Since the research toward high-power millimeter-wave generator becomes a tendency in high-power microwave, overmoded structure with the high-order mode has been a considerable interest because of its potential to increase power handling capacity (PHC). To expand the PHC of V-band transit-time oscillator and excite higher mode TM03, a dual annular multiple-beam cathode has been proposed. In the geometric structure of the dual annular multiple-beam diode, cathode rods around two concentric circles are uniformly placed on cathode base, and each circle has several graphite rods. Because of space charge shielding effect and fluctuation of electron beam, explosive emission current of inner and outer cathode circles is difficult to be balanced, and the electron beam transmission rate is not very high. To solve those two problems, the relative length (ΔL) and the relative radius (Δr) between the inner and outer cathode circles are optimized to obtain balanced currents of inner and outer beams and a good transmission rate. In this paper, the preliminary study of a dual annular multiple-beam cathode is carried out by optimizing the cathode structure. When ΔL and Δr are equal to 2.5 and 4 mm, respectively, the dual annular multiple-beams cathode can provide 2.71 kA uniform intense relativistic electron beams under 421 kV, and the magnetic field is 1.2T. As a simulation result, explosively emitted and the 95.2% of total beam current transmission rate can be reached.

With the development of the millimeter-wave technology, high frequency, high power, and compact structure are essential requirements on millimeter-wave HPM devices for future applications. The product of output microwave power and squared frequency (Pf2) is an important parameter to measure performance of HPM generators. Researchers make efforts on higher band devices in order to enhance frequency and power of HPM devices in decades.1–10 Furthermore, there are many restrictions in the development of HPM to higher frequency band, such as small PHC,11 high guiding magnetic field, and low efficiency. To solve those problems, especially small PHC of output cavities, overmoded structure with high-order operating mode in TTO starts becoming a scientific research hotspot because of its ability to increase PHC at a high frequency band.12–15 There are already some works in the overmoded coaxial structure of relativistic vacuum electron devices, which has been proved having higher PHC than the conventional hollow structure.16 On account that high-order mode is considered as the volumetric wave remarked by higher PHC than the surface wave, such as TM01,17 and a dual annular multiple-beam cathode can increase the diode current rather than increasing the diode voltage;18,19 a dual annular multiple-beam cathode is proposed to excite higher mode TM03 in an overmoded coaxial structure of relativistic vacuum electron devices in this article.

Compared to the single annual beam, there are some quality differences due to the different geometry construction of cathode. The multiple-beam cathode poles are cylindrical structure, and when compared to the annual beam cathode pole, it has smaller emitting area. So that the multiple-beam cathode poles may have worse transmission environment and even appear back electron because of the bigger emitting current density. To solve this problem, as much number of cathode poles as possible are set around cathode base. The current density is controlled below 50 A/cm2. Moreover, beam tunnel of multiple-beam cathode and annual beam cathode also has different influence for beam quality. When moving ahead under radial and axial magnetic fields, electrons have rotation motion and revolution motion on the basis of Lorentz force. The multiple-beam cathode has cylindrical beam tunnel, which has annular restriction when compared with circular beam tunnel of annul beam cathode. So that when moving ahead to collector, electrons have more likely to bombard tunnel's wall in the cylindrical beam tunnel. To solve this problem, the magnetic field needs to be designed carefully.

To some extent, the transmission of electron beams determines the experimental performance of devices.20,21 The beam–wave interaction efficiency, power, and pulse width of microwaves are greatly affected by beam quality. Usually, the transmission rate and the uniformity of current are used to describe the quality of IREB.

For obtaining good transmission of multiple-beam, some critical problems must be considered carefully.22–28 First, since dual annular multiple-beam cathode has two circles of cathode rods, the distribution of electric field is not uniform on tip of inner and outer cathode circles, which leads to inner cathode circle shielded by the space charge effect of the outer annular beam, and, thus, the explosive emission may not occur on the inner cathode circle. Second, the dual annular beam with different emission fields has significantly different energy of electron, which will excite undesired operating points in the HPM device. That is to say, microwave has an impure frequency spectrum containing more than one frequency components. These phenomena resulted from the performance of the two coaxial annular beams have been reported18,19 in an X-band backward wave oscillator. However, the dual annular multiple-beam cathode in a V-band TTO with a much smaller cathode size has not been investigated before. According to the aforementioned problems of multiple-beam emission, the transmission rate is always used to measure transporting performance of IREB. The preliminary study of the dual annular multiple-beam cathode of V-band TTO is presented in this paper to achieve balanced beam currents and good transmission rate of inner and outer cathode circles by means of theoretical analysis and simulation.

This article is organized as follows. Configuration of dual annular multiple-beam cathode of V-band TTO is presented in Sec. II. In Sec. III, current consistency and transmission rate of multiple beams are analyzed based on two analyzation details. One is theoretical analyses of space charge shield effect of inner and outer cathode rods; another is transmission rate with different radial distance between inner and outer cathode circles. Conclusions of this paper are given in Sec. IV.

To excite high-order mode of cavity, researchers usually used annular beam or multiple-beam to interact with electromagnetic wave in previous work.25–27 The annular multiple-beam cathode has advantages in obtaining higher electron beam current at a lower operating voltage level and suppressing TEM mode in drift tubes coupling between cavities. In addition, dual annular multiple-beam cathodes have advantages in exciting TM03 mode because beams are set at double peak of longitudinal electric field (Ez) at the radial direction.

In an HPM generator, an electron optical system is applied for producing, focusing, and transporting electron beam to beam-wave interaction area. To excite high-order microwave of resonant cavities, there are strict requirements of uniform, rigid, and steady electron beams. It is needed to optimize anode and cathode structures of diode to guarantee stable explosive electron emission.

The modeling of the dual multiple-beam diode is shown in Fig. 1, which is established by a 3D PIC simulation software CHIPIC. It contains metallic anode and total 48 cathode rods uniformly distributing in inner and outer circles. Graphite cathode rods are arranged at the same angular interval of 15° in each circle. According to the calculation of electron loading conductance and to reach a GW-level input energy, the accelerating voltage to provide beam energy for beam-wave interaction is around 320–450 kV to guarantee an IREB. The typical electron beam parameters chosen for PIC simulation voltage of diode are set to 400 kV in consider action of accelerator operating ability. As a consequence, the diode current is larger than 2.5 kA by simulation.

FIG. 1.

Three-dimensional configuration of a dual annular multiple-beam diode with TM03 mode cavities.

FIG. 1.

Three-dimensional configuration of a dual annular multiple-beam diode with TM03 mode cavities.

Close modal

The schematic of diode structure in the longitudinal XZ plane is shown in Fig. 2. The essential parameters are listed in Table I. As V-band cavities need to cutoff TM01 mode in the drift-tube, so that the width of beam tunnel should smaller than 2.5 mm. The diameters of cylinder cathode rods db and cylinder drift tube d in Fig. 2 are set to be 1.2 and 2.4 mm, respectively. As to suppressing lateral movement of electron in an extremely narrow beam tunnel, a longitudinal immersed solenoid magnetic field is also introduced with a magnetic field of 1.2T in the device. In Fig. 2, ΔL = Li-Lo stands for relative length of outer and inner cathode rods, which is the inner cathode rod length minus the outer one. Δr = rbo-rbi is relative radius of inner and outer cathode rods, which is the outer cathode circle radius minus the inner one. ΔL and Δr will influence the space charge shielding effect and transmission rate through shaping the electric field between anode and cathode.

FIG. 2.

Longitudinal cross section of dual annular multiple-beam cathode.

FIG. 2.

Longitudinal cross section of dual annular multiple-beam cathode.

Close modal
TABLE I.

Essential parameters of multiple-beam diode structure.

Par. L0 Li rbo rbi db d
Val. (mm)  12.5  15  32.5  28.8  1.2  2.4 
Par. L0 Li rbo rbi db d
Val. (mm)  12.5  15  32.5  28.8  1.2  2.4 

Fig. 3 shows a schematic diagram of overmoded coaxial five-gap buncher cavity of TM03 mode. The width of drift tube and cavity is dt = 0.6 and w = 2.2 mm. Because the TM03 mode has three peaks of Ez along the radial direction, the dual annular multiple-beam beams will be loaded at location no. 1 and location no. 2 for exciting the TM03 mode in cavities. The location no. 3 is aligned to a coupling channel for effectively coupling among five cavities.

FIG. 3.

Schematic diagram of TM03 mode cavity and Ez distribution at the radial direction.

FIG. 3.

Schematic diagram of TM03 mode cavity and Ez distribution at the radial direction.

Close modal
To analyze the explosive emission characteristics of the multiple-beam cathode, the cylindrical cathode rods can be approximated as some small spherical bulges. Before explosive emission happens, field-emission will be initially set up on cathode rod surface. According to the following field-emission model function:29 
J = A β E 2 φ t 2 y exp B φ 3 / 2 β E 1 θ y ,
(1)
where J is the current density of field-emission, φ is the work function of conductor surface, E is the surface electric field, and the value of constants A and B is 1.5414 and 6.8309, respectively. y is the reduced quantity of potential barrier by the Schottky effect. t2(y) and θ(y) are elliptic functions of Nordheim. β is the surface field enhancement factor.
Using k1 and k2, we replace the constants of formula (1)
k 1 = A φ t 2 y , k 2 = B φ 3 / 2 θ y .
(2)

Due to the beam physical emission mechanism of a dual annular multiple-beam cathode is much similar to that of a carbonate nanotube, their field enhancement factor is both related to shape and ratio of length to radius. To describe the space charge effect, the field enhancement factor defined by researchers is used to describe the emission of multiple-beam cathode rods.

β, the surface field enhancement factor, is defined as30,31
β = L / d b 2 ln L / d b 0.3 ,
(3)
where L is the length of cathode rod and db is a diameter of cathode rods.
Emission current can be expressed as
I = J S ,
(4)
where S = πdb2 is the area of cathode rod surface. By simultaneously combining equations (1)–(4), we can obtain
I = π k 1 L 2 β E d b ln L / d b 0.3 2 e k 2 d b 2 ln L / d b 0.3 L 2 β E .
(5)

It can be seen from Eq. (5) that under a certain voltage, the current of multiple-beam electron changes accordingly when L or db changes.

According to analysis of formula (5), the results of theoretical calculation are presented in Fig. 4, and relative parameters of formula (5) are shown in Table II. Under a stationary condition and 1.2 mm diameter of cathode rods, the normalized beam current I/I0 of a cathode rod increases constantly as the normalized length of the cathode rod L/L0 increases. It implies that the length of the cathode rode affects the beam current.

FIG. 4.

Theoretical calculation results (db = 1.2, L0 = 13 mm).

FIG. 4.

Theoretical calculation results (db = 1.2, L0 = 13 mm).

Close modal
TABLE II.

Parameters of formula (5).

Par. φ/eV A B L0 db
Val. (mm)  4.5  1.541 × 10−6  6.831 × 107  13  1.2 
Par. φ/eV A B L0 db
Val. (mm)  4.5  1.541 × 10−6  6.831 × 107  13  1.2 

It is obvious that the space charge shielding effect will happen on dual annular multiple-beam cathode. The outer cathode circle has significant effect on the inner one if the relative length of the outer and inner cathode rods ΔL is not carefully chosen.18 For convenience, we take an inner cathode rod and an outer one at the same angle of circle as a representative to investigate the space charge shielding effect.

The Ez vs relative length ΔL is simulated and shown in Fig. 5. It illustrates that when ΔL < 0, it means the outer cathode rod is longer than the inner one, and Ez will concentrate on the outer cathode rod as shown in Fig. 5(a).

FIG. 5.

Space charge effect and Ez vs the relative length of outer and inner cathode rods ΔL.

FIG. 5.

Space charge effect and Ez vs the relative length of outer and inner cathode rods ΔL.

Close modal

As a result, Io>Ii due to the space charge shielding field from the outer cathode rod appearing on the inner cathode rod. When ΔL ≥ 0, it means the inner cathode rod is longer than the outer one, Ez will gradually increase on tip of the inner cathode rod and weaken on the outer cathode rod as shown in Figs. 5(b) and 5(c). Therefore, ΔI will increase when the inner beam current Ii increases.

Figure 6 shows ΔI changes vs ΔL. It can be seen that at ΔL = 2.5 mm, when the inner cathode rod is 2.5 mm longer than the outer one, ΔI closely reaches to zero. It implies that the inner beam current is balanced with the outer one. If the inner and outer beam currents are almost same as each other, the interaction between microwave and beams can perform well to produce high-power microwave with a pure frequency spectrum.

FIG. 6.

Consistency of cathode current vs ΔL.

FIG. 6.

Consistency of cathode current vs ΔL.

Close modal

Because the TM03 mode has three locations in resonant cavity at the radial direction with concentration of peak axial electric field, for mode selection, it is efficient to set two electron beam circles at concentrated locations of 1 and 2 in Fig. 7 so that higher couple impedance and electron load conductance will be obtained.

FIG. 7.

Schematic diagram of cavity operating in the TM03 mode.

FIG. 7.

Schematic diagram of cavity operating in the TM03 mode.

Close modal

That is to say, a dual annular multiple-beam cathode has natural superiority in exciting TM03 mode compared to a one annular beam cathode because the dual annular multiple-beam cathode can be easily aligned at the locations where the electric field strength is concentrated. Furthermore, the dual annular multiple-beam cathode has a big advantage over an annular beam cathode because independent beam tunnels can transmit independently in tunnels without supporting conductor poles. Such an advantage will result in less microwave reflection of high-order mode cavities. In other words, it may bring less asymmetrically angular modes.

It can be seen in Fig. 7 that the TM03 mode has three locations of peak axial electric field at the radial direction shown in locations 1–3. To align the cathode circles to the peak axial electric field locations, the outer cathode circle of the dual annular multiple-beam cathode will be placed at location 1, and the inner one will be placed at location 2. Moreover, the coupling hole of microwave is set at location 3.

To some extent, the transmission of electron beams between cathode and anode in an HPM generator determines the experimental performance. The beam-wave interaction efficiency, power, and pulse width of microwaves are often greatly affected by beam quality. Uniformity of beams and transmission rate are used to describe the quality of intense relativistic electron beams (IREB). In addition to ΔL, Δr is also an important parameter to affect the fluctuation of electrons through variation of radial electric field (Er).

The distance Δr not only does represent the locations of inner and outer beams, but also is an important parameter affecting the beam transmission rate and the difference of inner and outer beam current. The beam transmission rate and the difference between the inner and outer beam current have a relationship of Er, and Er on cathode rod tips is closely related to Δr. Therefore, analysis and simulation are aiming at relationship between Er and Δr. Figure 8 shows Er on outer and inner cathode rod vs Δr. When Δr changes from 2 to 6 mm, the difference of the Er between the outer and inner cathode rod changes. The smallest difference of the Er appears when Δr comes to 4 mm.

FIG. 8.

Relationship between Er and Δr.

FIG. 8.

Relationship between Er and Δr.

Close modal

Good transmission rate of multiple-beam electron beam is always a target for explosive emission because electrons bombarding on tunnel wall will not only make loss of electron energy but also result in breakdown and pulse shortening for electrodynamic structure. Beam current is calculated by particle-in-cell simulation, and the changes of ΔI and transmission rate vs Δr from simulation are shown in Fig. 9.

FIG. 9.

Changes of ΔI and transmission rate vs Δr.

FIG. 9.

Changes of ΔI and transmission rate vs Δr.

Close modal

By means of simulation, the total beam current emitted from whole cathode containing inner and outer cathode circle is 2.71 kA, and a current value of 2.58 kA passes through the drift beam tunnel. Such a result illustrates that when ΔL = 2.5 and Δr = 4 mm, the beam transmission rate will reach 95.2% as shown in Figs. 9 and 10.

FIG. 10.

Simulation results of beam current.

FIG. 10.

Simulation results of beam current.

Close modal

In Ref. 21, the typical beam transmission rate of an X-band multiple-beam conventional klystron was measured experimentally to be 86%. In Ref. 32, the typical beam transmission rate of 92% in an X-band multiple-beam high-power klystron was obtained experimentally. Compared with X-band high-power microwave generator, the V-band TTO described in this article is with a very high frequency of around 60 GHz and a relatively small size. As a result, a beam transmission rate of 95% from simulation is reasonable.

Based on the research results described earlier, it can be inferred that the relative length of outer and inner cathode rods ΔL = 2.5 mm and the relative radius of inner and outer cathode rods Δr = 4 mm are the optimal parameters of the dual annular multiple-beam cathode. The cathode with such parameters will provide dual annular multiple-beam explosive emission with a balanced beam current and good transmission rate.

In the physical model of annular multiple-beam cathode, there is Coulomb force exist between each beam, and Lorentz force within each electron beam. Under microscopic circumstance, the ampere force between two beams can illustrate by some formulas. Two relevant formulas of above-mentioned forces can be expressed by
F 12 = q 1 q 2 4 π ε 0 R 12 2 ,
(6)
F q = q v × B .
(7)

In formulas (6) and (8), q is the quantity of electricity charge, ε 0 is the permittivity of vacuum, and R12 is the distance between two electrons of two beams, whose numerical value is about 7.5 mm. In formulas (7) and (9), v stands for the angular velocity of electron under the magnetic field, which is represented by the letter B in the end formular. Table IV shows the specific value of parameters in formulas (6) and (7).

Put the value of Table IV into the formulas will obtain the F12 and Fq, which is shown as follows:
F 12 = q 1 q 2 4 π ε 0 R 12 2 = ( 1.6 × 10 19 ) 2 4 π × 8.85 × 10 12 × ( 7.5 × 10 3 ) 2 4.09 × 10 24 ( N ) ,
(8)
F q = q v × B = q v B sin π 2 1.6 × 10 19 × 3 × 10 8 × 1.2 . ≈  5.76 × 10 8 N .
(9)

Comparing the values of F12 and Fq shows conclusion that the Lorentz force within beams of annular multiple-beam cathode is as much as about 16 orders of magnitude of the Coulomb force. So that the physical model of multiple-beam ignores the coupling between the multiple beams. Some details of emitted current and collected current of multiple-beam cathode are listed in Table III.

TABLE III.

Emitted and collected of beam current.

Parameters Value
Emitted current of one/all inner beam  56.5A/1.356 kA 
Emitted current of one/all outer beam  56.5A/1.356 kA 
Collected current of one/all inner beam  54.2A/1.300 kA 
Collected current of one/all outer beam  53.7A/1.288 kA 
All emitted current/collected current  2.712 kA/2.588 kA 
Parameters Value
Emitted current of one/all inner beam  56.5A/1.356 kA 
Emitted current of one/all outer beam  56.5A/1.356 kA 
Collected current of one/all inner beam  54.2A/1.300 kA 
Collected current of one/all outer beam  53.7A/1.288 kA 
All emitted current/collected current  2.712 kA/2.588 kA 
TABLE IV.

Parameters of formulas (6)–(9).

q (C) ε0 (F/m) R12 (m) v ( m / s ) B ( T )
1.6 × 10−19  8.85 × 10−12  7.5 × 10−3  3 × 108  1.2 
q (C) ε0 (F/m) R12 (m) v ( m / s ) B ( T )
1.6 × 10−19  8.85 × 10−12  7.5 × 10−3  3 × 108  1.2 

Under the conditions of magnetic field 1.2T, ΔL = 2.5 mm, and Δr = 4 mm, Fig. 11 shows the simulation results of beam trajectory at axial section, where the red color stands for the multiple-beam electron. It can be seen from Fig. 11, the electron beams transmit stably. To illustrate the balanced kinetic energy of inner and outer circle of multiple-beam electron, the space phase diagram of every electron is calculated by PIC simulation, as shown in Fig. 12. It can be seen that when electron beam enters the anode of axial position at 60 mm, the kinetic energy of inner and outer circle beams will reach an almost balanced kinetic energy. It is inferred that the electric field on inner and outer cathode rod is balanced.

FIG. 11.

Trajectory of particle at axial section of diode.

FIG. 11.

Trajectory of particle at axial section of diode.

Close modal
FIG. 12.

Kinetic energy of particles at inner and outer circle.

FIG. 12.

Kinetic energy of particles at inner and outer circle.

Close modal

In this paper, the optimization of a dual annular multiple-beam cathode applied to a V-band TTO based on the TM03 mode is presented. To deal with the space charge shielding effect between the inner and outer circles, a cathode consisting of 24 rods on inner circle and 24 rods on outer circle with a length difference is analyzed and designed. After optimizing structure size of cathode, the length of the outer cathode circle is 2.5 mm shorter than the inner cathode circle, and the radius of the outer cathode circle is 4 mm larger than that of the inner cathode circle. The inner and outer length of cathode rod are Li = 15 and Lo = 12.5 mm, respectively, and radii of inner and outer cathode are rbi = 28.8 and rbo = 32.8 mm, respectively. Under the condition of the ΔL = 2.5 and Δr = 4 mm, the balanced current of inner and outer beam can be obtained, and the total beam current transmission rate reaches 95.2% at the working voltage of 421 kV. In the future work, large PHC of a coaxial V-band relativistic TTO operating in the TM03 mode will be studied in more details by use of the dual annular multi-beam cathode.

This work was supported by the National Natural Science Foundation of China (No. 12205369), the Independent Research Foundation of College of Advanced Interdisciplinary Studies (No. 22-ZZKY-07), and Hunan Provincial Xiao-he Sci-Tech Talents Funding (No. 2023TJ-X62).

The authors have no conflicts to disclose.

Fanbo Zeng: Writing – original draft (equal). Jiande Zhang: Project administration (lead). Juntao He: Resources (lead). Junpu Ling: Methodology (equal).

The data that support the findings of this study are available within the article.

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