This paper presents an approach that greatly enhances both the output power and the conversion efficiency of the coaxial relativistic backward wave oscillator (CRBWO) by using coaxial dual annular cathodes, which increases the diode current rather than the diode voltage. The reasons for the maladjustment of CRBWO under a high diode voltage are analyzed theoretically. It is found that by optimization of the diode structure, the shielding effect of the space charge of the outer beams on the inner cathode can be alleviated effectively and dual annular beams with the same kinetic energy can be explosively emitted in parallel. The coaxial reflector can enhance the conversion efficiency by improving the premodulation of the beams. The electron dump on the inner conductor ensures that the electron beams continue to provide kinetic energy to the microwave output until they vanish. Particle-in-cell (PIC) simulation results show that generation can be enhanced up to an output power level of 3.63 GW and conversion efficiency of 45% at 8.97 GHz under a diode voltage of 659 kV and current of 12.27 kA. The conversion efficiency remains above 40% and the output frequency variation is less than 100 MHz over a voltage range of more than 150 kV. Also, the application of the coaxial dual annular cathodes means that the diode impedance is matched to that of the transmission line of the accelerators. This impedance matching can effectively eliminate power reflection at the diode, and thus increase the energy efficiency of the entire system.

Increasing output power and efficiency are important issues for the development of the relativistic backward wave oscillator (RBWO),1 which is one of the most promising high-power microwave (HPM) source devices.2 Advanced research has confirmed that RBWO with a coaxial rippled inner conductor, as shown in Fig. 1, which is known as a coaxial relativistic backward wave oscillator (CRBWO), can achieve high conversion efficiencies of up to 35%.3 In Fig. 1, the inner and outer conductors, which are connected equipotentially by the electron channel, serve as the anode, which enhances the quality of the electron beam.4 The introduction of the coaxial inner conductor can effectively reduce the space-charge depression of the beam's kinetic energy and increase the space-charge-limiting current.5 Theoretical analysis6 and numerical simulations3 of CRBWO predicted that its conversion efficiency would be higher than 35%, and this has been validated experimentally.7,8

FIG. 1.

Configuration of CRBWO.9 

FIG. 1.

Configuration of CRBWO.9 

Close modal

Although CRBWO is notable for its high conversion efficiency, the output powers obtained experimentally remain around the 1 GW level. This means that the output power of CRBWO is not as attractive as its conversion efficiency, particularly when compared with the output powers of 3-5 GW that have been produced experimentally by conventional hollow RBWOs in recent years.10,11 In Ref. 12, simulation results for CRBWO present high efficiencies of up to 43% in the X-band, while the output energy is approximately 1.0 GW. The highest experimental efficiency for CRBWO published to date is 36%, corresponding to an output power of 1.01 GW in the X-band.8 In lower bands, the maximum output power value obtained through numerical simulation is less than 1.7 GW with a conversion efficiency of more than 30%.13,14 However, in the corresponding experiments, the output power is 1.07 GW and the conversion efficiency is about 18%.15,16 In the previous numerical and experimental research on CRBWOs, we found that it was difficult to increase the output power with high conversion efficiency by raising the diode voltage.

However, the coaxial slow-wave structure (SWS) with double rippled conductors, as distinct from the hollow SWS with a single rippled conductor, provides a feasible way to increase the output power with high conversion efficiency by raising the diode current rather than the diode voltage. By comparing the performance of the coaxial SWS with a single rippled conductor with that of the device with the double rippled conductors, we found that the coaxial SWS with double rippled conductors has the higher conversion efficiency.17 This makes it possible to modulate and interact with the double annular beams via the double rippled conductors in CRBWO. Thus, we can increase the output power with high conversion efficiency by raising the beam current and making full use of the special structure of CRBWO. Increasing the beam current can result in augmentation of the input power. If we keep the conversion efficiency high in the case of a large beam current, then the output power will increase. To increase the diode current, CRBWO with coaxial dual annular cathodes is devised as illustrated in Fig. 2.

FIG. 2.

CRBWO with coaxial dual annular cathodes.

FIG. 2.

CRBWO with coaxial dual annular cathodes.

Close modal

This paper presents an efficient approach to enhance the output power of CRBWO with high conversion efficiency by application of coaxial dual annular cathodes. In Section II, the physical model of CRBWO with the coaxial dual annular cathodes is studied theoretically in detail, including the dispersion relation of the coaxial SWS, the diode structure with coaxial dual annular cathode, and the performance of the reflector and the collector of the spent electrons. Microwave generation enhancement by use of the coaxial dual annular cathodes is then validated and confirmed through numerical simulations in Section III. Finally, a short conclusion is given in Section IV.

The dispersion relation of the SWS qualitatively illustrates the coupling between the electron beams and the structure wave. In our previous research on CRBWOs, the inner and outer average radii of the coaxial SWS were r1 = 15.5 mm and r2 = 22.5 mm, respectively, the ripple depth was d1 = d2 = 2.0 mm, the ripple period was p = 16.0 mm, and the radius of the infinitely thin annular beam was rb = 18.75 mm, as shown in Fig. 3. We found that CRBWO as shown in Fig. 1 is good at generating microwave pulses with conversion efficiency of about 35% at a power level of 1 GW. However, when we raise the diode voltage, the output power and the conversion efficiency are reduced dramatically. This can be qualitatively explained by using the special dispersion relation of the coaxial SWS, as shown in Fig. 4, which is divided into fast-wave and slow-wave regions by the light line. In Fig. 4, the dispersion relation of the conventional hollow SWS, where the outer average radius, the ripple depth and the ripple period are the same as those of the coaxial SWS shown in Fig. 3, is also demonstrated for comparison. The fundamental harmonic branch of the TM01 mode in the conventional hollow SWS is located in the fast-wave region and its -1st harmonic branch is located in the slow-wave region. These beam lines are under the light line in any case. Therefore, the operating point, which is determined by the intersection of the wave structure with the beam lines, can be found, no matter how high the diode voltage becomes. However, in the coaxial SWS, both the fundamental and the -1st harmonic branches of the quasi-TEM mode lie in the slow-wave region.9 Thus, an electron beam with kinetic energy that is too high will deviate seriously from the π point of the quasi-TEM mode, as shown in Fig. 4, which indicates that the electron beams with energies of 700 keV and 800 keV are apparently far from the π point. This deviation of the beam line from the structure wave may result in inefficient wave-beam interactions with low conversion efficiency. Therefore, when we try to increase the output power of CRBWO by raising the diode voltage, the conversion efficiency and even the power may actually fall sharply.

FIG. 3.

Coaxial SWS with single annular cathode, as used in previous research. Inner and outer average radii are r1 = 15.5 mm and r2 = 22.5 mm, ripple depth is d1 = d2 = 2.0 mm, ripple period is p = 16.0 mm, and radius of infinitely thin annular beam is rb = 18.75 mm.

FIG. 3.

Coaxial SWS with single annular cathode, as used in previous research. Inner and outer average radii are r1 = 15.5 mm and r2 = 22.5 mm, ripple depth is d1 = d2 = 2.0 mm, ripple period is p = 16.0 mm, and radius of infinitely thin annular beam is rb = 18.75 mm.

Close modal
FIG. 4.

Dispersion relation of quasi-TEM mode in coaxial SWS of Fig. 3 and TM01 mode in conventional hollow SWS with the same outer average radius, the same ripple depth and the same ripple period.

FIG. 4.

Dispersion relation of quasi-TEM mode in coaxial SWS of Fig. 3 and TM01 mode in conventional hollow SWS with the same outer average radius, the same ripple depth and the same ripple period.

Close modal

The investigation of the dispersion relation of the coaxial SWS shows that raising the diode voltage can barely increase the output power with high conversion efficiency. We devoted our efforts to the development of a new CRBWO structure that enhances the output power with high conversion efficiency by increasing the diode current, as shown in Fig. 2. The coaxial SWS of CRBWO with the coaxial annular beams is shown in Fig. 5, where the inner and outer average radii are r1 = 42.0 mm and r2 = 50.5 mm, respectively, the inner and outer ripple depths are d1 = 1.5 mm and d2 = 2.0 mm, respectively, and the ripple period p = 14.0 mm. The dispersion relation indicates that the operating point is located close to the π point at about 9.2 GHz, as shown in Fig. 6. The beam line nearly coincides with the fundamental branch of the quasi-TEM mode. The coupling impedances of annular beams with different radii are compared in Fig. 7, from which we conclude that an annular beam with a larger radius will interact with the structure wave more efficiently because of the larger ripple depth on the outer conductor.

FIG. 5.

Coaxial SWS with coaxial dual annular cathodes. Inner and outer average radii are r1 = 42.0 mm and r2 = 50.5 mm, respectively, ripple depths are d1 = 1.5 mm and d2 = 2.0 mm, and ripple period is p = 14.0 mm.

FIG. 5.

Coaxial SWS with coaxial dual annular cathodes. Inner and outer average radii are r1 = 42.0 mm and r2 = 50.5 mm, respectively, ripple depths are d1 = 1.5 mm and d2 = 2.0 mm, and ripple period is p = 14.0 mm.

Close modal
FIG. 7.

Comparison of coupling impedances of annular beams with different radii.

FIG. 7.

Comparison of coupling impedances of annular beams with different radii.

Close modal
FIG. 6.

Dispersion relation of coaxial SWS, as depicted in Fig. 5.

FIG. 6.

Dispersion relation of coaxial SWS, as depicted in Fig. 5.

Close modal

The structure of the diode with the dual cathodes plays an important role in the operation of CRBWO with coaxial dual annular cathodes. In Fig. 2, the inner and outer cathodes are both at the same negative potential, and thus two coaxial annular beams with the same energy will be emitted explosively. However, two critical problems must be considered carefully.

First, because the inner cathode is coaxially embedded into the outer cathode, the inner cathode may be shielded by the space charge effect of the outer annular beam, and thus the explosive emission will not occur on the inner cathode.

Second, the improper diode structure is likely to result in a significant difference between the kinetic energies of the inner and outer annular beams. The significant difference in the kinetic energy means that the inner and outer beams interact with the structure wave at different operating points, which produces microwaves with an impure frequency spectrum containing two frequency components. The total electron beam energy, which consists of the potential energy and the kinetic energy, is determined by the diode voltage. However, the inner and outer annular beams, which have the same total energy, are transmitted at different radial positions and thus have different current intensities. This makes their potential energies differ from each other, and their kinetic energies also differ from each other. It is therefore possible that these electron beams with their different kinetic energies will excite electromagnetic fields that include different frequency components in the coaxial SWS. This phenomenon, which results from the performance of the two coaxial annular beams, has been reported recently.18 

We found that these two important issues have a strong dependence on the relative length lc and the transverse distance dc between the inner and outer cathodes, as shown in Fig. 2. The relative length lc has a significant effect on the shielding effect of the space charge of the outer annular beam. When lc < 0, i.e. when the inner cathode is shorter than the outer cathode, the electric field thus concentrates on the outer annular cathode, and the electric field on the inner cathode is therefore not strong enough, as shown in Fig. 8(a). This means that the explosive emission does not happen on the inner cathode. When lc ≥ 0, the electric fields on the tips of both the inner and outer cathodes are strong enough to produce coaxial annular beams, as shown in Fig. 8(b) and 8(c). Therefore, the relative length lc affects the currents of the coaxial annular beams, as shown in Fig. 9, which indicates that when lc < 0, the inner electron beam vanishes because of the space charge effect of the outer annular beam. In the case where lc ≥ 0, the inner cathode can be no longer completely shielded by the outer annular beam, and the inner annular beam is then emitted explosively. The inner beam current increases while that of the outer beam decreases with increasing relative length lc, and the total diode current of the inner and outer beams remains almost constant. This indicates that the diode structure in which lc ≥ 0 should be adopted to ensure explosive emission from both annular cathodes.

FIG. 9.

Effect of relative length lc on the beam currents with diode voltage of 650 kV.

FIG. 9.

Effect of relative length lc on the beam currents with diode voltage of 650 kV.

Close modal

To investigate the potential and kinetic energies of the inner and outer annular beams, the coaxial SWS structure with two annular beams is qualitatively approximated by using the physical model as shown in Fig. 7, in which the inner and outer radii of the coaxial waveguide are r1 and r2, respectively, and the radii of the infinitely thin annular beams from inner to outer are rb1 and rb2. The inner and outer conductors are connected equipotentially as an anode with zero potential, and thus the distribution is:

\begin{equation}\begin{array}{*{20}c}\left\{ \begin{array}{l@{\quad}l}\displaystyle \phi \left( r \right) = \frac{{\ln \frac{{r_1 }}{r}}}{{2\pi \varepsilon _0 \ln \frac{{r_2 }}{{r_1 }}}}\left( {\frac{{I_1 }}{{v_{01} }}\ln \frac{{r_2 }}{{r_{b1} }} + \frac{{I_2 }}{{v_{02} }}\ln \frac{{r_2 }}{{r_{b2} }}} \right) & r_1 < r < r_{b1} \\[12pt]\displaystyle \phi \left( r \right) = \frac{1}{{2\pi \varepsilon _0 \ln \frac{{r_2 }}{{r_1 }}}}\left( {\frac{{I_1 }}{{v_{01} }}\ln \frac{{r_1 }}{{r_{b1} }}\ln \frac{{r_2 }}{r} + \frac{{I_2 }}{{v_{02} }}\ln \frac{{r_2 }}{{r_{b2} }}\ln \frac{{r_1 }}{r}} \right) & r_{b1} < r < r_{b2} \\[12pt]\displaystyle \phi \left( r \right) = \frac{{\ln \frac{{r_2 }}{r}}}{{2\pi \varepsilon _0 \ln \frac{{r_2 }}{{r_1 }}}}\left( {\frac{{I_1 }}{{v_{01} }}\ln \frac{{r_1 }}{{r_{b1} }} + \frac{{I_2 }}{{v_{02} }}\ln \frac{{r_1 }}{{r_{b2} }}} \right) & r_{b2} < r < r_2 \\\end{array} \right. \\\end{array}\!\!\! ,\end{equation}
ϕr=lnr1r2πɛ0lnr2r1I1v01lnr2rb1+I2v02lnr2rb2r1<r<rb1ϕr=12πɛ0lnr2r1I1v01lnr1rb1lnr2r+I2v02lnr2rb2lnr1rrb1<r<rb2ϕr=lnr2r2πɛ0lnr2r1I1v01lnr1rb1+I2v02lnr1rb2rb2<r<r2,
(1)

where ɛ0 is the permittivity in the vacuum, and v01 and v02 are the electron velocities and I1 and I2 are the current intensities of the inner and outer annular beams, respectively. The electric potentials of the inner and outer annular beams, φ1 and φ2, respectively, taken from Eq. (1) are

\begin{equation}\begin{array}{*{20}c}\begin{array}{l}\displaystyle \varphi _1 = \phi \left( {r = r_{b1} } \right) = \frac{{\ln \frac{{r_1 }}{{r_{b1} }}}}{{2\pi \varepsilon _0 \ln \frac{{r_2 }}{{r_1 }}}}\left( {\frac{{I_1 }}{{v_{01} }}\ln \frac{{r_2 }}{{r_{b1} }} + \frac{{I_2 }}{{v_{02} }}\ln \frac{{r_2 }}{{r_{b2} }}} \right) \\[12pt]\displaystyle \varphi _2 = \phi \left( {r = r_{b2} } \right) = \frac{{\ln \frac{{r_2 }}{{r_{b2} }}}}{{2\pi \varepsilon _0 \ln \frac{{r_2 }}{{r_1 }}}}\left( {\frac{{I_1 }}{{v_{01} }}\ln \frac{{r_1 }}{{r_{b1} }} + \frac{{I_2 }}{{v_{02} }}\ln \frac{{r_1 }}{{r_{b2} }}} \right) \\\end{array} \\\end{array}.\end{equation}
φ1=ϕr=rb1=lnr1rb12πɛ0lnr2r1I1v01lnr2rb1+I2v02lnr2rb2φ2=ϕr=rb2=lnr2rb22πɛ0lnr2r1I1v01lnr1rb1+I2v02lnr1rb2.
(2)

Because the inner and outer annular beams have the same total energy, which consists of kinetic and potential energy, it is noted that

\begin{equation}\frac{{m_0 c^2 }}{{\sqrt {1 {-} \frac{{v_{01}^2 }}{{c^2 }}} }} {-} \frac{{e\ln \frac{{r_1 }}{{r_{b1} }}}}{{2\pi \varepsilon _0 \ln \frac{{r_2 }}{{r_1 }}}}\left( {\frac{{I_1 }}{{v_{01} }}\ln \frac{{r_2 }}{{r_{b1} }} {+} \frac{{I_2 }}{{v_{02} }}\ln \frac{{r_2 }}{{r_{b2} }}} \right) = \frac{{m_0 c^2 }}{{\sqrt {1 {-} \frac{{v_{02}^2 }}{{c^2 }}} }} {-} \frac{{e\ln \frac{{r_2 }}{{r_{b2} }}}}{{2\pi \varepsilon _0 \ln \frac{{r_2 }}{{r_1 }}}}\left( {\frac{{I_1 }}{{v_{01} }}\ln \frac{{r_1 }}{{r_{b1} }} {+} \frac{{I_2 }}{{v_{02} }}\ln \frac{{r_1 }}{{r_{b2} }}} \right).\end{equation}
m0c21v012c2elnr1rb12πɛ0lnr2r1I1v01lnr2rb1+I2v02lnr2rb2=m0c21v022c2elnr2rb22πɛ0lnr2r1I1v01lnr1rb1+I2v02lnr1rb2.
(3)

The difference between the electric potentials of the inner and outer annular beams can then be expressed as

\begin{equation}\Delta \varphi = \varphi _1 - \varphi _2 = \frac{{\ln \frac{{r_{b2} }}{{r_{b1} }}}}{{2\pi \varepsilon _0 \ln \frac{{r_2 }}{{r_1 }}}}\left( {\frac{{I_1 }}{{v_{01} }}\ln \frac{{r_1 }}{{r_{b1} }} + \frac{{I_2 }}{{v_{02} }}\ln \frac{{r_2 }}{{r_{b2} }}} \right)\end{equation}
Δφ=φ1φ2=lnrb2rb12πɛ0lnr2r1I1v01lnr1rb1+I2v02lnr2rb2
(4)

It is shown in Eq. (4) that the difference between the electric potentials Δφ can be reduced by reducing the transverse distance dc = rb2rb1 between the inner and outer cathodes. The difference between the electric potentials Δφ can also be reduced by making the inner and outer beam currents, I1 and I2, satisfy

\begin{equation}\left( {\frac{{r_{b1} }}{{r_1 }}} \right)^{I_1 } = \left( {\frac{{r_2 }}{{r_{b2} }}} \right)^{I_2 }\end{equation}
rb1r1I1=r2rb2I2
(5)

Although the analysis that combines Eq. (1), Eq. (2) and Eq. (4) relative to Fig. 10 provides an approximate investigation of the electron beam energies, it is indicated that the difference between the kinetic energies of the inner and outer annular beams is determined by both the radii and the current intensities of the inner and outer annular beams. The current intensities of the inner and outer annular beams are also significantly affected by the relative length lc, as shown in Fig. 9. In CRBWO with two coaxial annular beams, the difference between the electric potentials Δφ is expected to be as small as possible to ensure that the two annular beams have similar kinetic energies.

FIG. 10.

Approximate physical model of coaxial waveguide with two coaxial annular beams.

FIG. 10.

Approximate physical model of coaxial waveguide with two coaxial annular beams.

Close modal

Based on the theoretical study described above, the diode structure with coaxial dual annular cathodes is optimized numerically to avoid the shielding effect of the outer annular beam on the inner beam and to produce two annular beams with the same kinetic energy value. In this numerical optimization, we pay close attention to the relative length lc and to the transverse distance dc between the inner and outer cathodes.

According to the research results shown in Fig. 8 and Fig. 9, the structure where lc ≥ 0 is adopted. Through numerical simulations and optimization processes, we found that a diode structure with lc = 0 mm and dc = 2.5 mm can produce two annular beams with the same kinetic energy. In this case, the radial electric field between the inner and outer annular electron beams almost vanishes, as shown in Fig. 11(a), which indicates that the two coaxial annular beams have similar electric potentials, so that there is no radial electric field between them. The two coaxial annular beams also have similar kinetic energies, i.e., when lc = 0 mm and dc = 2.5 mm, Δφ = φ1− φ2 ≈ 0 can be achieved. If the relative length lc between the inner and outer cathodes increases with dc = 2.5 mm, then the current of the inner annular beam I1 increases, while that of the outer annular beam I2 is reduced, as shown in Fig. 9. In this case, Eq. (4) indicates that Δφ = φ1− φ2 < 0, and thus the radial electric field in the direction of -r between the coaxial annular beams appears in the diode region, as shown in Fig. 11(b), because the potential of the inner annular beam is lower than that of the outer beam. When the transverse distance dc between the inner and outer cathodes increases with lc = 0, the radial electric field between the two coaxial annular beams begins to appear as the two coaxial annular beams enter the anode, as shown in Fig. 11(c). We can predict that there will be an apparent difference between the kinetic energies of the two coaxial annular beams when they travel in the coaxial reflector and the coaxial SWS in the case where lc = 4 mm and dc = 2.5 mm, or in the case where lc = 0 mm and dc = 5.5 mm.

FIG. 8.

Distribution of scalar electric field with diode voltage of 650 kV when (a) lc < 0, (b) lc = 0 and (c) lc > 0.

FIG. 8.

Distribution of scalar electric field with diode voltage of 650 kV when (a) lc < 0, (b) lc = 0 and (c) lc > 0.

Close modal
FIG. 11.

Distribution of radial electric field with diode voltage of 650 kV when (a) lc = 0 mm and dc = 2.5 mm, (b) lc = 4 mm and dc = 2.5 mm, and (c) lc = 0 mm and dc = 5.5 mm.

FIG. 11.

Distribution of radial electric field with diode voltage of 650 kV when (a) lc = 0 mm and dc = 2.5 mm, (b) lc = 4 mm and dc = 2.5 mm, and (c) lc = 0 mm and dc = 5.5 mm.

Close modal

The results of the numerical simulation and optimization process are confirmed by the modulations of the two annular beams, as shown in Fig. 12. When lc = 0 mm and dc = 2.5 mm, the inner and outer annular beams have the same kinetic energies in the diode region and in the coaxial reflector, as shown in Fig. 12(a). Consequently, the coaxial beams are modulated to interact with the structure wave synchronously. When the relative length lc increases with dc = 2.5 mm, the two annular beams have significantly different kinetic energies in the diode region and in the coaxial reflector, as shown in Fig. 12(b). We can conclude from Fig. 12(b) that in the coaxial SWS, the positions of the peak values of the inner annular beam are staggered relative to the positions of the peak values of the outer beam. Thus, in this case, the modulations of the inner and outer annular beams that are shown in Fig. 12(b) are not as synchronous with each other as those shown in Fig. 12(a). This will lead to inefficient wave-beam interactions with low conversion efficiency. When lc = 0 mm and dc = 5.5 mm, the kinetic energies of the two annular beams are apparently quite different from each other. Thus, the modulations of the two annular beams are obviously detached from each other in both the diode region and the coaxial reflector, as shown in Fig. 12(c), and a considerable degree of chaos occurs in the bunching of the beams, especially at the end of SWS. As a result, these multiple kinetic beams will produce a microwave pulse with an impure frequency spectrum consisting of several components when lc = 4 mm and dc = 2.5 mm or when lc = 0 mm and dc = 5.5 mm.

FIG. 12.

Modulations of inner and outer electron beams with diode voltage of 650 kV when (a) lc = 0 mm and dc = 2.5 mm, (b) lc = 4 mm and dc = 2.5 mm, and (c) lc = 0 mm and dc = 5.5 mm.

FIG. 12.

Modulations of inner and outer electron beams with diode voltage of 650 kV when (a) lc = 0 mm and dc = 2.5 mm, (b) lc = 4 mm and dc = 2.5 mm, and (c) lc = 0 mm and dc = 5.5 mm.

Close modal

Based on the analysis and research described above, the diode structure with lc = 0 mm and dc = 2.5 mm should be adopted to provide explosive emission of dual coaxial annular beams with the same kinetic energy. By taking the difference between the smallest radius of the rippled conductor and that of the outer annular beam of 1.25 mm into account while also considering the beam thickness, the radii of the inner and outer annular beams are rb1 = 44.75 mm and rb2 = 47.25 mm, respectively.

The conventional reflector shown in Fig. 1 has been widely applied in RBWOs to prevent the microwaves generated in SWS affecting the performance of the diode and premodulating the electron beam.19–21 However, for the special structure of CRBWO, the coaxial reflector shown in Fig. 2, which can promote conversion efficiencies up to 43%, has been developed.12 The electric field distribution is illustrated in Fig. 13, and indicates that the stronger axial electric field can be excited on both the beam transmission lines in the coaxial reflector than in the conventional reflector.

FIG. 13.

Electric field distribution in (a) conventional reflector and (b) coaxial reflector.

FIG. 13.

Electric field distribution in (a) conventional reflector and (b) coaxial reflector.

Close modal

The electric fields on the inner and outer beam transmission lines in the conventional and coaxial reflectors are compared in Fig. 14 and Fig. 15. In our research, the axial widths of the two reflectors are both 10 mm. We found from Fig. 14 that on the inner beam transmission line at rb1 = 44.75 mm, the axial electric field in the coaxial reflector is much stronger than that in the conventional reflector, as shown in Fig. 14(a). On the outer beam transmission line at rb2 = 47.25 mm, the axial electric field in the conventional reflector becomes a little stronger than that at rb1 = 44.75 mm. However, the axial electric field in the conventional reflector is still much weaker than that in the coaxial reflector at rb2 = 47.25 mm, as shown in Fig. 14(b). The configurations and amplitudes of the electric fields on both beam transmission lines are very similar in the coaxial reflector. Also, at both rb1 = 44.75 mm and rb2 = 47.25 mm, the radial electric field in the coaxial reflector is a little weaker than that in the conventional reflector over most of the reflector range, as shown in Fig. 15. Therefore, the electric field distribution in the coaxial reflector is more beneficial for premodulation of both the inner and outer annular beams.

FIG. 14.

Axial electric fields in conventional and coaxial reflectors on (a) inner beam transmission line at rb1 = 44.75 mm, and (b) outer beam transmission lines at rb2 = 47.25 mm.

FIG. 14.

Axial electric fields in conventional and coaxial reflectors on (a) inner beam transmission line at rb1 = 44.75 mm, and (b) outer beam transmission lines at rb2 = 47.25 mm.

Close modal
FIG. 15.

Radial electric field in conventional and coaxial reflectors on (a) inner beam transmission line at rb1 = 44.75 mm, and (b) outer beam transmission lines at rb2 = 47.25 mm.

FIG. 15.

Radial electric field in conventional and coaxial reflectors on (a) inner beam transmission line at rb1 = 44.75 mm, and (b) outer beam transmission lines at rb2 = 47.25 mm.

Close modal

From a comparison of Fig. 13, Fig. 14 and Fig. 15, the coaxial reflector is better than the conventional reflector for the premodulation of the inner and outer annular beams. Also, when compared to the conventional reflector shown in Fig. 1, the coaxial reflector forms a narrow aperture anode that can notably lower the radial electric field in the diode region and enhance the quality of the electron beam.4 Therefore, the coaxial reflector, which has been proved to increase the conversion efficiency,12 should be applied to CRBWO with the coaxial dual annular cathodes.

The dump position of the electron beams has important effects on both the output power and the conversion efficiency. The dumping of the electron beams on the outer conductor requires radial expansion of the annular beam, which may destroy the uniformity of the annular beam. Also, dumping of the electron beams on the outer conductor is likely to induce the formation and oscillation of virtual cathodes at the output end of SWS.22 We therefore have both the inner and outer beams collected on the inner conductor, as shown in Fig. 2. The electron dump on the inner conductor prevents the radial expansion of the electron beam. The annular beams remain in a good modulation state where the electrons are collected. As a result, each electron can continue to contribute kinetic energy to the microwave amplification until it vanishes on the inner conductor.4 

The research results described above are validated and confirmed by particle-in-cell (PIC) simulation using the powerful CHIPIC code.23 In the numerical simulation, the generation of microwaves by CRBWO is enhanced by the use of the coaxial dual annular cathodes up to 3.63 GW at 8.97 GHz under a diode voltage and current of 659 kV and 12.27 kA, respectively, as shown in Fig. 16. The currents of the inner and outer annular beams with the same kinetic energies are Ii = 3.72 kA and Io = 8.55 kA, respectively. The conversion efficiency is 45% and the frequency spectrum is close to the theoretically predicted spectrum, as shown in Fig. 6, and is as pure as that shown in Fig. 16(a), which indicates that the electromagnetic fields generated by the inner wave-beam interaction and the outer wave-beam interaction are coherently combined together. The output power envelope, which is denoted by the red solid line in Fig. 16(b), oscillates with time, and this can apparently be attributed to over-bunching of the electron beams.24 The modulation of the beam current is uniform and consistent, as shown in Fig. 17.

FIG. 16.

(a) Output spectrum; (b) Instantaneous waveform and envelope of output power.

FIG. 16.

(a) Output spectrum; (b) Instantaneous waveform and envelope of output power.

Close modal
FIG. 17.

Modulation of beam current.

FIG. 17.

Modulation of beam current.

Close modal

The PIC simulation results indicate that the conversion efficiency remains higher than 40% over a wide diode voltage range, from 540–700 kV, as shown in Fig. 18. When the diode voltage exceeds the optimum value of 659 kV, the output power and the efficiency both fall sharply. This phenomenon corresponds to our analysis of the maladjustment of CRBWO for high diode voltages, as illustrated in Fig. 4. The output frequency varies by less than 100 MHz over the range from 460 to 740 kV. In particular, when the diode voltage is more than 581 kV, the output frequency is locked at 8.97 GHz. This is tentatively considered to be because of the merits of the quasi-resonating cavity.4,9 It is shown in Fig. 19 that the application of the coaxial dual annular cathodes greatly reduces the diode impedance. The numerical and experimental research results indicate that the diode impedance of CRBWO with a single annular beam is approximately 100 Ω.7,8 However, by using the coaxial dual annular cathode, the parallel impedance of the diode can be reduced to 40 Ω, while the efficiency remains higher than 30%. The diode impedance of 40 Ω can be matched to the characteristic impedance of the pulse-transmission line of our accelerator.25 CRBWO with the diode impedance of 40 Ω, which serves as the matched load for the accelerator, can eliminate power reflection at the diode caused by the impedance mismatch between the diode and the pulse-transmission line of the accelerator. The elimination of this mismatch can reduce waste of the pulsed power supplied by the accelerator.26,27 As a result, use of the coaxial dual annular cathodes not only increases CRBWO power efficiency, but also enhances the total energy efficiency of the HPM generation system.

FIG. 18.

Variation of generation with diode voltage.

FIG. 18.

Variation of generation with diode voltage.

Close modal
FIG. 19.

Variation of generation with anode-cathode (AK) gap.

FIG. 19.

Variation of generation with anode-cathode (AK) gap.

Close modal

This paper describes our efforts to increase the output power of CRBWO up to a level commensurate with that of the conventional hollow RBWO by using coaxial dual annular cathodes. The CRBWO structure is developed based on research into the dispersion of the coaxial SWS. The diode structure with the coaxial annular cathodes is studied and optimized numerically to avoid the effect of the inner annular cathode being shielded completely by the outer annular beam, and to ensure that the two annular beams have the same kinetic energy, which leads to coherent microwave generation with a pure frequency spectrum and high conversion efficiency. The coaxial reflector that was specially designed for CRBWO is used to enhance the conversion efficiency. The electron dump process on the inner conductor ensures that the electrons can continue to contribute kinetic energy to the microwave until they vanish on the inner conductor. The numerical simulation results proved that application of the coaxial dual annular cathodes promoted output power generation up to 3.63 GW with conversion efficiency of 45% at 8.97 GHz under the diode voltage and current of 659 kV and 12.27 kA, respectively. The conversion efficiency remains above 40% over a diode voltage range that is more than 150 kV wide, while the variation of the output frequency is less than 100 MHz over a voltage range of nearly 200 kV. Also, the application of the coaxial dual annular cathodes can reduce both the diode impedance and the power reflection at the diode caused by the mismatch between the diode impedance and that of the transmission lines of our accelerators. This reduction in the mismatch increases both CRBWO power efficiency and the total energy efficiency of the HPM generation system.

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