A 0.3 THz Multi-Beam Extended Interaction Klystron Based on TM10,1,0 Mode Coaxial Coupled Cavity

A new conceive of high-order mode multi-beam Extended interaction Klystron (EIK) at the terahertz (THz) band is proposed. The interaction circuit based on the coaxial coupled cavity at <inline-formula> <tex-math notation="LaTeX">${\mathrm {TM}}_{\mathrm {10,1,0}} - 2\pi $ </tex-math></inline-formula> mode is designed and analyzed. Benefiting from the larger transverse space of the coaxial coupled cavity operating at high-order mode, multiple electron beams are formatted to reduce the current density of each electron beam. The crucial parameters associated with the interaction are determined by theoretical calculation and optimized by three-dimensional (3D) Particle-in-Cell (PIC) simulation. The results of theoretical analysis and PIC simulation demonstrate that the EIK operates at <inline-formula> <tex-math notation="LaTeX">${\mathrm {TM}}_{\mathrm {10,1,0}} - 2\pi $ </tex-math></inline-formula> mode without mode competition and self-oscillation. Driven by ten individual electron beams with the voltage of 20 kV, the current of 0.1 A and the current density of 318 A/cm<sup>2</sup>, the maximum gain of 48 dB is achieved at the center frequency of 300 GHz with a 3-dB gain bandwidth of 200 MHz. And the corresponding output power and efficiency are 312.5 W and 1.56 %. The results indicate that the proposed EIK is a practical approach to generate high power terahertz waves using the thermal beam with a low compression ratio.


I. INTRODUCTION
The extended interaction device is a kind of vacuum electron device with high power, efficiency and reliability at millimeter and sub-millimeter wave frequencies, including Extended interaction Klystron (EIK) and Extended interaction Oscillator (EIO). Benefiting from the development of micro-fabrication technology and electronic optics technology, the extended interaction devices have been put into the applications that require excellent performance, ehigh stability and compact dimensions, such as satellite communication and the space-born radar [1]- [4]. However, there are many challenges in developing the extended interaction device at the terahertz band, the other factors such as conductivity, electron velocity spread and thermal dissipation become more significant than at lower frequencies [5], [6]. In addition, the dimension of the device decreases rapidly The associate editor coordinating the review of this manuscript and approving it for publication was Giovanni Angiulli . while the frequency increases to the terahertz band, which enormously increases the precision requirements of fabrication. The planar interaction circuits for millimeter wave EIKs such as ladder-type circuits and folded waveguides, are expected to be expanded to THz band, due to the promotion of high precision fabrication technology including Wire Electrical Discharge Machining (WEDM), lithographic (LIGA) and Nano Computer Numerical Control (CNC) milling [7], [8]. Moreover, the ladder-type extended interaction cavity operating at high-order mode is a potential solution [9].
Therefore, current challenges about the micro-fabrication of interaction circuits are perhaps trumped by the difficulties in producing small diameter, high current electron beams [5]. For a practical beam optics system optimized for a lifetime and electrical stresses, the current density emitted from the cathode must be kept within a reasonable range, and the compression of the beam is also limited. Thus, the beam current is inevitably restricted, which results in a significant reduction in output power and efficiency. One solution is to develop the sheet beam devices that are expected to break the limitation of beam current and raise the power level [10]- [12]. However, the propagation of the electromagnetic waves in the beam tunnel is difficult to be cut off completely, which causes the signal feedback between cavities and results in the undesired oscillation. Another potential approach is to use the multi-beam interaction system to reduce the current density of beams [13], [14].
In this paper, a new type of EIK operating at TM 10,1,0− 2π mode with ten electron beams is proposed to overcome the above restrictions. Compared with the conventional EIKs operating at fundamental mode, the proposed high-order mode EIK has a large structure to accommodate multiple beams, which is expected to reduce the current density of the beams and suitable for the present technology of the cathode.The current density of the electron beam is very high, and the current density of the cathode does not reach the current density of the electron beam. In Sec. II, the highorder mode interaction circuit including the coaxial coupled cavity and the multi-extraction output cavity with axial output circuit are designed, and their crucial parameters associate with the interaction are analyzed and determined. In Sec. III, the possibilities of mode competition and self-oscillation are evaluated. The main performances and the power capability of the proposed EIK are demonstrated by 3D PIC simulation in Sec. IV. Conclusions and discussions are given in Section V.

II. INTERACTION CIRCUIT DESIGN A. HIGH-ORDER MODE COUPLED CAVITY
The schematic of high-order mode coaxial coupled cavity with seven structural periods that is used as the bunching cavity is shown in Fig. 1, The cavity material is Oxygen-free copper, the main parameters of the coupled cavity are listed in TABLE 1. The coupled cavity consists of seven coaxial cavities, which are coupled periodically by ten coupling tunnels. In general, the higher order mode has a higher frequency in the same cavity, in other words, at the same resonant frequency, the cavity operating at higher order mode has a larger dimension than that operating at the lower order mode. Therefore, the TM 10,1,0 mode is selected as the transverse operating mode in pursuit of a larger dimension. In order to maintain the phase consistency of the axial electric  field in each period of the coupled cavity, the 2π mode is selected as the longitudinal operating mode. The axial electric field pattern of TM 10,1,0− 2π at 300 GHz is given by electromagnetic (EM) simulation using CST-MWS solver, as shown in Fig.2. According to the field pattern, ten cylindrical beam tunnels are arranged in where the axial electric field is extreme.
The gap coupling coefficient is an important parameter that reflects the interaction between the electron beam and the field. The case of a single gap is firstly considered. Because of the non-reentrant beam tunnel and the narrow gap, the amplitude of the axial electric field in the gap is approximately equal at any position. Referring to the research of [22], the square of the coupling coefficient M of a single gap is defined as where M z is the axial coupling coefficient; M r is the radial coupling coefficient; θ d is the gap transit angle; β e is the longitudinal propagation constant of the beam; γ e is the radial propagation constant of the beam; I 0 and I 1 are 0 th -order and 1 st -order first kind modified Bessel-functions, respectively; V 0 is the electron velocity, which is determined by the beam voltage V 0 . Thus, the coupling coefficient is a function of the gap length and the electron velocity. For a bunching cavity, to optimize the gain bandwidth, the gap length should be adjusted to maximize the gain bandwidth factor defined as F = M 2 (R/Q). Assuming that the beam tunnel radius R t is 0.12 mm and the beam radius R b is 0.1 mm, the relationship between the gain bandwidth factor F and the gap length d for different beam voltage V 0 is illustrated in Fig. 3(a). The cathode with a larger radius is used to compress to the radius of the electron beam.,and we have mentioned it in our design. The gain bandwidth factor can be improved by raising the beam voltage, and the gap length corresponding to the maximum value is larger. For an output cavity, the gap length should be optimized to the highest efficiency. The efficiency η of the output cavity with a single gap can be estimated by where: Z L is the gap impedance; Q L is the loaded quality factor; Q 0 is the intrinsic quality factor; Q ext is the external quality factor; I 0 is the DC beam current of individual beam; i 1 is the fundamental component of the induced current, it can generally reach 1.4 to 1.7 times of the I 0 for multi-stage bunching; m is the number of beams, in this paper, m is equals to 10. Assuming that Q ext = Q 0 , I 0 = 0.1 A, i 1 = 1.5I 0 , R t = 0.12 mm, R b = 0.1 mm, the relationship between the efficiency η and the gap length d for different beam voltage V 0 is given in Fig. 3(b). Similar to Fig. 3(a), the efficiency is improved by raising the beam voltage, and the optimum gap length increases with the beam voltage. However, the high beam voltage may lead to the breakdown of electron gun because of the small dimensions at THz band. Therefore, our EIK operates with the beam voltage of 20 kV, then the optimum gap length of the bunching cavity is 0.11 mm and that of the output cavity is 0.13 mm, as shown in Fig. 3(a) and Fig. 3(b), respectively.
However, in order to balance the efficiency and bandwidth, and considering the deviation of theoretical calculation, the gap length of all the cavities are optimized to 0.12 mm by P-IC simulation. If the operating beam voltage is determined, the period length L of the coupled cavity can be calculated by where ϕ is the phase shift of field, and ϕ = 2π when the coupled cavity operates at 2π mode; v p is the phase velocity of field. We make v p equal to v 0 , and the beam voltage at this condition is called synchronous voltage.

B. BEAM PARAMETERS
In theory, the gap coupling coefficient is largest if the beam radius R b equals to the beam tunnel radius R t . However, to prevent electrons bombard on the walls of the beam tunnels, the ratio of R b and R t need to be restricted. If the beam voltage V 0 is 20 kV, assuming R b = 0.83Rt, i 1 = 1.5I 0 , d = 0.13mm, the relationship between the estimated efficiency η of the output cavity, the beam current I 0 and the beam radius R b is given in Fig. 4(a), the relationship between the current density J 0 of beam, the beam current and the beam radius is given in Fig. 4(b). Fig. 4(a) indicates that the efficiency can be improved by increasing the beam current or decreasing the beam radius, but at the expense of an evident increase in current density of beam, as shown in Fig. 4(b). To achieve an efficiency of 1.5%, the current density of beam needs to exceed 1000 A/cm 2 . Besides, the radius of the beam tunnel should be small enough to cut off the propagation of electromagnetic wave in the tunnel while accommodating the electron beam with an appropriate current density. Therefore, the electron beam with small diameter and high current density is the key to improve the output power and efficiency of devices.
The cold cathode based on the field emission principle has an extremely high current density [16], [17], but the total current is low, and there are other problems such as inconsistent electron velocity and low reliability, thus, the thermalcathode with high current density is more appropriate for some time to come. Recently, high emission current density of up to 50-100 A/cm 2 have been achieved with a meaningful life by thermal cathode in the experiment [18], [19]. Besides, for the high power klystrons, a conventional approach is to use magnetic field compression to improve the beam current densities to over 100 A/cm 2 . However, a high compression ratio requires a strong magnetic field of the focus system. The most advanced designs have the ability to focus a 100 mA thermal beam through a tunnel diameter of 0.125 mm [6]. After considering various factors comprehensively, the parameters of both electron beams are set as V 0 = 20 kV, I 0 = 0.1 A, R b = 0.1 mm,and R t = 0.12 mm, thus the current density of the electron beam is J 0 = 318 A/cm 2 , which is feasible for the thermal beam with high emission current density and low compression ratio (<20). However, the efficiency is only 0.4 % under this condition according to Fig. 4(a), thus the efficiency single gap is difficult to be improved because of the limitation of beam current.

C. MULTI-EXTRACTION OUTPUT CAVITY
If the multi-gap coupled cavity operates at the 2π-mode, the characteristic impedance is proportional to the number of gaps, thus, it is an effective approach to improve the power and efficiency of devices. Referring to the research of [15], and introducing the radial coupling coefficient (M r ), the square of the multi-gap coupling coefficient can be defined as where: N is the number of gaps, and the value of β e L is an integral multiple of 2π when the cavity operates at 2π mode. Fig. 5(a) gives the square of the multi-gap coupling coefficient M versus the beam voltage V 0 . Around the synchronous voltage of 20 kV, the value of M 2 for different number N of gaps tends to a fixed value, which is equal to the coupling coefficient of a single gap. However, when the beam voltage deviates from the synchronous voltage, the variation of M 2 is more significant with the increase of N . Actually, the electrons lose part of kinetic energy and slow down when they pass through the gap, which is equivalent to the drop in beam voltage. The estimated efficiency η of the output cavity versus the number N of gaps is given in Fig. 5 Benefited with the linear growth of the R/Q of coupled cavity, the efficiency is improved significantly with the increase of the number of gaps, but ultimately saturates because of the decrease in M 2 caused by the deceleration of electrons. Therefore, it is meaningless to increase the number of gaps without restriction. Besides, increasing the number of gaps may cause the mode competition.
To overcome the above problem that a single multi-gap output cavity cannot extract energy from the beam efficiently, a multi-extraction output cavity is proposed, as shown in Fig. 6. The energies of beams are extracted twice by two TM 10,1,0 mode coaxial coupled cavities with five periods, and the output power signals are superimpose in the same  phase by the axial output circuit. The axial output circuit is consisted of the rectangular coupling holes with a width W h of 0.46 mm, the mode converters with a radius R c of 1.3 mm and the circular waveguide at TM 01 mode with a radius R g of 0.46 mm. The mode converters is used to convert the TM 1,0 mode in the coupling holes to the TM 0,1 mode in the circular waveguide, and they connect with the middle gap of each coupled cavity by the coupling holes arranged along the angular direction, which can minimize the distortion of the field pattern in the gaps caused by the output circuit. The heights of the coupling holes and the mode converters are equal to the gap length of 0.12 mm. External quality factor Q ext can be used to characterize the coupling between the cavity and the load, it is calculated by the group delay and given in Fig. 7. For circuit topology, the output ports of two identical output cavities are connected in parallel, thus, the Q ext of multi-extraction output cavity is half that of a single output cavity, as shown in Fig. 7(a). The width of the coupling hole has a significant effect on the external quality factor, as shown in Fig. 7(b), the Q ext decreases with the increase of the W h , which means that the coupling is improved. Fig.8(a) gives the guide wavelength of the output circular waveguide corresponding to the waveguide radius R g , and the external quality factor versus the waveguide radius is given in Fig.8(b), and it shows that the Q ext increases with the increase of the R g .
In order to superimpose the output power signals in the same phase, the phase of the field in each cavity should be identical, and the distance L c between the middle gaps of each  coupled cavity should be equal to the guide wavelength of the output waveguide. From (6), if the electron passes through a period length, the phase shift of the field is 2π, therefore, the drift distance L d between the coupled cavities should be an integral multiple of the period length, such as 0.544 mm, which is two times of the period length of 0.272 mm. However, the electrons have decelerated as they pass through the first coupled cavity, thus, the L d is slightly shorted to 0.51 mm according to PIC simulation. Then, we have the value of L c is 1.87 mm. Thus, the radius of the circular waveguide is set to 0.46 mm from Fig.8(a). According to the circuit theory, the load will obtain the highest power when the Q ext is equal to the Q 0 of 1100, thus, the width of the coupling hole is determined to be 0.46 mm according to Fig.7(b). Benefiting from the flexibility of the same phase by adjusting the distance between the coupled cavities and the guide wavelength of output waveguide, and no additional combiner is needed, which is beneficial to simplify the device.

III. STABILITY ANALYSIS
For the cavity operate in high-order mode, the risk of mode competition could not be ignored because of the numerous resonant modes coexist in the cavity. There are two transverse modes adjacent to the TM 10,1,0 mode are named as TM 9,1,0 and TM 11,1,0 according to the pattern of the axial electric field in the cross section. Moreover, the coupling between cavities induces a sequence of longitudinal resonant modes.
where β is the propagation constant of the field. Increasing the number of periods will lead to more resonant modes,   resulting in a fairly narrow frequency spacing between some modes. The dispersion characteristic of coupled cavity is an intuitive approach to identify the possibilities of mode competition. As an example, the dispersion curves of the TM 10,1,0 mode coupled cavity with seven periods is illustrated in Fig. 9. The TM 10,1,0 − 1/7π mode and TM 9,1,0 − 3/7π mode may interfere with the operating mode (TM 10,1,0 − 2π ) because of the narrow frequency spacing. TABLE 2 gives the resonant frequencies fr and the instantaneous bandwidths BW calculated by the intrinsic quality factor Q 0 . It is clear that the frequency spacing between the operating mode and either of the adjacent modes is wider than the sum of half-bandwidth of the two modes, which confirms that mode competition will not occur when the coupled cavity operating at TM 10,1,0 − 2π mode. The width of the coupling tunnel determines the coupling between the cavities, which affects the dispersion characteristics. The dispersion curves of TM 10,1,0 mode for different width of the coupling tunnel is given in Fig. 10. With the increase of the width of the coupling channel, the fluctuation of dispersion curve increases, which provides wider spacing of resonant frequency to prevent VOLUME 8, 2020 mode competition. However, the characteristic impedance will decrease with the increase of the width as a result of more and more electric fields exist in the coupling tunnels, thus the width of the coupling tunnel should be set at a compromise.
The electron beam will exchange energy with the RF field at the resonant modes near the beam line. Fig. 9 indicates that the TM 10,1,0 − 2π and the TM 11,1,0 − 2π mode exhibit the possibility of generating self-oscillation. Uncontrollable self-oscillation will destroy the pattern of field and disturb the bunching of the electron beam, resulting in the deterioration of gain and stability. To prevent the uncontrollable self-oscillation, the total power loss of the whole interaction circuit must be positive, which is judged by the stability condition expressed as, where G l is the equivalent circuit conductance of the cavity, G e is the conductance of the beam [20]. G l reflects the Joule-loss of the cavities, it is always positive. However, G e describes the power exchange between the electron beam and the RF field, it is positive when power transferred from the beam to the field is less than that absorbed by the beam, and becomes negative on the contrary. The normalized beam conductance is where G 0 = I 0 /V 0 is the DC beam conductance. Thus, the normalized beam conductance is a function of the beam voltage. Substituting (6) into (8),the normalized beam conductance of coupled cavity can be defined as where d |M z M r | 2 dβ e = 2 sin θ d 2 θ d 2 and  Fig. 11(a) gives the g e of proposed coupled cavity with seven periods at TM 10,1,0 − 2π mode and the TM 11,1,0 − 2π mode, obviously, the g e is positive at the operating beam voltage of 20 kV, thus, it is believed that the coupled cavity is stable. Fig. 11(b) indicates that, with the increase of period numbers, the change of G e becomes more significant.  Besides, the positive range of G e is reduced with the increase of period numbers. In other words, the coupled cavity with more periods requires a beam voltage with higher precision and stability, otherwise, the g e will be negative when the beam voltage is slightly higher than the operating voltage.

A. SIMULATION AND RESULTS
The complete interaction circuit of the TM 10,1,0 mode multibeam EIK proposed in this paper consists of an input cavity with the same structure as the single output cavity, the bunching section with four bunching cavities and a multiextraction output cavity containing two identical output cavities, as shown in Fig. 12(a). The conductivity of the cavity material is set to 5.8 × 107 S/m. The synchronous tuning of the cavities is utilized to raise the gain, and the important parameters of each cavity are given in TABLE 3. Same as the conventional klystron, the bunching section of the proposed EIK utilizes variable drift distances to enhance the bunching of the beams. In the case of small signals, the drift distances in the first two sections are approximately one-quarter of the reduced plasma wavelength for the optimal bunching process. To prevent the over bunching, so the drift distances in the next three sections should be shorter, which are determined by the PIC simulation. Finally, the drift distances between the cavities in the bunching section are arranged as 4.5, 4, 2, 2,and 1mm, respectively. Driven by ten individual DC electron beams with the beam voltage V 0 of 20 kV, the beam current I 0 of 0.1A and the beam radius R b of 0.1 mm, the performances of the proposed multi-beam EIK are demonstrated by the 3D PIC simulation using CST-PIC solver. The minimum focus magnetic field to maintain  the focus of the beam called Brillouin field can be given The theoretical calculation indicates that the interaction circuit satisfies the stable condition, however, even if there is no input signal, the rising edge of the DC electron beams inevitably excites a weak oscillation signal of the operating mode with a frequency of 300 GHz, as shown in Fig.13. The stable DC electron beams can be regarded as loads of the cavities because of the positive G e , thus the oscillation will gradually decay and remain a small amplitude, as shown in Fig.13(a). This controllable oscillation of operating mode does not destroy the stability of the device and the quality of the output signal.
The signal with power of 5 mW at desired center frequency of 300 GHz is injected into the device, and a snapshot of normalized axial electric field pattern at longitudinal sections of the whole interaction circuit is shown in Fig. 12(a). The electric field has the same pattern within each gap and becomes stronger cavity by cavity. Fig. 12(b) gives a snapshot of the normalized axial electric field pattern at cross sections of the output cavities. Obviously, the field pattern agrees well with TM 10,1,0 mode, thus, it is convinced that the device is operating at TM 10,1,0 − 2π mode. The steady output power signal is obtained after 8 ns, as shown in Fig. 14, the saturated output power is P out = 0.5 × 25 2 W=312.5W. As a contrast, the TM 10,1,0 mode multi-beam EIK with a single output  cavity is modeled and simulated under the same conditions, and its ouput power signal is also given in Fig. 14.
Obviously, the maximum output power of the EIK with a multi-extraction output cavity is nearly twice that of the EIK with a single output cavity. Therefore, the multi-extraction output cavity has been proven to improve the efficiency of EIK significantly. The frequency spectrum of the output signal is very pure within a wide band, as shown in Fig.15(a), which signifies that no model competition occurred. Therefore, the correctness of stability analysis in Sec. III is proved. Fig.15(b) shows the phase space of the electrons in the last bunching cavity and the multi-extraction output cavity. The  stepwise enhanced modulation of the beam is observed, and most electrons drastically decelerate in the output cavity, indicating that the energy of electrons is converted to the electromagnetic fields and coupled to the load. The gain curve of the input power and frequency is shown in Fig. 16. The maximum output power of 312.5 W is obtained for an input power of 5 mW at the center frequency of 300 GHz yielding a gain of 48 dB, and the 3-dB gain bandwidth calculated from Fig. 16(b) is about 200 MHz.

B. POWER CAPACITY ANALYSIS
The simulation results show that the proposed EIK has good performances, however, the device may not be able to withstand such large power because of the small size dimension and the restrictions of surface roughness. A strong electric field may lead to the breakdown of narrow gaps. According to the PIC simulation, the axial electric field in the middle gap of the first output cavity is strongest, and the voltage of the gap is given in Fig. 17(a), the voltage will increase to 397 V over time. The coupling holes can be regarded as the rectangular waveguides with a sectional dimension of 0.46 mm × 0.12 mm, and it is cut-off for the electromagnetic waves at 300 GHz. Complex reflection exists at the junction between the coupling hole and the coupled cavity, thus, it is difficult to calculate the power capacity of the coupling holes theoret-ically. But the voltage of the junction can be obtained by PIC simulation conveniently, as shown in Fig. 17(b), the voltage will increase to 380 V over time. Ideally, the electric field strength of the breakdown limit called E br is up to 1.3 MV/cm in the vacuum, however, because of the limitation of process conditions, the actual E br is much smaller. According to the research of R.
Hawley et al. [21], a conservative E br of 100 kV/cm is reasonable. Thus, the breakdown voltage of the gap with a length of 0.12 mm is 1.2 kV, which is much higher than 397 V and 380V, thus, the risk of gap breakdown is very low in the proposed EIK.
The power capacity of output circular waveguide operating at TM 0,1 mode is defined as where Z TM is the wave impedance of TM 0,1 mode, R g is the radius of the waveguide. Then we have the power capacity of the output circular waveguide at TM 0,1 mode is 160 kW, which is much higher than the maximum output power of 312.5 W. Synthesize the analysis above, it is believed that the power capacity of the proposed EIK is enough.

V. CONCLUSION
Conventional EIK is facing a series of difficulties when it extends to the THz band, the most prominent of which is the rapidly reduced dimension and the higher requirement of the current density of the electron beam. To deal with these problems, a novel TM 10,1,0 mode multi-beam extended interaction klystron is proposed in this paper, and the interaction circuit including the coaxial coupled cavity and the multi-extraction output cavity are designed and analyzed, and the crucial structure parameters are determined. The TM 10,1,0 − 2π mode is determined as the operating mode for larger dimensions and higher R/Q. There are ten electron beams are formatted with the voltage of 20 kV and the overall current of 1 A (0.1 A × 10), and the current density of each beam is 318 A/cm 2 , which is easily achieved by the electron gun that using thermal cathode with low compression ratio. The results of the 3D PIC simulation confirm that the device works at the TM 10,1,0 − 2π mode without mode competition and self-oscillation, and the modulation of the beams is intense. The maximum output power is 312.5 W with a gain of up to 48 dB at the center frequency of 300 GHz, the 3-dB gain bandwidth of 200 MHz is relatively narrow because of the high Q-factor cavities and the synchronous tuning. The results of PIC simulation indicate that the multi-extraction output cavity can significantly improve the output power and efficiency. The power capacity of the interaction circuit is also analyzed briefly, the results indicate that the power capacity has a lot of margins. In summary, the TM 10,1,0 mode multibeam EIK proposed in this paper provides a great potential approach for developing the high power terahertz sources.