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Abstract
In this paper, we use the method of high order TMn1 mode selection from the concept of narrow-band Smith-Purcell radiation (SPR) for powerful, over-mode, multi-gap extended interaction circuit designs toward millimeter wave and Terahertz (THz) region. As a core part, the multiple gaps interaction structure, equivalent to a subwavelength hole array (SHA), excites the narrow band SPR when an electron beam is injected. The SPR energy is collected by a pair of closed cavities, which satisfies (n-1) standing wave units. The SPR energy in the optimized cavity allows a high index n TMn1 mode operation to achieve the strongest Ez field and high characteristic impedance in a closed multi-gap resonant circuit. This provides an effective design to establish a stable high-order TMn1 mode that supports extended interaction circuits with large cross sections. A 0.46 THz extended interaction circuit, employing the novel high order TM51-2π mode operation output structure, has been designed to demonstrate the efficient beam-wave interaction in the proposed system. The method of TMn1 mode selection provides new insight into the understanding of the high-frequency extended interaction circuits by introducing the SPR concept, benefiting the development of millimeter wave and THz vacuum electron devices (VEDs).
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1. Introduction
Extended interaction devices [1,2], including oscillators and amplifiers are hybrid devices that combine the resonant property of klystrons and slow wave characteristics of coupled cavity traveling wave tubes (TWTs). Accordingly, extended interaction circuits are the key part for providing beam-wave interaction and typically consist of periodically placed multiple gaps and resonant coupling cavities. Due to the hybrid advantages of the klystron and TWTs, extended interaction circuits have essentially high characteristic impedance and high gain per unit length. Therefore, extended interaction circuits have shown great potential in achieving compact, efficient and powerful radiation in low millimeter wave region (below W-band).
Nowadays, an increasing interest is to push the frequency to high millimeter wave region and THz regime, and meanwhile realizing powerful, efficient radiation. High frequency and high-power specifications require an electron beam with properly high voltage and high current density. However, extended interaction circuits suffer from the reduction of the cross section and brings difficulties in achieving high power, because 1) the current density of a single beam is difficult to be increased [3], and 2) the gap voltage is limited due to the increased chance of breakdown and arcing [4]. It is significant to overcome the limitation of the gap voltage on the basis of keeping the gap and period length unchanged in contrast to conventional extended interaction circuits operated in the fundamental mode. Therefore, reducing the gap voltage is depended on enlarging the coupling cavities with fixed gaps.
With this idea, we have proposed specific circuits based on the TM31 mode regime in millimeter-wave and THz frequencies and studied their operating characteristics [5,6]. It is concluded that the mode selection of such high order TMn1 modes is critical to support their applications in extended interaction circuits. Conventional designs of such modes are based on 1) resonant condition of the standing wave with high index n across the cross section of the circuit and 2) synchronous condition for beam-wave interaction. For idea cases, which establish the single high order TMn1 mode in an extended interaction circuit, the cross section would be so oversized that there exist many resonant modes in the circuit. The mode competition would be a challenging issue for proper operation of the circuit.
In this paper, we introduce the concept of Smith-Purcell radiation (SPR) to design stable high order TMn1 mode and clarify the TMn1 mode selection from this concept for powerful extended interaction circuit designs. The idea is to exploit the concept of SPR generated from a SHA structure [7–11] to produce the SPR energy in radiation space at the desired frequency and collect the energy using closed cavities that are placed on both sides of the subwavelength holes array (SHA) structure. Based on this concept, the high order mode is formed and selected by combining the narrow band SPR excited by injecting an electron beam into the SHA and SPR energy in the closed cavities. The SPR energy is concentrated into the closed cavities with large cross sections and has peak electric fields at the desired frequency. It provides a high order TMn1 mode with strongest Ez field and highest characteristic impedance. This is promising in designing a stable high order TMn1 mode that has the potential to overwhelm other competing modes in the large cavities. It should be noted that the SHA structure is inherently a multi-gap resonant structure and the closed cavities can be equivalent to coupling cavities on both sides of the multi-gap structure in the insight of typical extended interaction circuits.
To clarify the high order TMn1 mode selection from the concept of SPR, we will review the conventional application of SPR in the SHA with open reflection mirrors, and on this basis, propose the method of the mode selection using such concept. Furthermore, an extended interaction circuit with the TM51-2π mode is designed and its beam-wave interaction capability is studied by using PIC simulations.
2. High order TMn1 mode selection based on the concept of the Smith-Purcell radiation
A SHA structure is chosen as a periodic multi-gap structure to generate SPR, and then a metal plate is used as a simple mirror surface to confine SPR in a limited space range to form a high-order mode. The basic model is shown in Fig. 1. The structure has the following features: The frequency range of SPR will be regulated by the long side (hole-x) of the SHA hole. The height of the metal plate (h-up) determines the field distribution of the collected SPR energy and accordingly, the specific modes with fields formation. The selection of the order of high TMn1 modes in the designed frequency range is depended on tuning the h-up.
First, we study the regulative effect of the hole-x on the SPR. A circular hole in the center of the SHA serves as an electron beam channel, as shown in Fig. 2(a). When electrons pass through the beam channel, compared to the traditional grating structure or SHA excited by electron moving above, four dipoles will be excited to oscillate [12]. These moving dipoles will oscillate and move along z direction, and their radiation will radiate through the holes to the upper half space and lower half space. Therefore, the sizes of the SHA have a significant impact on SPR [7].
Fig. 2. (a) A schematic of the structure of the electron passing through a SHA. (b) The effect of the hole-x on the radiation spectrum. The green region is the main frequency band of SPR for the hole-x of 350 µm.
In the analysis using simulation domains, we set the surrounding space of SHA as an open boundary, which is equivalent to placing SHA in an infinite space, thus facilitating the observation of its radiation field. When the SHA is excited by an electron bunch with 30 keV, the simulation results show the relation between the hole-x and SPR, as shown in Fig. 2(b). When the hole-x is decreased from 400 µm to 300 µm, the peak value of the frequency increases from 414 GHz to 517 GHz. For other parameters of the subwavelength hole array, neither hole-y nor hole-z will affect the peak frequency position of the narrow-band radiation.
In typical Orotrons, the metal mirror together with the grating will form an open quasi-optical cavity, which has obvious characteristics such as sparse modes and high-quality factor. The formula that the electron speed v meets the period length and frequency is $v = cL/{\lambda _n}$ [13], where c is light speed, L is the period and ${\lambda _n}$ is the working wavelength. As a result, the SPR of 90o is selected, but other radiation angles and frequencies are not selected. Then, the distance between mirrors h has to be adjusted for the microwave generation in a FP resonator operating at TEM00n mode with index n, where n is an integer describing the half wavelength number of the mode: $h = n{{{\lambda _n}} / 2}$. Therefore, for a quasi-optical cavity, we must make the height of the cavity adjustable to match the speed and period by moving one metal mirror.
Here, the metal plates and SHA are used to support the higher-order mode at the desired frequency. The high-order mode is named as TMn1 and the index n is determined by the variation in the distribution pattern of the axial electric field along the transverse direction.
To make the higher-order mode have a large characteristic impedance, we hope that the higher-order mode has a strong axial electric field distribution in the SHA. When n is odd, the symmetric structure can be used, such as TM51 mode of symmetric structure. The asymmetric structure will be used when n is even. Here, we use symmetric structure so that a strong axial electric field is excited in the SHA along the z-direction. So, we will focus on the TMn1 modes, where n is odd.
When the hole-x is 350 µm, the peak range of SPR as shown in the green region in Fig. 2(b) falls between 430 GHz and 480 GHz. If we select the operating frequency is 466 GHz, h-up will be adjusted to obtain a high-order mode. In general, this size is closely related to half wavelengths, h-up≅q(λ/2) where index q = n/2. Figure 3 shows that when h-up are 150 µm, 490 µm, 840 µm and 1190 µm, the corresponding modes are TM11-2π, TM31-2π, TM51-2π and TM71-2π respectively. Here, if we design the structure to operate at TM51-2π mode, h-up should be about 840 µm.
Fig. 3. The relation between the fields distribution along transverse direction of different modes with specific h-up at a certain frequency (467 GHz).
It is noted that this kind of structure has good performance when it is enclosed on both sides along the x-direction. The slab-x that is defined as the width of the cavity along the x-direction, plays a critical role in forming closed cavities acting as collecting the SPR energy. Therefore, the effect of the slab-x on the characteristic impedance (R/Q) and the frequency of the TM51-2π mode is calculated, as shown in Fig. 4. It can be seen that when the slab-x is 800 µm, the maximum value is obtained for the R/Q. At the same time, it can be seen that slab-x has little effect on the frequency. This means tuning the slab-x can make the TM51-2π mode have large R/Q, on the basis of maintaining its resonant frequency in the frequency band with the green region, as shown in Fig. 2(b). It is significant to exploit this point to make the TM51 mode overwhelm other modes in the structure with large cross section at the desired frequency. The frequency band with the green region is largely determined by hole-x and accordingly, can be tuned to satisfy frequency specifications.
Fig. 4. R/Q variations and frequency of the TM51-2π mode with the width of the cavity (the size along the x-direction of the closed cavity, slab-x).
3. Electromagnetic characteristics of the extended interaction circuit based on the mode selection
The SHA structure is actually a multi-gap resonant structure and the closed spaces on the both sides of the SHA structure can be regarded as coupling cavities in typical extended interaction circuits. Figure 5 shows the basic model of the circuit. Based on the idea of mode selection from the SPR, the circuit is designed to operate in the TM51-2π mode at THz band. The structural parameters of the circuit are listed in Table 1. The dispersion diagram of the circuit is shown in Fig. 6. The green, blue, pink and black lines refer to TMs, TM11, TM31, TM51 mode respectively. TMs is a surface wave mode, while TM11, TM31 and TM51 are cavity modes. The red lines are the electron beam lines. These beam lines intersect with several modes, demonstrating the synchronous conditions of these modes are satisfied with the electron beam injection at the voltage of about 30 kV, as shown in the yellow area.
Fig. 5. The sketch map of Z = 0 plane and X = 0 plane of complete structure. The values of the parameters in figure correspond to Table 1.
Table 1. PARAMETERS
In Fig. 6, the yellow region represents the region where mode competition may occur. We can divide mode competition into two categories, one is the competition between adjacent longitudinal modes caused by frequency proximity, such as the competition of TM51-π/19 and TM51-2π. And the other is the competition of synchronous modes intersecting with the electron beam, such as TM11-10π/19, TM31-7π/19 and TM51-2π mode competition. For the first type, they correspond to different synchronization voltages, and we can adjust the voltage to avoid competition. The second type of the mode competition will be the main problem to be solved for EIO operating at high order mode. The focus of our work is to use the narrow-band radiation characteristics of SHA to suppress the excitation of low-order modes. The specific process is as follows.
We analyzed the mode competition among TM11-2π, TM31-2π, and TM51-2π. According to the previous analysis, for the case of the hole-x of 350 µm, the frequency of the peak range of SPR is from 430 GHz to 480 GHz when the beam voltage is 30 kV, as shown with the green band in Fig. 7. If h-up is 840 µm, the frequency of the TM11-2π, TM31-2π and TM51-2π mode is 255.345 GHz, 385.135 GHz and 467.014 GHz, respectively. The resonant frequency of the desired TM51-2π mode is located in the designed frequency range with the green region (within 430 GHz and 480 GHz). When h-up is changed to be 490 µm, the frequency of the TM11-2π, TM31-2π, and TM51-2π mode is shifted to 341.569 GHz, 466.697 GHz, and 645.388 GHz, respectively. For this case, only the TM31-2π mode is resonated in the green region within 430 GHz and 480 GHz. Figure 7 shows the desired mode will be located in the green region by adjusting h-up, and accordingly, the circuit selects this mode at the desired frequency. This supports the high order TMn1 mode selection with different high index n on the basis of tuning the frequency of the TMn1 mode to be located within the green region.
Fig. 7. The resonant frequency of TM11-2π, TM31-2π,TM51-2π is related to h-up. The green band represents the radiation band for the SHA.
In order to verify our idea, we calculated the Ez field distributions along the center line in the electron beam channel of these three modes for the case of h-up = 840 µm, as shown in Fig. 8(a). Obviously, the Ez field strength of TM51-2π mode is strongest, which predicts this mode would overwhelm other modes and it may be excited when its synchronous condition is satisfied.
Fig. 8. (a) The axial electric field distribution along the center line in the electron beam channel for the case of h-up = 840 µm. (b) The axial electric field distribution along the center line in the electron beam channel for the case of h-up = 490 µm.
For the case of h-up = 490 µm, the Ez field along the center line in the electron beam channels of the three modes are also compared, as shown in Fig. 8(b). The Ez field of the TM31-2π mode becomes stronger than that of the TM11-2π and TM51-2π mode. This means the TM31-2π mode is easier to be excited. The comparison of the two results in Fig. 8 shows that when h-up is adjusted so that the frequency of TMn1-2π mode falls in the green region (as shown in Fig. 2(b)) with the peak value of the SPR range, the field strength of this mode would be stronger than that of the other TMn1-2π modes. The green regions as shown in Figs. 6,7, and 8 correspond to Fig. 2(b), which supports the high order TMn1 mode selection.
4. Analysis of beam wave interaction
In order to examine the capability of the beam-wave interaction of the circuit, we design a simplified output structure (a coupling hole and a standard WR-3 waveguide) for the designed circuit operated in the TM51-2π mode and conducted PIC simulations by CST [14]. A distinguishing and important characteristic of the oscillation regime is the existence of a definite minimum beam current below which the cavity will not oscillate. Based on the small-signal theory [6], the starting current is mainly determined by the characteristic impedance R/Q, the load quality factor Ql, and the normalized electron conductance ge, which is expressed as [15–19]
It can be seen from Eq. (1) that when the negative peak value of the normalized electronic conductance, ge, is the maximum, the starting current can be obtained. Figure 9 shows the variation of the normalized ge with the beam voltage for the TM51-2π, and other possible competing modes, such as the TM51-π/19, TM51-2π/19, TM31-7π/19, TM11-10π/19 modes. It implies that the oscillation region of the TM51-2π would not suffer from the interferences of the other modes. Based on the structural parameters listed in Table 1, the value of R/Q is 910 Ohm, Ql is 566. Accordingly, the starting current of the designed circuit is about 0.02 A.
Fig. 9. Normalized ge as a function of the beam voltage for the TM51-2π, and other possible competing modes, such as the TM51-π/19, TM51-2π/19, TM31-7π/19, TM11-10π/19 modes.
When the current is larger than the calculated value above, the oscillation is predicted to be started. When considering surface loss in the circuit, the starting current will be increased. For the Cu with conductivity 5.8 × 107 S/m, the start-up time changes less and the output power continues to increase after the current is greater than 0.1 A in simulation. Therefore, the current of 0.115 A at the beam voltage of 29 kV is used to drive the designed circuit. A constant focusing magnetic field of 1 Tesla is applied to confine the electron beam.
PIC simulation results are shown in Fig. 10–11 in detail. To allow even lower frequency signals to pass through the coupling hole (600 µm × 100 µm × 100 µm), the coupling hole we have chosen has a longer long side. In order to ensure that the excited low-order mode can also be output, the output waveguide WR-3, whose size is 863.6 µm × 431.8 µm, and corresponding cutoff frequency is 173 GHz, was selected for the over mode output to verify that our structure can effectively suppress low order modes. Figure 10(a) shows that a stable signal is obtained after 20 ns. Figure 10(b) shows that the FFT diagram of the operating frequency is 466.31 GHz. This is consistent with the frequency of the TM51-2π mode predicted in the dispersion curves shown in Fig. 6 and indicates that only one mode is excited. Figure 11(a) shows that the operating mode is TM51-2π. The phase space plot of the electrons is shown in Fig. 11(b). It shows that an efficient beam-wave interaction occurs in the designed circuit. These simulation results show preliminarily the interaction capability of the designed circuit operated in a single mode driven by the given electron beam.
Fig. 10. Simulation results of the designed circuit with a simplified output structure. (a) Time signal received at the waveguide port. (b) The frequency spectrum of the port signal.
Fig. 11. (a) The Ez field distribution at 50 ns. (b) Electron energy distribution along the Z-axis at 50 ns.
We conducted PIC simulations with different electric conductivities for the metal material used in the circuit. Figure 12 shows the relationship between the output power and the starting current as a function of conductivity. It can be seen from the Fig. 12 that the electrical conductivity has an obvious effect on the starting current, and the starting current continues to increase with the decrease of conductivity. In addition, the output power decreases with the decrease of the electrical conductivity for the cases the beam currents are set as the starting currents.
Fig. 12. The relationship between the output power and the start-up time as a function of conductivity.
Next, we investigated the effects of operating voltage and coupling cavity height on output power and frequency, as shown in Fig. 13. Unlike the Orotron, the operating frequency remains almost unchanged when the voltage is changed, so there is no need to adjust the height of the cavity. When the coupling cavity height h-up varies between 800 µm to 880 µm, the frequency gradually decreases and the device still operate at TM51-2π mode, which is consistent with the narrowband radiation frequency range and trend of SHA in Fig. 7. This characteristic significantly reduces requirements for cavity size processing.
Fig. 13. (a) The output power and frequency versus operating voltage U. (b) The output power and frequency versus coupling cavity height h-up.
The parameters of electron beam and magnetic field have a significant impact on the electron beam circulation rate, which affect the output power of the device significantly. The Fig. 14(a) shows the effect of the cathode radius rc (the electron beam radius) on the output power and electron beam circulation rate, it has almost no effect on frequency. The Fig. 14(b) shows the influence of the magnetic field Bz on the output power and electron beam circulation rate.
Fig. 14. (a) The output power and electron beam circulation rate versus (a) the cathode radius rc, (b) the magnetic field Bz.
In addition, the number of the period is also an important parameter. An appropriate number of periods will result in smaller starting current and larger output power. The relationship between the number of periods n and the output power and electron beam circulation rate is shown in Fig. 15. When we choose a number of 31 and current of 115 mA, the output power is 40 W. Meanwhile, when the current is 60 mA, an output power of 14.5 W can be obtained.
Combined with the analysis of the narrow-band radiation of SHA and PIC simulation results, the main advantages for the proposed structures are as below:
On one hand, the radiation produced by the electron beam passing the SHA is not restricted to 90°, easing the restriction of cavity geometric parameters. On the other hand, the mechanism of the narrow-band radiation of SHA can help suppress lower order modes, therefore allowing stable operation at the high order mode. Moreover, by replacing the open cavity in traditional Orotron with a closed cavity, the dissipated energy has been reduced, and the output power has been further increased.
5. Conclusion and discussion
High-order mode operation is an effective way for VEDs to operate in THz. In this paper, based on the SPR in the SHA, an effective method to design VEDs operating at high-order modes is proposed. It finally realizes the THz vacuum radiation source operating at the higher-order mode and overcome the frequency limits of conventional fundamental mode version with a smaller cavity in higher frequency. Particularly, the mode competition issue will also be improved because of the characteristic of SPR. Therefore, this method may become an effective method to push the development of the electron beam driven devices toward THz frequency.
Funding
National Natural Science Foundation of China (62171098, 92163204, 61921002); Fundamental Research Funds for the Central Universities (ZYGX2020ZB013); Natural Science Foundation of Sichuan Province (2022NSFC0512).
Disclosures
The authors declare no competing interests (including both financial and non-financial interests).
Data availability
All data that support the findings of this study are included in the manuscript. Source data for those graphs may be obtained from the corresponding authors upon reasonable request.
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