Design and Non-Uniform Current Analysis of a 0.35-THz Extended Interaction Oscillator Based on Pseudospark-Sourced Multiple Sheet Electron Beams

Abstract

A novel method, which combines a multiple-beam extended interaction oscillator (EIO) with pseudospark-sourced (PS) sheet electron beams, is applied to generate high-power terahertz sources. For a multiple-beam EIO, the beam cross-section is significantly improved by replacing the commonly used pencil electron beams with sheet electron beams. The PS electron beams have the advantage of high current density and operate without a focus magnetic field. The volume of the cavity is larger when the EIO operates in the TM₃₁-3π mode than in the conventional TM₀₁-2π mode at the same operating frequency. The EIO operating at the terahertz frequency has a larger cavity volume, which means greater power capacity and lower manufacturing difficulty. For a PS multiple-beam EIO, the non-uniformity of electron beam currents is a common problem. In order to study this problem, an original high-order mode EIO driven by PS multiple sheet electron beams is presented with enhanced output power at 0.35 THz. The authors analyze electron beams with different currents through particle-in-cell (PIC) simulations. Simulation results show that the EIO can operate stably even in the case of non-uniform PS electron beam currents. When each current is 1.4 A, simulation results show the EIO’s output power of 4.9 kW at 0.35 THz. Considering the low conductivity of 1.1 × 10⁷ S/m, the efficiency is still 1.42%.

1. Introduction

It is well known that vacuum electronic devices (VEDs) can provide electromagnetic radiation sources, which are characterized by a greater power and higher efficiency than solid-state devices at terahertz frequency [1,2]. High-power terahertz radiation sources have a powerful attraction for molecular spectroscopy, high-resolution radar, plasma diagnostics, and air warning systems [3,4,5,6,7,8,9,10,11,12]. Among various VEDs, extended interaction devices (EIDs) are widely used due to their high gain per unit length. A high unity gain can significantly reduce the size of the interaction circuit, making the EIDs more compact than conventional VEDs [13,14]. The EIDs were initially designed to combine the wide bandwidth of the traveling wave tubes with the high efficiency of the traditional coupled-cavity klystrons, so the EIDs have both advantages. EIDs generally include extended interaction oscillators (EIOs) and extended interaction klystrons (EIKs).

In the slow-wave structure, to ensure the beam–wave interaction efficiency of electrons, the diameter of the cylindrical electron beam is generally on the order of the wavelength of operation [15]. Therefore, the beam current is undoubtedly reduced for a single-beam EIO using a thermionic cathode electron gun at the terahertz frequency. The current reduction is mainly caused by the relatively small current density of the electron beam emitted by the thermionic cathode electron gun and the relatively small size of the electron beam at the terahertz frequency. The current density of a thermionic cathode is generally less than 100 A/cm². For an EIO operating at 0.35 THz, the diameter of its cylindrical electron beam is usually on the order of 0.1 mm. Therefore, increasing the cathode current density and enlarging the cross-sectional area of the electron beam tunnel are urgent problems to be solved.

The pseudospark discharge system is a new type of plasma cathode discharge. Its remarkable advantage is that it can emit high-current-density electron beams [16]. This characteristic meets the requirement of high-frequency VEDs. Pseudospark-sourced (PS) electron beam’s current density can exceed tens of thousands of amperes per square centimeter, which is much higher than the current density that a thermionic cathode electron gun can produce [17]. In addition to the advantage of high current density, the pseudospark discharge system does not require an external magnetic field to focus the electrons because it generates a plasma channel to restrict the movement of electrons. The discharge system does not require a high vacuum environment; the air pressure is generally 50–500 mTorr. The above advantages make the pseudospark discharge system perform exceptionally well compared to other electron guns.

EIO can use single sheet beams or multiple pencil beams to increase the cross-section of electron beams. Another way to further increase the cross-section of the electron beams is to use multiple sheet beams. EIO can operate in either the fundamental mode or the high-order mode, depending on the electric field distribution of the mode. The cavity volume of the TM₃₁-3π mode is larger than that of the traditional TM₀₁-2π mode when two EIOs operate at the same frequency [18,19]. EIOs operating at terahertz frequencies are small, so the larger size dramatically reduces the manufacturing difficulty. Therefore, it is definitely a wise choice to use EIO operating in high-order 3π mode at terahertz frequencies.

For a multiple-beam device, it is crucial to know how the non-uniformity of currents in electron beams affect the EIO capability [20]. Due to various reasons, the current value in each electron beam may be different, and some electron beams have no current in actual experiments. The PS discharge produces an electron beam with a short pulse length. The whole discharge process is speedy. The velocity spread of the PS electron beams leads to different electron beam current densities. Can the multiple sheet beam EIO operate stably in the case of non-uniform electron beam currents? It is a question worth studying urgently.

A 0.35 THz TM₃₁-3π mode EIO driven by PS multiple sheet electron beams is designed to verify the above ideas. The structure of this paper is organized as follows. In Section 2, an introduction to PS multiple sheet electron beams is given. Section 3 includes a preliminary design of a 0.35 THz EIO driven by PS multiple sheet electron beams. Section 4 is about PIC simulations and the problem of non-uniform beam currents. Conclusions are given in Section 5.

2. Introduction of the PS Multiple Sheet Electron Beams

Since Christiansen and Schultheiss discovered PS discharge in the 1970s [21], this new type of plasma discharge has been widely used to generate high current density and high-energy electron beams in research. In addition, PS discharge also has the advantages of brightness (10¹² A/m² rad²) and fast current rise (10¹² A/s) [22,23]. The PS discharge operates on the left of the lowest point of the Paschen curve. The PS discharge generally uses high-voltage breakdown to generate electron beams in low-activity gases, such as noble gases and nitrogen. It is challenging for a traditional thermionic cathode electron gun to generate a sheet electron beam and multiple pencil electron beams, not to mention multiple sheet electron beams. There is a high requirement for the magnetic system in a thermionic cathode electron gun. However, the PS discharge generates a plasma channel, which focuses on the electrons. Therefore, PS discharge is very suitable for producing multiple sheet electron beams.

The PS discharge device is characterized by simplicity and robustness. Figure 1 is a schematic of the PS discharge system with a post-acceleration. On the far left of the figure is a hollow cathode, which is the critical component for PS discharge. A negative high-voltage DC power supply is connected to the hollow cathode, and a high-voltage resistor R₁ is placed between them to limit excessive current. The voltage on the hollow cathode can be measured by the high-voltage probe 1. The hollow cathode and the anode are separated by an insulator. The breakdown voltage of a single-gap discharge system is relatively low. Of course, the breakdown voltage can be increased by alternately placing insulators and intermediate electrodes between the hollow cathode and the anode. There is another insulator between the anode and the grounded flange, and a high-voltage resistor R₂ is connected between them. The negative high voltage creates a plasma discharge in the gap between the hollow cathode and anode. The gap between the anode and the grounded flange accelerates the electron beam to a higher energy. In the process of acceleration, resistor R₂ plays an important role [24]. Before the conductive stage of the pseudospark discharge, the voltage on resistor R₂ is much smaller than the voltage between the hollow cathode and anode, so most of the voltage drops on the pseudospark discharge gap. When a voltage breakdown occurs between the cathode and the anode, the voltage between the hollow cathode and anode becomes almost zero, and then the total voltage is applied to the resistor R₂. The voltage on resistor R₂ can accelerate the generated electron beam. The accelerated electron beam forms the shape required by the EIO after passing through a collimator with six of the same rectangular apertures. The size of each aperture is 0.34 mm × 0.08 mm. A Rogowski coil wound around the electron beam is used to measure the electron beam current. Working gas, such as argon, nitrogen, and helium, is injected slowly through a mechanical needle valve, which controls the gas inlet. A vacuum pump discharges the gas to the outside, maintaining a stable state of the entire closed environment through the cooperation with the mechanical needle valve.

The PS discharge is divided into three main stages: the Townsend discharge, the hollow cathode discharge, and the conductive discharge. In the Townsend discharge phase, when a negative high voltage is applied to the hollow cathode, some seed electrons are generated in the hollow cathode. The seed electrons collide with the background gas molecules in the cathode, thus ionizing the gas molecules and producing more electrons and positive ions. During the hollow cathode discharge phase, the movement of the electrons and ions further increases the frequency of collisions with gas molecules. As a result, the number of electrons and ions in the cavity rises, and the electron beam current gradually increases. In the third stage of PS discharge, a large number of electrons move toward the anode, and the first electrons continue to ionize the background gas. A high-current density and high-brightness electron beam are obtained. The generated electron beam enters the beam tunnel of EIO after being accelerated.

3. Design of a 0.35 THz EIO Based on PS Multiple Sheet Electron Beams

As shown in Figure 2 and Figure 3, the main components of the EIO are coaxial coupling cavities and an output structure. Eleven identical coaxial coupling cavities are connected with a hollow columnar coupling structure to form an eleven-gap slow-wave structure. The period of the slow-wave structure is 0.47 mm, and the gap length is 0.18 mm. Six-sheet beam tunnels (with a cross-section size of 0.36 × 0.1 mm) are evenly distributed on the slow-wave structure. A standard waveguide and a rectangular coupling hole are connected to the middle of the slow-wave structure.

Figure 4 is the dispersion diagram of the proposed EIO with different TM modes. For the TM₃₁-3π mode operation, the corresponding synchronous beam voltage is 38.8 kV. As can be seen from the diagram, the multiple sheet beam EIO operates at a frequency of 353 GHz in TM₃₁-3π mode, much higher than the frequency of 235 GHz in TM₀₁-2π mode. For the same device, the 3π high-order mode can provide a higher frequency than the traditional 2π fundamental mode. EIO operating in high-order mode can generate higher-frequency electromagnetic waves without reducing the device’s size.

In Figure 4, the orange, green, blue, and red curves represent TM₀₁, TM₁₁, TM₂₁, and TM₃₁ modes, respectively. Each marker on the curves represents a mode. The intersections of the 38.8 kV beam line with the markers indicate the possible mode competitions at this operating voltage. Figure 5 shows the electric field contour diagram of TM₀₁, TM₁₁, TM₂₁, and TM₃₁ modes in the x–y plane. It can be seen in Figure 5a,d that the region of the higher electric field completely covers the six-sheet beam tunnels. In contrast, the higher electric field regions in the TM₁₁ and TM₂₁ modes do not cover all beam tunnels in Figure 5b,c. It means that the electron bunching is weak, and the energy of the electrons cannot be effectively transferred to the electromagnetic wave in some beam tunnels. Therefore, the primary mode competition is only possible between TM₀₁ and TM₃₁ modes. It can be seen from Figure 5d that the distribution of the six-sheet beam tunnels coincides with the contour of the electric field in the TM₃₁ mode. Compared with the multiple pencil beam tunnels, the multiple sheet beam tunnels can be adjusted in size to fit the electric field distribution on the x–y plane. In this method, the beam–wave interaction efficiency is greatly improved.

Beam-loading conductance G_e represents the energy exchange between the electron beams and the external circuit. The normalized beam-loading conductance G_e is [25]

$$G_e=\frac{1}{8}\frac{{\beta }_e}{{\beta }_q}\left[{\left|M\left({\beta }_e-{\beta }_q\right)\right|}^2-{\left|M\left({\beta }_e+{\beta }_q\right)\right|}^2\right]$$

(1)

where M (β_e − β_q) and M (β_e + β_q) can be expressed as

$$M\left({\beta }_e\pm {\beta }_q\right)=\frac{{{\displaystyle \int }}_{-\infty }^{\infty }E\left(z\right)e^{j\left({\beta }_e\pm {\beta }_q\right)z}dz}{{{\displaystyle \int }}_{-\infty }^{\infty }\left|E\left(z\right)\right|dz}$$

(2)

where E(z) is the electric field along the center of the beam tunnel in the EIO cavity, and β_e and β_q are the DC beam propagation constant and the reduced plasma propagation constant, respectively. Whether the EIO oscillates or not, the value of G_e plays a decisive role. A positive G_e means that the electron beams absorb energy from the external circuit. By contrast, if G_e is a negative value, the electron beams exchange energy to the high-frequency electromagnetic field, thereby causing oscillation. When the value of G_e reaches the negative peak, it means that the mode is most likely to oscillate at this beam voltage. In order to ensure that there is no mode competition within the range of the operating voltage, the G_e of the operating mode should be as close as possible to the negative peak value, while the G_e of other competition modes ought to be guaranteed to be positive.

As shown in Figure 4, in addition to TM₃₁-3π mode, the markers of TM₀₁-(2π + 1) mode and TM₃₁-(3π − 1) mode are also adjacent to the 38.8 kV beam line. Once the voltage condition around 38.8 kV is met, undesired mode competitions may occur. Figure 6 shows the trends of G_e versus voltage for different modes. In Figure 6, the voltage as the abscissa is divided into three regions. In region Ⅱ (in the range of 39–41.5 kV), the G_e of TM₃₁-3π mode is negative, and the other two are positive. It means that TM₃₁-3π mode can operate in the 39–41.5 kV voltage range without mode competition. In region Ⅰ (in the range of 35–39 kV), the G_e of the TM₀₁-(2π + 1) mode is negative and the smallest of all, so it dominates in this voltage range. In region Ⅲ (in the range of 41.5–45.5 kV), the G_e of both TM₃₁-3π and TM₃₁-(3π − 1) modes is negative. It implies that competition may occur between the two modes. The designed multiple sheet beam EIO operates well and has no mode competition in the voltage range of 39–41.5 kV.

4. PIC Simulation of the 0.35 THz EIO Driven by PS Multiple Sheet Electron Beams and Analysis of Non-Uniform Beam Currents

In order to verify the multiple sheet beam EIO performance, the device was simulated with the CST Particle Studio. The operating voltage is set at 41 kV, and the current of the six-sheet beams is the same at 1.4 A. Considering that the EIO operates in the terahertz frequency, the non-ideal surface losses become a factor that cannot be ignored; therefore a very low conductivity of 1.1 × 10⁷ S/m is used in the simulation. Figure 7 shows a stable 4.9 kW output power signal lasting 30 ns. The device operates at 349.91 GHz. The corresponding efficiency is 1.42%. The pulse generated by the PS discharge is on the order of ten nanoseconds, so the EIO’s oscillation startup time should be short enough. The oscillation startup time of the EIO is about 3.7 ns, which meets the requirement of fast oscillation of a pseudospark discharge system. It can be seen from the frequency spectrum diagram that there is no mode competition in the TM₃₁-3π mode at the operating voltage of 41 kV, which once again verifies the stability of the device operating in this high-order mode.

Figure 8 shows the output characteristics of the multiple sheet beam EIO as a function of beam voltage. From the previous analysis of Figure 6, it is found that there is no mode competition when the device operates in the range of 39–41.5 kV, and the PIC simulation also confirms the previous conclusion. When the operating voltage is less than 39 kV, the undesired non-TM₃₁-3π mode is excited, and the output power and efficiency are low. In the 39–41.5 kV range, the output power and efficiency increase almost linearly. In the case of constant current, the total input power of the device increases with the increase in electron injection voltage. In the same voltage range, the efficiency of the device also increases gradually. Therefore, the EIO’s output power becomes higher and higher. As shown in Figure 8, when the beam voltage is 42 kV, the output power is only 0.02 kW, and the device suffers from severe mode competition. The output power reaches its peak at 41.5 kV. The oscillation startup time of the EIO is the shortest when the beam voltage is 40.5 kV or 41 kV.

In the previous PIC simulations, the current value of each sheet beam is 1.4 A, and the current density is around 5000 A/cm². The output characteristics of the device at other current densities were also simulated. As shown in Figure 9, oscillation startup time, output power, and efficiency are analyzed as a function of beam current density when the beam voltage is 41 kV. The oscillation startup time shows a decreasing trend with the increase in current density. Both output power and efficiency increase with higher current density. However, the efficiency gradually tends to saturation. When the current density exceeds 8000 A/cm², the influence of the space charge effect is increased because of the excessive input current. The mutual repulsion between electrons becomes stronger, affecting the electron bunching and leading to unstable output power.

In order to verify the stability of the multiple-beam EIO with non-uniform electron beam currents, the device is simulated with different beam currents. The authors start with the non-uniformity of one electron beam. As shown in Figure 10a, the current of electron beam Ⅰ is set to 5I₀, and the current of other electron beams is set to I₀. I₀ is a constant with a value of 1.4 A. Figure 10b is a trajectory diagram of electrons. It can be found that the electrons in the six-sheet beams are bunching. The energy distribution of the electrons of beam Ⅰ is no different from that of other beams.

All simulation parameters are set the same as before, and only the current in beam Ⅰ is increased by five times. Figure 11 shows that the port of the standard waveguide obtains a stable 6.9 kW output power signal lasting 15 ns. As the total input current becomes larger, the output power increases from 4.9 kW to 6.9 kW compared to before. The efficiency of the device is 1.2%, which is slightly lower than the previous 1.42%. The oscillation startup time is shortened to 1.4 ns. The operating frequency is almost the same as before, at 349.7 GHz. Based on the above data, it can be known that the stability of the EIO is good and that the efficiency and frequency are almost unchanged.

In order to further study the influence of non-uniformity of electron beam currents on the multiple-beam EIO, the current of the electron beams from Ⅱ to Ⅵ remains set to a constant I₀, while the current value of the electron beam Ⅰ increases from 0 to 8I₀. Figure 12a shows the variation trend of the multiple-beam EIO’s output power with the electron beam Ⅰ current. When the current of the electron beam Ⅰ is 0 (that is, no electrons pass through the beam at all), the EIO has an output power of 4 kW at 349.99 GHz. Even if one of the electron beams has no current at all, the EIO can still operate stably. In the region where the changing current increases from 0 to 2I₀, the changing trend of the output power can be approximately regarded as a straight line. This part is called the linear region. When the current value of the electron beam Ⅰ is between 2I₀ and 5I₀, the slope of the trend curve becomes smaller and smaller. This period is called the transition region. When the changing current exceeds 5I₀, the output power does not change with the varying current. This part is called the saturation region. The same result can also be obtained when the electron beam with increased current is changed from beam Ⅰ to beam Ⅱ–Ⅵ.

The non-uniformity of one electron beam has been analyzed. How does the non-uniformity of the two electron beams affect the performance of the multiple sheet beam EIO? In order to research the effect, current values of beams Ⅰ and Ⅱ increase at the same time, while the currents of other electron beams are constant. As shown in Figure 12b, the output power of the EIO also experiences a linear increase at the beginning and then gradually reaches the transition region. The difference is that before the curve comes to the saturation region, the output power of the EIO is no longer stable. As analyzed in Figure 9, excessive input current increases the influence of the space charge effect. When any two electron beams increase the current simultaneously, the output power also changes similarly. The authors also simulated the non-uniformity of three or more electron beams and found that the change rule of the output power is similar to that of two beams.

In the application of multiple-beam devices, the actual problem with current non-uniformity is definitely more complicated than the situation we analyzed above. For example, the beam current may be different for each electron beam, or some electron beams may not even have electrons passing through. A large amount of simulation data shows that the EIO’s efficiency is roughly equal in the linear region. In other words, the output power is only related to the total input current in the case of constant beam voltage. The device operates well even with zero current in one or two beams. When the entire current value of the six-sheet beams is constant, the output power is almost the same. When the device operates in the transition region, the efficiency gradually decreases, and the influence of the current non-uniformity on the device begins to appear. The distribution of current in the electron beam has an effect on the output power. PS discharge systems can generally generate a maximum current density of 10⁴ A/cm² [26]. In the previous PIC simulation, the current density is 5 × 10³ A/cm². The maximum current is only 2I₀, so the multiple-sheet beam EIO operates in the linear region. In summary, the TM₃₁-3π mode multiple sheet beam EIO does not have high requirements on the current uniformity of the six-electron beam.

5. Conclusions

This article presents the design of an original TM₃₁-3π mode EIO driven by PS multiple sheet electron beams. Multiple sheet electron beams greatly increase the cross-section of electron beams and improve the input current. This is the first time the combination of multiple sheet beams and EIOs is realized. The multiple sheet electron beams have the problem of difficulty in achieving magnetic field focusing, but the PS electron beams do not require an external magnetic field. The device’s operating mode is TM₃₁-3π mode instead of the traditional TM₀₁-2π mode, significantly improving the EIO cavity’s size. At terahertz frequencies, a large cavity size can effectively improve the power capacity and reduce the manufacturing difficulty. The analysis of the EIO dispersion curve and beam-loading conductance shows that the TM₃₁-3π mode can overcome the mode competition and operate stably. PS multiple electron beams may suffer the problem of current non-uniform. The designed multiple sheet electron beam EIO can still operate normally under the condition of non-uniform beam currents. Even if the current of two electron beams is 0, the multiple-beam device can still generate terahertz radiation stably. PIC simulation results show that the EIO driven by PS multiple sheet electron beams have an output power of 4.9 kW at ~0.35 THz when the background material conductivity is 1.1 × 10⁷ S/m. The beam–wave interaction efficiency is 1.42%. The output power and efficiency of the device are vastly improved compared to the previous 0.35 THz EIO driven by a PS sheet electron beam [27]. Its output power is 1.8 kW, and beam–wave interaction efficiency is 0.88%.