In this article, the diocotron modes developing in a coaxial diode and the resonant magnetron modes forming in a six-vane A6 relativistic magnetron both fed by a split cathode are studied by Particle-in-Cell simulations. The split cathode is a novel type of cathode which sustains a column longitudinally oscillating electrons in a coaxial diode (smooth bore) or a magnetron over a non-emitting axial cathode conductor. The connection between the longitudinal oscillatory dynamics and the diocotron or magnetron mode development, as well as whether the diocotron modes affect the magnetron modes, is investigated.
I. INTRODUCTION
Relativistic magnetrons (RMs) were suggested in the early 1970s and were intensively investigated both theoretically, numerically, and experimentally due to their potential to generate high power microwave (HPM) pulses of several tens of ns duration, exceeding 1 GW power and high electronic efficiency.1 However, RMs and many other similar devices suffered from HPM pulse shortening caused by the evolution of explosively formed cathode plasma, the nonuniform radial expansion of which leads to mode competition and premature termination of microwave generation. The explosive plasma formed on the surface of a solid cathode expands toward the anode, changes the cathode–anode distance, and consequently the resonance conditions to the point when the generation of microwaves terminates. Several ideas were proposed to solve this problem, such as the transparent cathode,2 the virtual cathode (VC),3 and the magnetic mirror produced VC.4 The transparent cathode, even with a reduced emission surface, does not cancel plasma evolution inside the anode space. The proposed VC concepts are too complicated to be realized in practice.
Recently, the split cathode was suggested as an electron source which overcomes the above-mentioned problems.5 The split cathode is a source of electrons in which the emitting cathode and the plasma are outside the magnetron volume, while the electrons are trapped in this volume by simple means which do not require changing the design of the magnetron or adding magnetic field sources.
A split cathode is seen in Fig. 1 where it is coaxial with a regular anode (part 7 in Fig. 1) (smooth bore coaxial diode) or a magnetron anode. The split cathode consists of an annular emitting cathode placed at a distance outside the upstream end of an anode (part 3 in Fig. 1) and a reflector (part 2 in Fig. 1) placed at a distance from the downstream end of the anode, which suppresses axial electron flow further downstream. These two parts are connected by a central conducting rod (part 5 in Fig. 1) so that the cathode, reflector, and the rod are at the same potential. For an actual picture of a system, see Fig. 1 in Ref. 6 in which experiments comparing the behavior of a split cathode are compared to that of a solid cathode. Figure 2 in Ref. 6 clearly shows that pulse shortening, which appears when a solid cathode is used to feed a magnetron disappears when it is replaced by a split cathode. It has been further confirmed experimentally that using a split cathode in an axial output magnetron, no pulse shortening develops during a ∼200 ns duration voltage pulse.7
The annular electron column is magnetized by an external axial magnetic field. The space charge of the column surrounding the rod screens it so that the explosive emission from it is not very likely. A second reflector can be added upstream or the cathode holding the annular emitter can be made of sufficiently large radius (part 1 in Fig. 1) so that no axial electron flows upstream. A second cathode can be added downstream (part 4 in Fig. 1) so that the split cathode becomes symmetric. In addition to trapping all the electrons, the split cathode has the advantage that the explosive plasma forms outside the space between the central rod and the anode and remains outside until it axially and radially expands and reaches the anode.8
The annular electron column is produced by electrons emitted from a thin circular explosive emission edge. These electrons are first accelerated toward the anode and propagate inside the space between the anode and the rod. Then, as electrons approach the reflector, they decelerate to zero velocity and return toward the cathode; that is, electrons oscillate in the axial potential well and rotate in the azimuthal direction in the transverse radial electric and axial magnetic fields. Thus, for both the solid and split cathode, the electrons start at full potential and zero kinetic energy. In the split cathode, though, they start outside the anode and enter the anode interaction space at a lower potential level while gaining axial kinetic energy. This affects the operation of a magnetron fed by a split cathode for which the generation of microwaves will start for an applied magnetic field less than that required for a solid cathode.
During the applied high voltage pulse, electron emission continues and the potential well fills with increasing electron charge. If the magnetic field is sufficiently high, so that it keeps the electrons from flowing toward the anode, it can, in time, reach the limit of the accumulated charge allowed by the repulsive Coulomb forces. For very high magnetic fields, since electrons cannot escape from the potential well and do not expand sufficiently to be collected on the anode, a squeezing process begins.9,10 Namely, the increasing electron density is counteracted by a reduction in the kinetic energy of the electrons, a situation known as a squeezed state.11,12
Numerical study of the space charge dynamics inside an annular electron beam in its squeezed state revealed the formation of diocotron instability leading to the formation of a rotating periodic azimuthal modulation of the density13 (electron bunches). It was found that the time of these bunches appearance and their number depend on the value of the magnetic field. This set of rotating periodic bunches was observed in recent experiments studying the longitudinally oscillating annular electron column contained in a split cathode feeding a coaxial diode.14 The dynamics of the diocotron instability was studied over many years, mostly theoretically and numerically in situations different from those existing in the pure high energy electron plasma axially oscillating inside a split cathode fed coaxial diode.15
The electron dynamics in a split cathode is different than that in a conventional solid cathode fed relativistic magnetron. To understand the electron dynamics in any E × B device and relativistic magnetrons, in particular, numerous studies were performed over the last 50 or so years, a list of which is too long to be included here. Moreover, the presence of periodic diocotron bunching in the electron flow rotating azimuthally and the relationship between these bunches and the spokes developing in the slow wave structure as a function of the insulating magnetic field were also studied.16–19 Because of counterstreaming, the electron flow cannot be considered to be a Brillouin flow.20 The dynamics of the annular axially oscillating electrons in a split cathode is unique and studying the appearance of the azimuthal diocotron instability in this system attracted our interest.
In the present article, the connection between the periodicity (the diocotron mode) of the diocotron charge bunches in a smooth bore anode (a coaxial diode) fed by a split cathode and the electromagnetic modes developing in a relativistic magnetron of the same length and inner diameter are investigated by numerical means. Relatively low magnetic fields and voltages of a few hundred kV, for which it was found that such magnetrons radiate high power microwaves best, are considered. The Hull and Buneman–Hartree limits should exist for a split cathode device too. That is, there must be a lower bound to the magnetic field for minimal magnetic insulation and an upper bound too where best synchronization exists between the electron azimuthal drift and electromagnetic wave phase velocities. On the other hand, the formulas developed for both these limits for either a smooth bore coaxial diode or a magnetron are not applicable in a split cathode fed device. The Hull type limit should be at a lower magnetic field value because the electrons are entering the anode space with the potential energy smaller than the cathode potential. The same is also relevant to the Buneman–Hartree limit. However, higher magnetic field values are considered too, in an attempt to understand the effect of squeezing on the diocotron and magnetron modes. The questions addressed are (1) Is there a connection between the longitudinal and azimuthal motion in a split cathode fed device (coaxial diode or magnetron)? and (2) Is there a connection between the diocotron mode developing in a coaxial diode and the magnetron mode? An analytical model for the behavior presented has not been developed so far, but these simulations are sufficiently important to be presented. All simulations were performed using the MAGIC Particle-in-Cell (PIC) code.21
II. A COAXIAL DIODE FED BY A SPLIT CATHODE
A coaxial diode, or a smooth bore device, fed by a split cathode as seen in Fig. 1, is considered in this section. The split cathode consists of a cathode (1 in Fig. 1), which also acts as the upstream reflector, and a downstream reflector (2 in Fig. 1), both of 20 mm radius. There are two emitters on each side (3 and 4 in Fig. 1), each consisting of a 10 mm long, 8 mm outer radius, and 1 mm width hollow cylinder. The distance between the axial edge of each emitter from the anode is 10 mm. Here, only the upstream emitter is allowed to emit a space-charge-limited current and the non-emitting downstream emitter is kept only for symmetry. The 50 mm long anode (7 in Fig. 1) has inner radius rin = 20 mm. The connecting rod (5 in Fig. 1) radius is 2.5 mm. The plasma forming at the edge of the emitter and its expansion are neglected.
The entire system is immersed in a constant axial magnetic field and fed by a voltage applied on the upstream open boundary of Fig. 1, rising in 1 ns to 400 kV. In this section, a relatively high magnetic field of 2 T is considered. In Fig. 2, some results of the corresponding PIC simulations are presented. For this case, up to the time the simulations were carried out, no current flows to the anode and the total accumulated charge in the rod-anode volume between the upstream and downstream anode ends is still increasing (see Fig. 2 black curve). One can see that the space charge increases quickly during the first 5 ns followed by a slower filling rate up to ∼25 ns which then reduces to a more gradual charge increase. For this relatively high magnetic field, the time at which the simulation has been stopped is not yet the limit of the magnetic insulation. In Fig. 2, the emitted current and the wedge charge are also drawn. The wedge charge for all coaxial diode cases is defined as the electron charge contained in a 20° wedge volume extending radially from the axis to the anode inner radius and containing a 10 cm long region of the anode length centered at the anode longitudinal center.
After ∼8 ns (the time it takes the voltage to settle at 400 kV), the emitted current in Fig. 2 starts to sinusoidally oscillate up to ∼23–24 ns, followed by a time interval ∼23–44 ns in which the oscillations become periodic peaks. The wedge charge oscillates along this entire time interval while the oscillation period seems to increase with increasing time.
Figure 3(a) shows results of the time–frequency analysis of the wedge charge signal observed in Fig. 2. The frequency changes gradually from fw ∼1.4 GHz near ∼11 ns, through ∼1.2 GHz near 22 ns, ∼1 GHz near ∼24 ns, ∼0.9 GHz near 35 ns, ending at ∼0.6 GHz at ∼40–60 ns. The emitted current oscillations start also at a frequency of fe ∼1.4 GHz and are similar to fw up to ∼22 ns from where it jumps to fe ∼0.6 GHz at ∼25 ns almost the same as fw from ∼40 ns and on. Both contain some weaker intermittent signals at ∼1.2–1.1 GHz at t > 25 ns.
The emitted current oscillations reflect the longitudinal charged cloud oscillations which are the result of temporary over-injection of the charge in the potential well and the reaction of the already accumulated charge to this increase. This can be seen in the [vz, z] phase space snapshots at various increasing times shown in Fig. 4.
At early times, the phase space island is almost empty with most of the electrons moving close to a single electron's route in the potential well reaching maximum velocity near the center of the anode. By the times reached in the first row of Fig. 4 (∼11.5 ns), a period is distinguished at the center of which (11.49 ns in Fig. 4) charge is almost equally concentrated near the emitter and the reflector. This charge concentration causes the reduction in the emitted current to minimum. At the two endpoints of this period, the space charge is spread and the charge near the emitter indicates maxima of the emitted current. The period of the oscillation in the first row of Fig. 4 corresponds to a longitudinal frequency fz ∼1.4 GHz, close to that observed in the emitted current oscillations, fe, in Fig. 3(b). The nature of the space charge density oscillations at the later time seen in the second row of Fig. 4 (∼21.5 ns) is similar. Also, at the period endpoints, the phase space island is now divided by charge concentration at the center. The period of the space charge oscillations seen in the second row of Fig. 4 corresponds to a frequency of ∼1.1 GHz, close to that observed in Fig. 3(b). In the third row in Fig. 4, the space charge density concentrates mainly either close to the emitter or close to the reflector (maximum emitted current). This space charge concentration is accompanied by squeezing (reduction in vz) over an increasing region of phase space in time (i.e., fourth row in Fig. 4). The third row's space charge density oscillation period in Fig. 4 corresponds to a frequency of ∼0.63 GHz and the fourth row to ∼0.57 GHz.
It is, thus, evident that the emitted current oscillations frequency is strongly related to the space charge density's longitudinal oscillations. The reduction in the frequency [see Fig. 3(a)] is the result of the space charge accumulation between the cathode and the reflector. A sharp decrease in the frequency at t ∼25 ns is related to a partially (over a part of the longitudinal space) squeezed state formation with larger electron density and decreased kinetic energy. For a finite axial magnetic field amplitude, this will continue to a limit where the space charge will start to expand to the anode and release over-injected charge as anode current.
In Fig. 5, snapshots of the azimuthal space charge density distribution at the center of the anode are drawn at times within the time periods of each row of Fig. 4. Note that the times of the snapshots correspond to the times pointed out by the arrows in Fig. 3 and also to the intervals in each row in Fig. 4. The reduction in the azimuthal wedge charge oscillation frequency has been connected in the above-mentioned discussion to the reduction in the longitudinal charge oscillations. At the same times when these changes occur, Fig. 5 shows that the diocotron mode (the number of charge packages) reduces too. The rotation frequency of a charge package, fR, depends on the value of and is given as , where fw is the wedge charge oscillations frequency. For between 5 and 4, one obtains charge smearing which makes it difficult to depict a constant mode. The consequence of this is the gradual frequency decrease seen in Fig. 3(a).
The above-mentioned results indicate that the longitudinal oscillations of the electrons determine the diocotron mode number, . When the electron transit time between the two cathodes, , and the diocotron frequency, relate as , then an electron extracted from a bunch at an axial location returns to a bunch after an axial oscillation period (generally not the same bunch because of the azimuthal bunch rotation). So, at early times (small accumulated charge, Figs. 2 and 3, ns), mode is synchronous with the longitudinal oscillations. The accumulation of negative space charge between the cathodes reduces the axial electron velocity and, consequently, decreases the longitudinal oscillations frequency. Correspondingly, the mode number decreases from to (Fig. 3, ns). Later, the nature of the cloud of oscillating electrons changes—a squeezed state with high charge density and low energy appears and shares the phase space with an electron hole or holes, a region with lower charge density and higher energy.22 The presence of the electron–hole is a characteristic feature of the early stage of the squeezed state formation.14,23 This hole and squeezed state region move longitudinally (see Fig. 4), changing periodically the longitudinal distribution of the electron density. The period of this space charge oscillation is not defined by the individual electron velocity. As it was shown in Ref. 14, this hole, when considered as a perturbation of the squeezed state, propagates in the form of a space charge wave with a phase velocity ∼1.5 times smaller than individual electrons velocity. This means that the frequency of the synchronous diocotron oscillation should also reduce by the same proportion. Indeed, as shown in Fig. 3, the level of squeezing is accompanied by the reduction in the diocotron mode number, from to through as an intermediate state.
III. A COAXIAL DIODE AND A 6-VANE MAGNETRON FED BY A SPLIT CATHODE IN A WEAK MAGNETIC FIELD
The magnetic field of 2 T used in Sec. II does not allow the axially oscillating electron space charge to expand and reach the anode during a reasonable time for a magnetron to operate. A split cathode fed magnetron operates well for axial magnetic fields 0.15–0.4 T.2 In this section, a coaxial diode of the same geometry as that used in Sec. II but at a magnetic field of 0.25 T and a magnetron consisting of six 40° vanes separated by six 20° cavities of outer radius rout = 40 mm at the same magnetic field amplitude are studied. This magnetron is a regular A6 magnetron and it will be referred to below as an A6-Mag.
For the coaxial diode, a large anode current turns on almost immediately (∼4 ns) which limits the voltage at 350 kV, less than the applied 400 kV [see Fig. 6(a)]. In Fig. 6(b), the total accumulated charge, the wedge charge, and the emitted current are drawn (compare with Fig. 2 at 2 T). The wedge charge oscillates with ±2.5 nC at fw ∼5.0 GHz, but oscillations are not visible in the total charge and the emitted current.
Snapshots of the azimuthal space charge density distribution at the anode center at early and later times can be seen in Fig. 7 where a rotating diocotron mode, = 3, which starts as early as ∼8 ns can be discerned. The longitudinal phase space seen at 30 ns in Fig. 8 does not change much from as early as ∼8 ns and there is no indication to the formation of a squeezed state.
In contrast to the high magnetic field case discussed in Sec. II, here the diocotron mode formation and its azimuthal frequency are not connected to longitudinal oscillations of the accumulated charge. Figure 8 demonstrates a steady-state distribution of the electron density with no indications of the presence of a local squeezed state and its oscillations (see Fig. 4). The reason for this is that at this low magnetic field, sufficient charge is released to the anode by the radial electric field and the steady state contains too little charge in the cathode–reflector–rod–anode volume. Thus, the emitted current does not overcharge this volume for squeezing to set on. (Note again the binning in the histogram near the emitter.)
Let us now compare the coaxial diode's behavior to that of an A6-MAG fed by a split cathode at the same applied voltage and 0.25 T. The only difference between the two cases is that six vanes with 40 mm deep 20° azimuthal cavities are carved in the smooth bore anode. Figure 9(a) displays the voltage, the anode current, and the wedge charge, and Fig. 9(b) shows the emitted current and the total accumulated charge. These should be compared to the corresponding coaxial diode results in Fig. 6. The wedge volume here differs from that defined for the coaxial diode in Sec. II in that it extends radially to r = 40 mm to include the charge in one of the 20° cavities of the A6-MAG.
The anode current's average amplitude (∼640 A) is twice as large as that of the coaxial diode (∼300 A) and the voltage is lower, ∼300 kV [compare to Fig. 6(a)]. When the anode current reaches its maximum (∼10 ns), the voltage and the total charge drop. The anode current oscillates at a frequency of ∼5 GHz whereas the wedge charge is at 2.47 GHz [Fig. 9(a)]. The total charge and the emitted current [Fig. 9(c)] oscillate at a frequency of ∼5 GHz. Even the voltage oscillates at this frequency but on a very small scale.
The longitudinal phase space for the magnetron is very similar to that seen in Fig. 8 and with little change in time, i.e., no squeeze state forms in the A6-MAG either. This means that the longitudinal electron oscillations have no role in the magnetron's characteristics. The wedge charge oscillates too with an amplitude modulation which flattens in time.
In Fig. 10, snapshots of the azimuthal charge density distribution at the center of the magnetron are drawn at 10 and 80 ns. In a solid cathode fed magnetron, the electrons which do not contribute to the microwave field remain close to the cathode surface or, ideally, disappear and only spokes remain. Here, there is no “cathode” and the electrons contributing to the electromagnetic wave are spokes or diocotron charge packages at the corners of a charge triangle. Note that two lines appear on the sides of the charge triangle. This is because the emitter has two sharp edges at its inner and outer radius and there is increased emission along these two circles. Also note that the charge is not necessarily uniformly distributed along the triangle sides. One should compare the snapshots in Fig. 10 with those for the coaxial diode in Fig. 7. For both cases, an = 3 diocotron shape emerges which in the A6-MAG develops into a magnetron mode.
The distribution in Fig. 10 suggests the existence of an electromagnetic π-mode which is confirmed by the azimuthal electric field appearing at the entrance of each of three cavities separated by a 2π/3 azimuthal angle. The time dependence of this field is shown in Fig. 11.
The charge triangles in Fig. 10 rotate as is evident from the azimuthal electric field, Eϕ (Fig. 11), the anode current, and the wedge charge (Fig. 9) oscillations. Spectral analysis shows that Eϕ oscillates at a fixed frequency of 2.47 GHz. For the wedge charge, two frequencies appear and do not change in time, a dominant frequency at 2.47 GHz and a smaller amplitude frequency at ∼5 GHz. Remember that fw = 5 GHz for the corresponding coaxial diode [see Fig. 6(b)]. The wedge charge and azimuthal electric field oscillations are at the same lower frequency which indicates the effect of the slow wave structure which does not exist in the coaxial diode.
The results of this section for low magnetic field show that the diocotron mode = 3 appears in the coaxial diode (Fig. 7) and the magnetron π mode (Fig. 10) appears at the same early time as the azimuthal electric field sets on (Fig. 11), while the axial electron oscillations have no effect on the azimuthal dynamics.
IV. INTERMEDIATE MAGNETIC FIELDS
In this section, the behavior of the coaxial diode and the A6-MAG are compared for applied axial magnetic fields in the range 0.3 T < Bz < 0.5 T. Assuming a potential of 214 kV (calculated separately) for electrons inside the anode space at a radius of 8 mm, the Hull magnetic field limit for the A6-MAG is ∼0.20 T, which means that the case studied in Sec. III is for conditions only slightly above this limit. In Fig. 12, the azimuthal electric field in one of the magnetron cavities is drawn for different values of the applied axial magnetic fields.
One should note that Eϕ reaches values of ∼500 kV/cm for 0.25 T (Fig. 11) and increases up to ∼800 kV/cm for 0.40 T. For magnetic fields 0.4 and 0.45 T, the maximum amplitude of Eϕ drops to ∼200 and ∼250 kV/cm, respectively. Also, note that the delay time to the onset of Eϕ for 0.25–0.35 T is less than 10 ns, which then increases to ∼23 ns for 0.40 and 0.45 T and it increases to ∼60 ns for 0.50 T.
In Fig. 13, the time dependence of the input voltage and the total current for different values of the magnetic field is drawn. The voltage rises to its maximum value in ∼4 ns while the total current first peaks at the value of the initial displacement current before electron emission starts. Then, for 0.25–0.35 T, the voltage drops as the total current increases at almost the same time ∼8–10 ns. This time is the same as the delay times of Eϕ. At 0.40 T, the delay time increases considerably and for 0.45 and 0.5 T, the total current has decreased considerably. Note that as the magnetic field increases the steady state voltage increases while the total current decreases so that the input power from 0.25 to 0.40 T decreases from ∼175 to ∼120 MW. At the same time, the maximum value of Eϕ increases, that is, the efficiency of the magnetron increases with increasing magnetic field within this range of magnetic field amplitudes.
Figure 14 displays snapshots of the azimuthal charge density distribution at the magnetron center for the different magnetic fields close to the end of the corresponding delay in the microwave generation appearance time and at late times.
For Bz ≤ 0.4 T, close to the delay time's end, a charge density triangle is evident representing a diocotron mode = 3 which then develops into a magnetron π-mode with spokes of increasing size. For Bz ≥ 0.45 T, the π-mode is replaced by a 3π/2 mode and a diocotron mode = 4 is also discerned at the end of the long delay times.
In the coaxial diode for 0.3 T ≤ Bz ≤ 0.40 T, no distinct and persistent diocotron mode appears at any time so the triangular shapes for the corresponding magnetron during the delay time are a property of the magnetron and have nothing in common with the coaxial diode. Also, the accumulated space charge longitudinal oscillations are insufficient to have any effect on the diocotron mode in a coaxial diode (as is the case for very high magnetic fields in Sec. II). These longitudinal charge oscillations start to be important for Bz ≥ 0.45 T and becomes evident in Fig. 15 where the charge distribution at the axial center of the coaxial diode for these magnetic fields can be seen. A diocotron mode = 4 develops by the end of the delay times for the appearance of Eϕ for the corresponding A6-MAG. For 0.45 T diocotron modes, = 4 and 5 are competing at first but mode nD = 4 takes over and persists (Fig. 15). By 0.5 T, = 4 is strong and persistent. This mode is the same as that seen in the A6-MAG for the same values of the magnetic field (see Fig. 14).
The oscillation frequency of Eϕ in Fig. 12 for the A6-MAG is ∼2.5 GHz for 0.25 ≤ Bz ≤ 0.40 T, it reduces slightly to 2.4 GHz for 0.45 and 0.5 T. For the coaxial diode, a wedge charge azimuthal oscillation can be picked up only for 0.45 and 0.5 T at fw ∼ 2.7 GHz. For these magnetic field values, as accumulation of the electrons increases and squeezing intensifies, oscillations in the emitted current appear too at fe ∼0.6 GHz, but in contrast to the high magnetic field case (Sec. II) these oscillations do not synchronize or affect the azimuthal charge packages rotation. The oscillations of the emitted current for the magnetic field of 0.45 T are at the same frequency fe ∼ 0.6 GHz as for the strong magnetic field of 2 T (see Fig. 2), which confirms that this frequency is related to the electron space charge longitudinal oscillation, the frequency of which is independent of the magnetic field strength.
V. DISCUSSION AND SUMMARY
It is questionable as to what parameters are to be used for the split cathode-fed magnetron to estimate the Buneman–Hartree limit.24 However, the upper limit of the magnetic field can be estimated from the results seen in Fig. 14 to be 0.4 > Bz > 0.45 T. Above this limit, the magnetron still operates but at much lower values of Eϕ, increasing turn on time delays and a different mode.
Regarding the connection between the longitudinal and azimuthal motion of the space charge in a split cathode fed device, it can be stated that such a connection exists in a coaxial diode for sufficiently high axial magnetic fields. The results of simulations show that for high magnetic fields, the longitudinal oscillations of the space charge determine the diocotron mode excited in the system.
For low magnetic field values relevant to the magnetron's operation, the diocotron mode and the magnetron mode are the same only when the magnetic field is sufficiently weak so that excess charge does not accumulate in the potential well but is directly collected as anode current. Though electrons are oscillating longitudinally, no space charge oscillations develop and there is no relation between the azimuthal and axial electron motion.
As the magnetic field increases between the Hull and Buneman–Hartree limits, no distinct or persistent diocotron modes develop in a coaxial diode. The magnetron operates well with the split cathode with increasing efficiency and unchanging frequency.
When the magnetic fields are close or slightly above the Buneman–Hartree limit, the magnetron still operates producing much lower electromagnetic fields after increasingly long time-delays to microwaves generation. Diocotron modes do develop and persist and are of the same symmetry as the magnetron modes. At the same time, the longitudinal charge oscillations have little or no effect on the wedge charge oscillations.
ACKNOWLEDGMENTS
At the Technion, this work was supported by Technion (Grant No. 2029541) and ONRG (Grant No. N62909-21-1-2006) and at the University of New Mexico by AFOSR (Grant No. FA9550-19-1-0225) and ONR (Grant No. N00014-19-1-2155). The authors thank the Referees for their positive response and their careful and critical evaluation of the manuscript.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
John Leopold: Conceptualization (equal); Investigation (equal). Yury P. Bliokh: Conceptualization (equal); Investigation (equal); Writing – review & editing (equal). Yakov E. Krasik: Conceptualization (equal); Investigation (equal); Writing – review & editing (equal). Artem Kuskov: Conceptualization (equal); Investigation (equal); Writing – review & editing (equal). Edl Schamiloglu: Conceptualization (equal); Investigation (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.