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Abstract
Vacuum electronic devices utilizing free-electron-based mechanisms are a crucial class of terahertz radiation sources that operate by modulating electron beams. In this study, we introduce what we believe is a novel approach to enhance the second harmonic of electron beams and substantially increase the output power at higher frequencies. Our method employs a planar grating for fundamental modulation and a transmission grating operating in the backward region to augment the harmonic coupling. The outcome is a high power output of the second harmonic signal. Contrasting with traditional linear electron beam harmonic devices, the proposed structure can achieve an output power increase of an order of magnitude. We have investigated this configuration computationally within the G-band. Our findings indicate that an electron beam density of 50 A/cm2 at 31.5 kV can produce a 0.202 THz center frequency signal with an output power of 4.59 W. As the electron beam voltage is adjusted from 23 kV to 38.5 kV, the output signal frequency shifts from 0.195 THz to 0.205 THz, generating several watts of power output. The starting oscillation current density at the center frequency point is 28 A/cm2, which is significantly lower in the G-band compared to conventional electron devices. This reduced current density has substantial implications for the advancement of terahertz vacuum devices.
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1. Introduction
Terahertz waves have demonstrated their unique properties across a diverse range of applications. This increase in application scope has, in turn, led to a growing demand for terahertz sources. Emerging technologies for generating terahertz radiation involve both photonics and electronics; however, these approaches are closely tied to the development goals of high power output and miniaturization. These advances provide a foundation for our proposed method in second harmonic wave generation to further improve terahertz sources. Currently, vacuum electron devices represent the most viable solution for achieving high power and miniaturization within the sub-terahertz band radiation. The transition of conventional vacuum electron devices to the terahertz band faces significant challenges due to the dimension congruence effect and limitations in electron beam performance.
The electron beam will carry a certain frequency information after being modulated in the high frequency structure. By means of Floquet's theorem, this frequency can be seen as a multiplication of an infinite number of multiplier electron beams. We call the multiplier signal the corresponding ordinal harmonic of the electron beam. The electron beam in devices operating in a harmonic manner is generally modulated by the fundamental frequency, and the electron beam harmonics are subsequently used to output a frequency multiplier signal. This unique mode of operation reduces the requirement for electron beam emission density in these devices. Thus, the vacuum electron devices operating at harmonic is a significant approach for generating THz radiation.
Various harmonic-operating vacuum electron devices, such as the gyrotron, harmonic traveling wave tube, and super radiation orotron, have been explored both theoretically and experimentally. The gyrotron overcomes the dimension congruence effect, facilitating terahertz radiation generation more efficiently than other devices. Presently, gyrotrons can output frequencies exceeding 1 THz under fundamental and harmonic conditions [1,2]. However, due to the size limitations of superconducting magnets, achieving miniaturization for gyrotrons in the short term remains challenging. Besides the harmonic gyrotron, the harmonic traveling wave tube has also attracted significant attention. Theoretical findings suggest that its output power level is comparable to that of the input signal [3]. In recent years, experimental studies have demonstrated that the highest power harmonic traveling wave tube operates in the G-band, with corresponding pulsed output power in the hundreds of milliwatts range. Utilizing specialized structures to make Smith-Purcell (SP) radiation coherent and to achieve control over superradiation is an essential research direction for overcoming the dimension congruence effect and generating tunable, coherent terahertz radiation [4–9].
Numerous theoretical studies have been conducted to investigate the amplification of electron beam harmonic components [10]. Specifically, the electron beam can be pre-modulated by the self-excited oscillation at the fundamental frequency grating, extracting the harmonic components of the pre-modulated frequency via the second harmonic frequency grating, and subsequently obtaining the harmonic components or the super-radiation of the harmonics. The low power output of terahertz sources with harmonic output is primarily attributed to the weakness of the higher harmonic components of the electron beam [11–15].
In these studies, the stability of device output and output methods are often not considered from a practical standpoint. Most existing research is qualitative, demonstrating the feasibility of extracting and amplifying the higher electron beam harmonic components to achieve higher power terahertz radiation using specialized structures [16,17]. However, these two-section grating structures tend to increase the device length, which is unfavorable for the electron beam circulation rate.
2. Simulation investigation on the second harmonic enhancement
In summary, based on the aforementioned research, we propose a novel structure for electron beam harmonic coupling and amplification, referred to as the harmonic coupled and enhanced oscillator (HCEO).
2.1 Description of the principle of implementation
As depicted in Fig. 1, the HCEO is composed of a planar grating, a transmission grating, a coupling cavity, and an output waveguide. The electron beam travels between the planar grating and the transmission grating. Initially, the electron beam is modulated by self-excited oscillations on the surface of the planar grating. Subsequently, the harmonic components carried by the modulated electron beam are coupled to the transmission grating, where they interact and amplify the second harmonic. The final harmonic signal is output through the coupling cavity and waveguide. In this study, we calculated and analyzed the G-band HCEO using CST, with the dimensional parameters detailed in Table 1.
Table 1. G-band HCEO simulation parameters
2.2 Calculation and analysis of HCEO
The crux of this structure's design lies in the double-multiple relationship between the intersection of the voltage lines and the dispersion curves of the two gratings. The dispersion relationships for both types of gratings are illustrated in Fig. 2. The 31.5 kV electron beam line corresponds to a frequency of 0.101 THz at the intersection with the planar grating and 0.202 THz at the intersection with the extended interaction mode of the transmission grating, indicating a frequency-doubling relationship. As observed from the dispersion relation, the dispersion is not severe in the interval between the intersection of the electron beam line and the dispersion curve at 23-38.5 kV. These factors contribute to the HCEO's broad tunable bandwidth with the corresponding parameters described in the manuscript. Since both gratings operate in the backward region, the output waveguide was positioned near the emitting side of the electron beam. In order to avoid energy coupling between the two gratings, we need to optimize the reflection grating and transmission grating so that the eigenmodes within the working frequency band do not overlap.
Generally, the high-frequency field is primarily concentrated near the electron beam. In a series of previous studies, harmonic gratings were planar gratings, and the output design of the harmonics was not considered. The transmission grating can exist in both surface wave mode and extended interaction mode. The surface wave mode is confined to the grating surface, while the extended interaction mode enables the coupling of harmonic energy near the electron beam to the waveguide using the coupling cavity. By adjusting the parameters of the coupling cavity to control the working state of the transmission grating, the excitation of the harmonic extended interaction cavity mode is ultimately achieved.
Figure 3 displays the electric field distribution of the two gratings in both the Eigenmode simulation and PIC simulation. Figure 3(a) and Fig. 3(b) depict the same surface mode of the planar gratings. The phase information for the signal of the fundamental electric field in the planar grating is consistent. Figure 3(c) and Fig. 3(d) illustrate the electric field distribution of the transmission grating, exhibiting an extended interaction mode. Notably, despite the size differences between the coupling cavity and the electron beam tunnel, the field distribution does not display asymmetric characteristics.
Fig. 3. Electric field distribution obtained using Eigenmode and PIC simulations. (a) The surface mode simulated by using Eigenmode simulation. (b) The surface mode excited by the electron beam. (c) The extended interaction mode simulated by using Eigenmode simulation. (d) The extended interaction mode excited by the electron beam.
Figure 4 presents the electric field distribution during the operation of the structure. The electric field in the figure contains harmonic and fundamental components with comparable intensities. The calculated results in Fig. 3 exhibit the same phase information as the field distribution in Fig. 4. This suggests that after the electron beam has interacted with the planar grating to oscillate, the harmonics are coupled to the transmission grating and amplified, ultimately resulting in the output of harmonic electromagnetic energy.
2.3 Solution of rectangular electron beam tunnel for shunting electromagnetic field
During the simulation, it was discovered that the output power remained weak even when the electron beam was sufficiently modulated. Subsequently, the fundamental and harmonic field distributions in the structure were simulated, as shown in Fig. 5(a) and Fig. 5(c). It was observed that the beam tunnel diverted most of the electromagnetic energy. In this case, the output power at the central frequency point is 1.28 W. The electron beam tunnel at both ends of the interaction structure can be considered as a rectangular waveguide with a cross-sectional size of 1.8 mm * 0.15 mm. Its transmission characteristics were calculated, and the results are displayed in Fig. 6(a). The transmission parameters of the electron beam indicate that the S11 of the electron beam channel is in a fully conductive state around -120 dB, near the two frequency points of 0.101 THz and 0.202 THz. Consequently, the fundamental and harmonic energy remain high in the electron beam tunnel, resulting in low output power.
Fig. 5. Surface modes and harmonic electric field distribution obtained from PIC calculations. (a) The surface mode simulated by PIC simulation without reflector. (b) The surface mode simulated by PIC simulation with reflector. (c) The harmonic simulated by PIC simulation without reflector. (d) The harmonic simulated by PIC simulation with reflector.
Fig. 6. (a) S-Parameters of the electron beam tunnel. (b) S-Parameters of the electron beam tunnel with reflector.
To address the issue mentioned above, specifically the decrease in output power due to the diverted electromagnetic energy, it is necessary to localize the electromagnetic energy within the interaction structure using specific measures. Therefore, we designed a reflector loaded at both ends of the interaction structure, which reflects the electromagnetic energy coupled to the electron beam tunnel at both ends back into the structure. As illustrated in Fig. 6(b), the reflector consists of two rectangular cavities, where the lateral dimensions of the rectangular waveguide are the same as those of the electron beam tunnel. The operating bandwidth interval of the reflector is related to its structural parameters, which can be adjusted to obtain the corresponding transmission parameter reflector.
By adjusting the rectangular resonant cavity parameters, the reflector achieves a value of 0 dB for S11 and less than -20 dB for S21 around 0.101 THz and 0.202 THz. Under these parameter conditions, the reflector can achieve both fundamental and harmonic localization. The internal field distribution of the structure after adding the reflector is shown in Fig. 5(b) and Fig. 5(d). The fundamental and harmonic electric fields are primarily distributed near the planar grating and transmission grating, the field intensity in the electron beam tunnel at both ends is significantly weakened, and the field intensity of the waveguide is considerably higher, accomplishing the objective of increasing output power through local field energy. The G-band HCEO with a reflector has an output power of 4.59 W at 0.202 THz.
2.4 Analysis of the second harmonic enhancement of electron beam
In addition to the reasons analyzed above, the degree of enhancement of the electron beam's second harmonic can also have a significant impact on the output power. As can be deduced from the operating principle of the HCEO, the strength of the second harmonic coupled to the transmission grating greatly influences the enhancement of the second harmonic. Once the parameters of the planar grating are determined, the degree of fundamental modulation of the electron beam is also established. The harmonic intensity of the electron beam increases with deeper fundamental modulation but decreases with excessive fundamental modulation of the electron beam. The relative position of the transmission grating to the planar grating determines the second harmonic coupling of the transmission grating to the electron beam.
As illustrated in Figs. 7–9, we analyzed the output power and electron beam modulation obtained from the transmission grating in three position cases. These three figures correspond to the three cases of insufficient electron beam fundamental modulation, electron beam fundamental modulation transition, and optimal intensity of fundamental modulation. In Fig. 7, the insufficient fundamental modulation intensity of the electron beam leads to low second harmonic intensity of the electron beam coupled to the transmission grating. Consequently, the second harmonic enhancement of the electron beam is insufficient, resulting in weak output power.
Fig. 7. Insufficient fundamental modulation position: (a) Time-domain power output diagram; (b) Electron beam phase space diagram.
Fig. 8. Excessive fundamental modulation position: (a) Time-domain power output diagram; (b) Electron beam phase space diagram.
Fig. 9. Optimal second harmonic output power condition: (a) Time-domain power output diagram and frequency distribution; (b) Electron beam phase space diagram.
In Fig. 8, the transmission grating is located at the position where the electron beam is excessively clustered. At this position, the output power exhibits a time-domain distribution that first increases and then decays to some extent. This is because, during the process of deepening the fundamental modulation until the excessive clustering of the electron beam, there is a certain point that allows the transmission grating to couple to the maximum second harmonic intensity at this position. As the fundamental modulation continues to deepen, the fundamental electron beam at this position becomes excessively clustered, and the second harmonic of the electron beam weakens. The weakening of the second harmonic intensity leads to a reduction in the signal output power at the corresponding frequency.
After design optimization, a G-band HCEO with a center frequency of 0.202 THz was obtained. Figure 9 shows the electron beam modulation when the output frequency is at 0.202 THz. The electron beam modulated by the fundamental frequency is further modulated by the higher harmonic, and the two modulated frequencies show a multiplicative relationship. This demonstrates the principle of second harmonic coupling enhancement of the HCEO. The simulation takes into account the ohmic loss of the material. A single frequency point of the output spectrum was obtained by FFT processing of the output power shown in Fig. 9(a), and a stable power output of 4.59 W is achieved at 0.202 THz.
In the HCEO, the enhancement of the electron beam harmonics is carried out with the modulation of the fundamental wave. As a result, the requirements for the electron beam in this oscillator are much lower compared to devices that directly modulate the electron beam at the output frequency. Calculations were performed for the oscillation current at the electron beam energy of 31.5 kV, and the results shown in Fig. 10 were obtained. The minimum oscillation current density for the G-band HCEO presented in this paper is 28 A/cm2, which yields an output power of 533.54 mW. The value of the oscillation starting current for this HCEO is significantly lower than the starting current of other oscillators operating with an extended interaction mode [18,19]. This makes the HCEO a more attractive option for generating THz radiation, particularly in applications where lower starting current densities are desirable.
The HCEO's adjustable bandwidth is a result of both gratings operating in the backward region. As shown in Fig. 11, the operating bandwidth calculation results indicate that the output frequency of the device can be adjusted within a range from 0.195 THz to 0.205 THz as the electron beam voltage changes from 23 kV to 38.5 kV. This tunable bandwidth offers flexibility in the device's applications, allowing it to adapt to different frequency requirements within the specified range. This feature makes the HCEO particularly attractive for various applications in THz technology, where the ability to adjust the output frequency can be crucial for specific tasks or experiments.
3. Conclusion
In summary, this study presents a novel structure called the harmonic coupled and enhanced oscillator (HCEO), which aims to enhance the electronic beam harmonic coupling and addresses the challenge of generating high-power harmonics. To achieve this goal, a reflector was designed and incorporated into the interaction structure based on the device's unique characteristics. As a result, high-power harmonic output can be obtained with a small structure length and low current density.
The HCEO device successfully achieved watt-level output power within its designed operating bandwidth range, demonstrating its potential for use in various applications requiring high-power harmonics. By combining a novel structure, optimized reflector design, and tunable bandwidth, the HCEO offers a promising solution for generating high-power harmonics in a compact and efficient manner, making it an attractive option for future advancements in THz technology and related applications.
Funding
National Key Research and Development Program of China (2017YFA0701000, 2020YFA0714001); National Natural Science Foundation of China (61921002, 61988102, 62071108); Natural Science Foundation of Sichuan Province (2022NSFSC0513, 2022NSFSC0514); Fundamental Research Funds for the Central Universities (ZYGX2020J003, ZYGX2020ZB007); Key Laboratory of THz Technology, Ministry of Education, China.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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