. 6. The beam position shown by the dotted line corresponds to the beam radius employed in the gyrotron, which is intentionally a little larger than the optimum radius of the operating mode. Such a choice of beam position strongly reduces the coupling to the counterrotating parasites. (Initially, a smaller beam radius was employed, but simulations of the startup scenario [37] and some experimental results showed that under -mode suppresses the desuch conditions, the parasitic sired mode in the region of parameter space where maximum efficiency would otherwise be expected.) The simulations presented here, in which the excitation of six modes by a beam with a certain velocity spread was studied, were done with the use of the self-consistent, multifrequency code MAGY. (The original version of this code is described in [38].) Our experience with such simulations has been that, to 847 achieve accurate results, the time step should not exceed 0.1 ns, a small fraction of the cavity fill time. The time step chosen in the simulations was, thus, 0.05 ns. It was next determined that for the cavity geometry in question, on the processors available, it takes about 10 h of real-time to simulate 100 ns of mode evolution, when six modes are being considered. Clearly, at such speeds, it would be impossible to simulate complete 100 s voltage rise in long-pulse gyrotrons. Instead, we employed two time-saving techniques. First, we began our simulations at about 50 kV (rather than zero), choosing the initial voltage to be slightly below the voltage at which the modes’ growth rates become positive. Second, we divided the voltage rise into 2 kV steps, simulating the mode evolution at each voltage value for 100 ns (a time long enough for these modes to reach steady-state in most cases, as will be discussed), and using the final values of the mode amplitudes and phases from the previous run as input data for the subsequent run. This choice of voltage steps and duration of each run corresponds to a voltage rise time coefficient (introduced above) equal to 20-kV/ s. This coefficient is, for comparison, approximately 80- and 0.8-kV/ s for MIT short-pulse experiments and CPI long-pulse experiments, respectively. Our choice of steps and lengths of runs should, thus, be adequate for modeling slower startup scenarios such as that employed in the CPI long-pulse experiments. Although these simulations employ a series of instantaneous voltage steps, past experience has suggested that such an approach is acceptable if small enough steps are used (c.f., for instance, [39] where a 60-ns voltage ramp was chosen or [40], where 5-kV steps in voltage have been considered). In our case, 2- kV steps accurately described the sequence of events in the mode excitation and competition during the voltage rise; calculations with smaller steps performed in some runs yielded the same results. The simulations were performed using a typical set of operating parameters: 80-kV 40-A electron beam produced by a diode-type electron gun (in the absence of space charge neutralization, the beam voltage is 75.04 kV and the corresponding orbital-to-axial velocity ratio is 1.33; intermediate values of the beam voltage, current and orbital-to-axial velocity ratio for lower cathode voltages were calculated using a particle trajectory code). The resonator and output waveguide geometry and the magnetic field profile were chosen according to the gyrotron design. The maximum value of the magnetic field in the interaction region was about 55.3 kG. Results of the simulations are shown in Fig. 7. This figure is divided into two parts because of the length of the simulations; and the vertical scale for the two parts is different. The figure shows the radiated power in all six modes at the output cross section, where the simulations were ended. This cross section is about 3 cm downstream from the end of the straight section forming the resonator. The results shown in Fig. 7 demonstrate an extremely interesting and complex sequence of events. First, at low voltages, the higher frequency and counterrotating modes, which are the corotating -modes, are excited, but the counterrotating mode wins this competition. This mode remains dominant up to voltages of about 64 kV, but its power level is on the order of a few milliwatts. Then, the central modes of both triplets are excited 848 Fig. 7. Startup scenario for a 140-GHz, CPI gyrotron. The beam voltage varies in 2 kV steps. For each voltage the simulations are conducted for 100 ns time intervals. The first and second parts of the figure correspond to the voltages from 54 to 66 kV and from 68 to 80 kV, respectively. at 64 kV. At 70 kV, the operating mode starts to suppress the counterrotating rival, however, as that rival is suppressed, the oscillations of the lower frequency parasitic mode begin to grow. Then, in the range of voltages between 72 and 76 kV, the three corotating modes coexist with comparable power levels. The triplet of parasitic counterrotating modes is not completely suppressed here, but the power in these modes is at least two orders of magnitude lower than the power in the corotating triplet of modes; for instance, at 76 kV, the power of the modes in the corotating triplet ranges from 10 kW to more than 100 kW. Finally, starting at 78 kV, the desired operating mode begins to suppress all others, and at the final voltage, this mode reaches a power level of 1 MW, while the power of each of the remaining five parasites does not exceed 1 W. Note that in the region of coexistence of two triplets (when the voltage ranges from 72 to 76 kV) none of the modes oscillates with a constant amplitude. The amplitudes of all modes oscillate slightly, which is typical for regimes of automodulation (see, e.g., [25]). The period of these oscillations is about IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004 2.5 ns, which corresponds to about a 400-MHz automodulation frequency. This modulation frequency is much higher than one can expect from estimating the nonequidistance of the spectrum of modes in one triplet. As follows from the values of eigen-numbers of these modes, the latter is only about 9 MHz. The observed higher frequency modulation can be explained by the coexistence of two triplets, in which the frequencies of the corresponding modes are slightly separated. For instance, the eigen-numbers of the central modes in the two triplets are and . So, the beating between these modes should occur with a frequency on the order 140 GHz 280 MHz, which is quite of close to the 400 MHz observed in the simulations. The discrepancy between 280 and 400 MHz can be attributed to frequency pulling effects, i.e., the effects of the beam current on mode frequencies; as known, the value of frequency pulling depends on the position of a mode frequency in the cyclotron resonance band. So, for different modes the frequency pulling is different. While the predicted operation at the final operating parameters is in the desired mode, at the desired power level, it should be noted that at some steps in the simulation, the mode amplitudes did not in fact reach a steady-state before the subsequent step. In addition, because some steps exhibit oscillation in multiple modes, there is some concern that these simulations seem to indicate that the gyrotron will not reach a desirable steady-state if operated at lower voltages. For these reasons, we elected to check some stages of our simulation using longer computational runs. First, we checked, the results for 60 kV, where the steady-state was not reached in 100 ns. In a longer -mode, run (400 ns, instead of 100 ns), the dominant as well as other modes, reached a steady-state similar to those demonstrated in the shorter runs at 62 kV. (Note that in these calculations, one of the modes exhibited automodulation beating with a much lower frequency of about 25 MHz, which can be associated with the nonequidistance of the mode spectrum within the triplet, as discussed above.) Next, we conducted a longer simulation at 64 kV, where clearly a 100-ns time interval was not long enough to reach a steady state. Results of a 700-ns long run for this voltage are shown in Fig. 8(a). It is interesting that during the first 300 ns, when the power of two competing and ) increases, other modes exhibit modes ( steady-state operation at low-power levels. However, later, when the operating mode starts to suppress its rival, the damping of the -mode allows the appearance of automodulaparasitic tion oscillations in all modes, including the operating one. So, at a given voltage, it takes more than 0.5 s to reach steady-state, and in this state the desired operating mode suppresses all parasites. A further increase in the voltage causes only an increase in the power of the operating mode to the 1 MW level, as shown in Fig. 8(b), while the power of all parasites does not exceed 1 mW. It should be noted that in these long runs the highest power even -mode did not exceed of the most dangerous parasitic 1 W, while in short runs some parasitic modes reached a 100-kW power level (c.f. Figs. 7 and 8). These effects demonstrate that the predicted behavior can be quite sensitive to the details of the startup scenario if the voltage is not changed slowly relative to the rise times of the modes under consideration (as discussed above). NUSINOVICH et al.: STARTUP SCENARIOS IN HIGH-POWER GYROTRONS Fig. 8. Evolution of modes in the final stage of the startup scenario allowing additional time for the modes to reach steady-state. (a) Results of the long run at 64 kV. (b) Increase in the power of the desired mode with the further increase in voltage up to the nominal level. IV. CONCLUSION Our results lead to a few important conclusions. First of all, they show that it is necessary to consider the time scale of the voltage rise when attempting to predict the outcome of a startup scenario, because the sequence of modes, that can be excited, and their final power levels can be quite different depending on whether the voltage rises slowly or rapidly relative to the rise times of the modes themselves. This means, in particular, that the results of short-pulse tests of gyrotrons operating in high-order modes may not be reproducible in long-pulse tests of the same tubes. Second, these results show that to predict the final power levels in the various modes, it is necessary to track the behavior of the oscillator starting from a voltage, at which the first mode can be excited from the noise level, up to the nominal voltage. Third, in the case of exciting many modes, it is often necessary to consider the temporal mode evolution during time intervals much longer than the 849 saturation time, which we estimated above for a single-mode excitation. Indeed, our simple formula for the saturation time, which was derived in Section III-A, yields for the parameters of our simulation an approximate saturation time of 10 ns, while our simulations have shown that in some cases even 100 ns is not enough to reach a steady-state. Clearly, this delay in reaching the steady-state is associated with mode interaction processes. Our results also show that a simultaneous treatment of two triplets was absolutely necessary for determining details of mode excitation and interaction. Even more, it is not clear, in the cases like the one shown in Fig. 7 for voltages ranging from 72 to 76 kV, whether it was sufficient to consider only three modes in a triplet. It seems possible that in the case of modes coexisting at such high-power levels, there could be additional equidistant modes excited, like in the case considered in [9]. So, it may be desirable to consider, for example, five corotating modes instead of three, in order to be sure that the results of our simulations are adequate to the real device. The observed interactions among triplet modes arise not only from the effect of mode suppression, but also from the parametric interaction between the modes. In order to better understand the relationship between these nonlinear effects, it would be worthwhile, in parallel with numerical simulations, to develop quasi-analytical methods for analyzing these physical processes. Finally, it should be emphasized that the results of our simulations agree qualitatively with experiments, in which the CPI’s 140-GHz gyrotron demonstrated reliable operation in the desired mode at power levels over 900 kW [41]. Such operation was obtained at a slightly higher magnetic field than the design value, to avoid -mode, and the obtained efficiency excitation of the was, therefore slightly lower than predicted. The cause for this difference is unclear, but may include such possible factors as an azimuthally nonuniform electron emission from the cathode, misalignment of the beam relative to the cavity, higher than expected electron velocity spread, or microwave reflections from the gyrotron load. Further simulations, employing smoothly increasing voltages rather than discrete steps, exploring different operating parameters (such as velocity spread and magnetic field), and including additional neighboring modes, may help to identify the source of the observed differences between simulation and experiment. In spite of the success of our study and corresponding successful experiments at CPI, the fact remains that there is little freedom in manipulating the operating parameters during the voltage rise in gyrotrons, especially when diode-type electron guns are used. Therefore, the excitation of the desired mode and its evolution to the regime of high-efficiency operation is possible only when the spectral density of dangerous parasitic modes does not exceed a certain critical level. When the goal is to increase the radiated power and/or operating frequency, which requires the operation at even higher order modes, it might become impossible to provide a proper startup scenario in gyrotrons with conventional cylindrical resonators. Then, to achieve this goal, one would need to use resonators with improved selective properties. Coaxial resonators with properly tapered and/or slotted inserts are one such class of devices. Recently, the use of such resonators has allowed researchers 850 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004 to realize operation at the -mode in a 2-MW 165-GHz gyrotron [42]. In conclusion, let us note that almost all present-day highpower gyrotrons employ diode-type magnetron injection guns, because they eliminate the need for an additional power supply for the modulating anode, and simplify operation of the gyrotron. As noted above, this simplification limits the ability to tailor the startup scenario. As was shown already in the first paper on the startup scenarios [5], if in such gyrotrons there is any parasitic mode, whose frequency is slightly higher than the operating one and whose coupling impedance to the beam is close to that for the operating mode, this parasite can be excited first. This conclusion is, however, valid for the cases when the beam current at voltages corresponding to the cyclotron resonance with this parasite is close to its nominal value. If, in the initial stage of the voltage rise, when the electron emission is space charge limited, the current increases with the voltage slower, this parasitic excitation can be avoided. In notations used in Section I, the corresponding condition can be written , where is the voltage, which correas sponds to the cyclotron resonance with this parasite. Of course, this requires some modifications in the electron gun design or possible use of different emitters. An alternative means for introducing additional flexibility, which can be used in long enough pulses, would be to introduce a low-inductance magnetic coil to adjust the magnetic field during the voltage rise and partially compensate for the resultant change in the electron-cyclotron frequency with the rising voltage. Small changes to the magnetic field (on the order of a few percent) are likely to be sufficient to alter the startup scenario and avoid problematic parasitic modes. V. SUMMARY In this paper, a number of issues, which are the most important for the successful excitation of a desired mode in a gyrotron oscillator operating at one of the high-order modes, have been discussed and analyzed. It is shown that for correct analysis of the gyrotron startup scenario in the case of operation in such modes as the , it is necessary to take into account at least six modes forming two quasi-equidistant triplets, one of which corotates and another counterrotates with the gyrating electron beam. (Note that this result agrees with recent results of multimode simulations done for a 2-MW 170-GHz coaxial gyrotron operating in the mode [43].) It is also shown that the sequence of excitation starting from the first modes, which can be excited from the noise level during the voltage rise, is important for correctly identifying the expected mode excitation at the nominal operating point. The results obtained are in approximate agreement with experimental data. This demonstrates the utility of multifrequency, nonstationary codes for validating the design of high-order mode gyrotron cavities, and interaction processes there. The fact that experimental tests matching the simulated conditions resulted in excitation of a parasitic -mode, rather than the desired mode (which was recovered in experiments by operating at a slightly higher magnetic field) indicates that, while our understanding of the complex in- teractions of multiple modes with the electron beam and with each other have been improved significantly, further analysis of these interactions is required before the physical behavior of a particular gyrotron can be predicted with confidence. ACKNOWLEDGMENT The authors wish to thank J. Rodgers for a stimulating discussion. REFERENCES [1] E. V. Appleton and B. van der Pol, “On a type of oscillation-hysteresis in a simple triode generator,” Phil. Mag., vol. 43, p. 177, 1922. [2] B. van der Pol, “On oscillation hysteresis in a triode generator with two degrees of freedom,” Phil. Mag., vol. 43, pp. 700–719, 1922. [3] G. S. Nusinovich, “Mode interaction in gyrotrons,” Int. J. 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Temkin, “Generalized nonlinear harmonic gyrotron theory,” Phys. Fluids, vol. 29, pp. 561–567, 1986. [15] M. E. Read, G. S. Nusinovich, O. Dumbrajs, G. Bird, J. P. Hogge, K. Kreischer, and M. Blank, “Design of a 3-MW 140-GHz gyrotron with a coaxial cavity,” IEEE Trans. Plasma Sci., vol. 24, pp. 586–595, June 1996. [16] M. A. Moiseev and G. S. Nusinovich, “Concerning the theory of multimode oscillations in a gyromonotron,” Radiophys. Quantum Electron., vol. 17, pp. 1305–1311, 1974. [17] L. A. Weinstein, General theory of resonance electron oscillators, in High-Power Electronics, Moscow, U.S.S.R., vol. 6, Nauka, pp. 84–129, 1969. [18] N. G. Basov, V. N. Morozov, and A. N. Oraevskiy, “Nonlinear mode interaction in lasers,” Sov. Phys. JETP, vol. 22, pp. 622–628, 1966. [19] D. V. Kisel, G. S. Korablev, V. G. Pavelyev, M. I. Petelin, and S. E.Sh. E. Tsimring, “An experimental study of a gyrotron operating at the second harmonic of the cyclotron frequency with optimized distribution of the high-frequency field,” Radio Eng. Electron. Phys., vol. 19, pp. 95–100, 1974. [20] O. Dumbrajs, T. Idehara, Y. Iwata, S. Mitsudo, I. Ogawa, and B. Piosczyk, “Hysteresis-like effects in gyrotron oscillators,” Phys. Plasmas, vol. 10, pp. 1183–1186, 2003. NUSINOVICH et al.: STARTUP SCENARIOS IN HIGH-POWER GYROTRONS [21] A. V. Gaponov, M. I. Petelin, and V. K. Yulpatov, “The induced radiation of excited classical oscillators and its use in high-frequency electronics,” Radiophys. Quantum Electron., vol. 10, pp. 794–813, 1967. [22] G. S. Nusinovich, O. V. Sinitsyn, M. Yeddulla, L. Velikovich, T. M. Antonsen, Jr., A. N. Vlasov, S. Cauffman, and K. Felch, “Effect of the radial thickness of electron beams on mode coupling and stability in gyrotrons,” Phys. Plasmas, vol. 10, Aug. 2003. [23] G. S. Nusinovich and M. Botton, “Quasi-linear theory of mode interaction in gyrotrons with azimuthally inhomogeneous electron emission,” Phys. Plasmas, vol. 8, pp. 1029–1036, 2001. [24] G. S. Nusinovich, A. N. Vlasov, M. Botton, T. M. Antonsen, Jr., S. Cauffman, and K. Felch, “Effect of the azimuthal inhomogeneity of electron emission on gyrotron operation,” Phys. Plasmas, vol. 8, pp. 3473–3479, 2001. [25] G. S. Nusinovich, “Review of the theory of mode interaction in gyrodevices,” IEEE Trans. Plasma Sci., vol. 27, pp. 313–326, Apr. 1999. [26] G. P. Saraph, T. M. Antonsen, Jr., G. S. Nusinovich, and B. Levush, “A study of parametric instability in a harmonic gyrotron: Designs of third harmonic gyrotrons at 94 GHz and 210 GHz,” Phys. Plasmas, vol. 2, pp. 2839–2846, 1995. [27] A. L. Goldenberg and M. I. Petelin, “The formation of helical electron beams in an adiabatic gun,” Radiophys. Quantum Electron., vol. 16, pp. 106–111, 1973. [28] S. E.Sh. E. Tsimring, “Synthesis of systems for generating helical electron beams,” Radiophys. Quantum Electron., vol. 20, pp. 1550–1560, 1977. [29] J. M. Baird and W. Lawson, “Magnetron injection gun (MIG) design for gyrotron applications,” Int. J. Electron., vol. 61, pp. 953–967, 1986. [30] W. B. Herrmannsfeldt, SLAC Rep. 226, Stanford, CA, Nov. 1979. [31] M. I. Petelin and V. K. Yulpatov, “Linear theory of a monotron cyclotronresonance maser,” Radiophys. Quantum Electron., vol. 18, pp. 212–219, 1975. [32] G. S. Nusinovich, “Linear theory of a gyrotron with weakly tapered external magnetic field,” Int. J. Electron., vol. 64, pp. 127–135, 1988. [33] V. L. Bratman, M. A. Moiseev, M. I. Petelin, and R. E. Erm, “Theory of gyrotrons with a nonfixed structure of the high-frequency field,” Radiophys. Quantum Electron., vol. 16, pp. 474–480, 1973. [34] M. Yeddulla, G. S. Nusinovich, and T. M. Antonsen, Jr., “Start currents in the gyrotron,” Physics of Plasmas, vol. 10, pp. 4513–4520, 2003. [35] K. E. Kreischer and R. J. Temkin, “Single-mode operation of a highpower, step-tunable gyrotron,” Phys. Rev. Lett., vol. 59, pp. 547–550, 1987. [36] K. Felch, M. Blank, P. Borchard, P. Cahalan, S. Cauffman, T. S. Chu, and H. Jory, “Progress update on CPI 500 kW and 1 MW, multisecondpulsed gyrotrons,” in Proc. 3rd IEEE Int. Vacuum Electronics Conf., IVEC-2002, Monterey, CA, Apr. 23–25, 2002, pp. 332–333. [37] M. Yeddulla, G. S. Nusinovich, A. N. Vlasov, T. M. Antonsen, Jr., K. Felch, and S. Cauffman, “Startup scenario in a 140 GHz gyrotron,” in Proc. 27th Int. Conf. Infrared Millimeter Waves, R. J. Temkin, Ed., San Diego, CA, Sept. 22–26, 2002, pp. 9–10. [38] M. Botton, T. M. Antonsen, Jr., B. Levush, K. T. Nguyen, and A. N. Vlasov, “MAGY: A time-dependent code for simulation of slow and fast microwave sources,” IEEE Trans. Plasma Sci., vol. 26, pp. 882–892, June 1998. [39] S. H. Gold and A. W. Fliflet, “Multimode simulation of high-frequency gyrotrons,” Int. J. Electron., vol. 72, pp. 779–794, 1992. [40] G. S. Nusinovich, M. E. Read, O. Dumbrajs, and K. E. Kreischer, “Theory of gyrotrons with coaxial resonators,” IEEE Trans. Electron Devices, vol. 41, pp. 433–438, Mar. 1994. [41] T. S. Chu, M. Blank, P. Borchard, P. Cahalan, S. Cauffman, K. Felch, and H. Jory, “Development of high-power gyrotrons at 110 GHz and 140 GHz,” in Proc. 4th IEEE Int. Vacuum Electronics Conf., Seoul, Korea, May 28–30, 2003, pp. 32–33. [42] B. Piosczyk, A. Arnold, G. Dammertz, O. Dumbrajs, M. Kuntze, and M. Thumm, “Coaxial cavity gyrotron—Recent experimental results,” IEEE Trans. Plasma Sci., vol. 30, pp. 819–827, 2002. [43] B. Piosczyk, S. Alberti, and A. Arnold et al., “A 2 MW, CW, 170 GHz coaxial cavity gyrotron for ITER,” in IAEA Technical Meeting on ECRH Physics and Technology for ITER, Kloster Seeon, Germany, July 14–16, 2003. 851 Gregory S. Nusinovich (SM’92–F’00) received the B.Sc., M.Sc., and Ph.D. degrees from Gorky State University, Gorky, U.S.S.R., in 1967, 1968, and 1975, respectively. In 1968, he joined the Gorky Radiophysical Research Institute, Gorky. From 1977 to 1990, he was a Senior Research Scientist and Head of the Research Group at the Institute of Applied Physics, the Academy of Sciences of the U.S.S.R., Gorky. From 1968 to 1990, his scientific interests included developing high-power millimeter- and submillimeter-wave gyrotrons. He was a Member of the Scientific Council on Physical Electronics of the Academy of Sciences of the U.S.S.R. In 1991, he immigrated to the U.S. and joined the Research Staff at the Institute for Plasma Research (presently, the Institute for Research in Electronics and Applied Physics), University of Maryland, College Park. Since 1991, he has also served as a Consultant to the Science Applications International Corporation, McLean, VA, the Physical Sciences Corporation, Alexandria, VA, and Omega-P, Inc., New Haven, CT. He is the author of Introduction to the Physics of Gyrotrons (Baltimore, MD: The Johns Hopkins University Press, 2004). He has authored and coauthored more than 150 papers published in refereed journals. His current research interests include the study of high-power electromagnetic radiation from various types of microwave sources. Dr. Nusinovich is a Member of the Executive Committee of the Plasma Science and Application Committee of the IEEE Nuclear and Plasma Sciences Society. He is a Fellow of the American Physical Society (APS). In 1996 and 1999, he was a Guest Editor of the Special Issues of the IEEE TRANSACTIONS ON PLASMA SCIENCE on High-Power Microwave Generation and on Cyclotron Resonance Masers and Gyrotrons, respectively. Presently, he is an Associate Editor of the IEEE TRANSACTIONS ON PLASMA SCIENCE. Oleksandr V. Sinitsyn was born in Kharkiv, Ukraine, in 1978. He received the B.S. degree in radio physics from V. Karazin Kharkiv National University, Kharkiv, Ukraine, in 2000 and the M.S. degree in electrical engineering from the University of Maryland, College Park, in 2002. He is currently working toward the Ph.D. degree in electrical and computer engineering at the University of Maryland. His research interests include theory and design of high-power microwave sources. Leonid Velikovich was born in Moscow, Russia, in 1981. He received the B.Sc. degree in computer science from the University of Maryland, College Park, in 2003. His research interests include numerical methods and computational geometry. Muralidhar Yeddulla received the B.S. degree in electronics engineering from Bangalore University, Bangalore, India, in 1994 and the M.S. degree in microwave engineering from Banaras Hindu University, Varanasi, India, in 1997. He is currently working toward the Ph.D. degree at the University of Maryland, College Park. He worked as a Member of the Research Staff at Bharat Electronics, Bangalore, India, from February 1997 to July 2000. His research interests include studies on mode competition in high-power gyrotrons, peniotrons, and autoresonant devices. 852 Thomas M. Antonsen, Jr. (M’82–SM’96) was born in Hackensack, NJ, in 1950. He received the B.S. degree in electrical engineering in 1973, and his M.S. and Ph.D. degrees in 1976 and 1977, all from Cornell University, Ithaca, NY. He was a National Research Council Post Doctoral Fellow at the Naval Research Laboratory, from 1976 to 1977, and a Research Scientist in the Research Laboratory of Electronics at MIT, Cambridge, from 1977 to 1980. In 1980, he moved to the University of Maryland, College Park, where he joined the faculty of the Departments of Electrical Engineering and Physics in 1984. He is currently a Professor of physics and electrical and computer engineering. He has held visiting appointments at the Institute for Theoretical Physics (UCSB), the Ecole Polytechnique Federale de Lausanne, Switzerland, and the Institute de Physique Theorique, Ecole Polytechnique, Palaiseau, France. He served as the Acting Director of the Institute for Plasma Research, University of Maryland from 1998 to 2000. His research interests include the theory of magnetically confined plasmas, the theory and design of high-power sources of coherent radiation, nonlinear dynamics in fluids, and the theory of the interaction of intense laser pulses and plasmas. He is the author and coauthor of over 200 journal articles and co-author of the book Principles of Free-Electron Lasers. Dr. Antonsen has served on the editorial board of Physical Review Letters, The Physics of Fluids, and Comments on Plasma Physics. He was selected as a Fellow of the Division of Plasma Physics of the American Physical Society in 1986. In 1999, he was a co-recipient of the Robert L. Woods award for Excellence in Vacuum Electronics Technology, and in 2003 he received the IEEE Plasma Science and Applications Award. Alexander N. Vlasov (M’95–SM’03) was born in Lipetskaya Oblast, Russia, in 1958. He received the B.Sc. (Honors), M.S., and Ph.D. degrees in physics from Moscow State University, Moscow, U.S.S.R., in 1981, 1986, and 1987, respectively. After graduation, he was a Research Scientist from 1986 to 1991, an Assistant Professor from 1991 to 1995, and an Associate Professor since 1995 with the Department of Physics, Moscow State University. From 1991 to 1998, he was a Visiting Scientist with the Institute for Plasma Research, University of Maryland, College Park. Since 1999, he has been with Science Applications International Corporation, McLean, VA. His current research interests include the theory, computer simulations, and design of high-power sources of coherent microwave and millimeter-wave radiation. View publication stats IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004 Stephen R. Cauffman (M’85) was born in Iowa City, IA, in 1969. He received the B.S. degree in physics from Stanford University, Stanford, CA, in 1991 and the Ph.D. degree in plasma physics from Princeton University, Princeton, NJ, in 1997. For his doctoral dissertation, he studied emission in the ion cyclotron range of frequencies driven by fusion products in deuterium-tritium plasmas in the Tokamak Fusion Test Reactor. Since 1997, he has been with Communications and Power Industries (CPI), Palo Alto, CA, working on the design of high-power gyrotrons. Kevin Felch (M’84) was born in Denver, CO, on December 7, 1952. He received the A.B. degree in physics from Colorado College, Colorado Springs, in 1975 and the Ph.D. degree in physics from Dartmouth College, Hanover, NH, in 1980. His graduate work involved the design and construction of a 400-kV 30-A electron-beam system and the use of dielectric-loaded waveguides in combination with the electron beam for the generation of millimeter-wave Cherenkov radiation. During 1980, he held a Postdoctoral position at the Ecole Polytechnique, Palaiseau, France. His work there involved the design, construction, and operation of a free-electron laser experiment. In 1981, he joined Varian Associates, Inc., as a Member of the Gyrotron Engineering Department. His work at Varian (now Communications and Power Industries, Palo Alto, CA) has involved the development of high-power gyrotron oscillators and amplifiers. Since 1996, he has served as Team Leader for gyrotron activities at Communications and Power Industries. Dr. Felch is a member of the Plasma Physics Division of the American Physical Society and the Plasma Science Society of the IEEE. In 1999, he received the Robert L. Woods Award for achievements in the field of vacuum electronics. In 1985, he was a Guest Editor for a Special Issue of the IEEE TRANSACTIONS ON PLASMA SCIENCE.