Challenges in the Design and Development of Slow-Wave Structure for THz Traveling-Wave Tube: A Tutorial Review

Abstract

As solid-state devices continue to advance, vacuum electron devices maintain critical importance due to their superior high-frequency power handling, long-term reliability, and operational efficiency. Among these, traveling-wave tubes (TWTs) excel in high-power microwave (HPM) applications, offering exceptional bandwidth and gain. However, developing THz-range TWT slow-wave structures (SWSs) presents significant design challenges. This work systematically outlines the SWS design methodology while addressing key obstacles and their solutions. As a demonstration, a staggered double vane (SDV) SWS operating at 1 THz (980–1080 GHz) achieves 650 mW output power, 23.35 dB gain, 0.14% electronic efficiency, and compact 21 mm length. Comparative analysis with deformed quasi-sine waveguide (D-QSWG) SWS confirms the SDV design’s superior performance for THz applications.

1. Introduction

Vacuum electron devices, such as klystron, magnetron, traveling-wave tube (TWT), and gyrotron, are designed to generate and amplify high-power microwave signals [1,2]. They were the only high-power sources across the microwave range before solid-state devices emerged. While solid-state devices are used for lower-power and higher-frequency applications, vacuum electron devices excel in efficiency, temperature resilience, bandwidth, and reliability, making them essential for high-power microwave systems [3]. For the broader bandwidth and higher gain, TWTs are better options. These days, the interest in operation of the TWTs at terahertz (THz) frequency band (0.1–3 THz) is increasing due to their THz applications in 6G communications, medical imaging, defense, satellite communications, particle accelerators, plasma diagnostics and material characterization shown in Figure 1 [4].

In TWTs, the slow-wave structure (SWS) is a critical component that significantly influences the device’s performance. The helix-shaped SWS is widely favored due to its high interaction impedance and broad bandwidth. However, designing and fabricating a helix SWS becomes increasingly challenging at terahertz frequencies because of the extremely small dimensions. Various SWS configurations, such as folded waveguides [5,6], sine waveguides, and staggered double vanes [7], have gained prominence due to their compatibility with micro-fabrication techniques, making them easier to develop. Compared to pencil electron beams [8,9], sheet electron beams (SEBs) [10] offer advantages such as the ability to carry higher electron beam current and provide a larger interaction surface with the SWS.

In recent years, in the THz frequency range, the development of these configurations are growing rapidly. In 2018, Yanyan Tian et al. [11] designed a ridge-loaded folded rectangular groove waveguide (RLFRGWG) at 0.34 THz with output power of 39 W, gain of 26.99 dB and with an efficiency of 0.59%. This design provides a higher output power with wider bandwidth by applying a pencil electron beam. Zhigang Lu et al. [12] in 2020 designed a double tunnel sine waveguide (DTSWG) of sheet beam at 215 GHz with maximum power of 405 W and electron efficiency of 9.8%. The sine waveguide structures like flat-roofed sine waveguide [13], sine groove waveguide, and piecewise sine waveguide [14] are very efficient in the THz region. Due to higher demand at 1 THz frequency, in 2021, Ruichao Yang et al. [15] developed and tested a flat-roofed sine waveguide (FRSWG) SWS and achieved a simulated-result output power of 720 mW, and a gain of 28.57 dB at a frequency band of 1.02–1.04 THz. In 2022, Ruichao Yang et al. [16] modified the FRSWG SWS into deformed-quasi sine waveguide (D-QSWG) SWS with a simulated output power of 300 mW and gain of 25 dB, developed using the nano-CNC machining. Fabricating SWS at higher frequencies is a challenging task. In 2023, Zechuan Wang et al. [17] designed a staggered double segmented grating (SDSG) SWS by combining sine waveguide (SW) with staggered double grating (SDG) SWS with an output power of 32.8 W, electronic efficiency of 2.84% at 340 GHz. It has advantages like wide bandwidth, higher coupling impedance, lower ohmic loss, and easy fabrication. In 2025, Junyoung Lee et al. [18] designed and cold tested a staggered double vane (SDV) SWS with a wider 3dB bandwidth of 40 GHz by developing a diamond-shaped Bragg resonator.

Although numerous designs are available for fabricating slow-wave structures (SWSs) in the THz frequency band, the discussion of design methodologies and their associated challenges remains a critical area that needs to be addressed. Among the SWS configurations mentioned, the staggered double vane (SDV) [19,20,21] combined with a sheet electron beam stands out as a superior choice. At higher frequencies, the SDV emerges as a promising alternative for fabrication due to its numerous advantages. It provides higher interaction efficiency under a sheet electron beam, enables an increased electron beam aspect ratio, and offers high power-handling capability. Additionally, SDV structures feature enhanced thermal stability, ease of fabrication, and lower phase velocity, making them highly versatile for a wide range of applications. Based on the above discussion, the objectives of this study are as follows:

• To outline a step-by-step design methodology for SWSs in traveling-wave tubes (TWTs) operating in the THz frequency band.
• To identify the challenges in designing SWSs and propose potential solutions to overcome them.
• To present a case study by designing a staggered double vane (SDV) structure operating at 1 THz frequency.

In Section 2, a step-by-step design methodology for SWS at THz frequencies is presented. Section 3 addresses the challenges encountered during the design and development of SWS, along with potential solutions. Section 4 provides a case study by examining an SDV SWS operating at 1 THz frequency. Finally, Section 5 concludes the review.

2. Design Methodology of SWS at THz Frequency

It is a complex and multidisciplinary process to design a slow-wave structure (SWS) for a traveling-wave tube (TWT) at terahertz (THz) frequency. The design process involves optimizing the selected structure to achieve efficient beam–wave interaction, high gain, high efficiency, and broader bandwidth, while simultaneously addressing challenges such as optimizing the design and input parameters, fabrication tolerances, thermal management, and power handling effects. The design methodology and step-wise design approach is shown in Figure 2 and the design approach is explained in the three following steps: initialization, characteristics analysis, and prototype and validation.

2.1. Initialization

The design process begins by defining key parameters including the frequency range (e.g., 0.3–3 THz), bandwidth (10–20% of center frequency), gain target (20–50 dB), and output power requirements. The slow-wave structure (SWS) geometry such as helical, coupled-cavity, planar, or folded wave-guide is then selected based on theoretical modeling (equivalent circuit or field theory), simulation results (e.g., CST/HFSS), compatibility with the available beam voltage, and fabrication constraints (e.g., lithography tolerances, NANO-CNC machining, or 3D-printing capabilities). This systematic approach ensures optimal performance while accounting for practical implementation challenges.

2.2. Characteristics Analysis

This section analyzes key SWS characteristics including dispersion, transmission, beam–wave interaction, and thermal/structural performance. Dispersion studies evaluate RF field distribution, phase velocity ($v_p$), and interaction impedance to optimize geometry, ensuring a flat dispersion curve across the target bandwidth for consistent operation. The design must synchronize $v_p$ with electron beam velocity ($v_e$) while maximizing interaction impedance for efficient energy transfer. These combined analyses enable performance-driven SWS optimization. Transmission characteristics analyze RF signal propagation through the SWS, evaluating three critical parameters. The return loss ($S_{11}$) quantifies impedance matching, where lower values (<−10 dB) minimize reflections and maximize power transfer, and the insertion loss ($S_{21}$) measures the amount of input RF signal lost as it travels through the SWS. A lower insertion loss means more RF power is available for amplification, while higher insertion loss reduces the overall gain and efficiency of the TWT, and the bandwidth identifies the operational frequency range where these parameters meet the specifications. Hence, poor $S_{11}$ causes reflected power losses, while higher $S_{21}$ directly reduces TWT gain and efficiency. Together, these metrics determine the effective pass band and power handling capability of the SWS.

The beam–wave interaction in a SWS of TWT is the fundamental energy-transfer mechanism enabling RF signal amplification. This interaction analysis determines key performance metrics like output power, gain, and electronic efficiency. Critical design parameters include the electron beam’s voltage/current (providing interaction energy) and axial magnetic field strength (ensuring beam focusing). As the beam traverses SWS, thermal effects arise from beam interception and ohmic losses, necessitating the careful design of the beam tunnel and electron gun with low fill factor to minimize these impacts. Thermal and structural analysis is critical for SWS design, addressing operational challenges from heat accumulation and mechanical stresses. Thermal modeling evaluates heat generation (from beam interception and ohmic losses) and dissipation pathways to prevent performance degradation, while structural simulations assess vibration resistance and thermal expansion tolerance to avoid deformation. Together, these analyses ensure operational reliability, extended lifespan, and safety by maintaining structural integrity under extreme conditions [22,23].

2.3. Prototype and Validation

At THz frequencies, fabrication tolerances are extremely critical, as they are in the micrometer range. Various fabrication techniques, such as NANO-CNC technology [24,25], micro-machining [26], 3D printing [27,28], etching and lithography [29], are available to manufacture the device. After fabricating the device, cold tests (without the electron beam) are conducted to measure the RF characteristics of SWS, such as the dispersion curve, interaction impedance, return loss, and insertion loss. Following this, hot tests (with the electron beam) are performed to measure gain and bandwidth, output power and efficiency, as well as beam interception and thermal performance. The design methodology of the SWS from initialization to prototype and validation is shown in detailed in Figure 2.

3. Challenges and Possible Solutions in the Design of SWS for THz-TWT

During the design of the slow-wave structure (SWS) at THz frequency, there are many challenges to overcome. The design challenges and the possible solutions are explained in detail as follows.

3.1. Finding the Design Parameters of SWS

The design parameters such as period of the cell, beam tunnel height, slot height, and vane width define the shape of the SWS. Designers often face significant difficulties at this stage. Unlike in cross-field devices such as magnetrons, there are no precise mathematical expressions to establish even the ranges of design parameters for TWTs. To investigate the design parameters and objective functions—such as operating frequency, output power, bandwidth, and efficiency—four research methodologies are typically employed: equivalent circuit method [30,31], field equations theory [32,33], computer simulation, and fabrication. However, each method has its limitations. Field theory involves solving complex transcendental equations, making it a highly intricate methodology. Optimization using simulation tools like CST or HFSS often relies on trial and error, consuming significant memory and time. Fabricating a device without first confirming the design parameters through simulation or theory is generally not advisable. Thus, designers must navigate this challenge carefully to achieve an optimal SWS design.

To address these challenges, there are three primary optimization methods available: the conventional method, the CST or simulation-based optimization method, and genetic or AI-based algorithms as shown in Figure 3. Traditional trial-and-error approaches are inherently time-consuming for SWS optimization. While CST-based methods improve upon this, their iterative nature still demands excessive computational resources and manual tuning, often resulting in sub-optimal local solutions and slow convergence. Modern AI-driven optimization techniques overcome these limitations by enabling efficient global optimization with reduced computational costs and automation. However, these methods require pre-existing datasets or initial simulations to establish performance baselines for effective optimization. This optimization approach can handle the non-linear problems, multi-objective functions, and has global search capability [34,35,36,37]. Some genetic algorithms include the non-dominated sorting genetic algorithm-II (NSGA-II), particle swarm optimization (PSO), cellular genetic algorithm (CGA), standard genetic algorithm (SGA), ant colony optimization (ACO), etc.

For example, when using NSGA-II, the process begins by generating an initial population (random design parameters) of solutions ($P_t$, $Q_t$) along with the respective objective functions. The best designs are grouped and ranked according to Pareto dominance ($F_1$, $F_2$, $F_3$). Solutions for the next generation are selected ($P_{t+1}$), prioritizing those in higher-ranked fronts. New solutions are generated through crossover and mutation, then combined with the parent population. The merged group undergoes sorting and diversity checks before selecting the top performers for the next generation. This cycle repeats until achieving optimal results. The schematic diagram for the NSGA-II algorithm is shown in Figure 4. Recent studies have successfully applied genetic algorithms to optimize various vacuum electron devices, including magnetron anode, depressed collector [38], helix TWT [39], and the pill-box dielectric window.

AI-based optimization employs various advanced algorithms to enhance device performance, including Random Forest (RFA) for decision-making, Neural Networks (BPNN, CNNs, and GNNs) for pattern recognition and prediction, Reinforcement Learning (RL) for adaptive system control, and Physics-Informed Neural Networks (PINNs) for incorporating domain-specific physical constraints into the learning process. These techniques enable more efficient and accurate optimization compared to traditional methods. Convolutional Neural Networks (CNNs) [40] are a powerful type of AI that works especially well for designing SWS structures in THz devices. Think of them like smart filters that scan over a design, picking out important patterns and simplifying them step by step. First, they detect small details, then combine them to understand bigger features, and finally make decisions based on what they have learned. This makes CNNs perfect for tasks where spatial layout matters—in optimizing the complex shapes needed in THz technology. The representation of the CNN is shown in Figure 5. This will be helpful in finding the design parameters quickly and accurately. While these AI algorithms are still in development, some studies [41,42,43] have successfully applied them to optimize vacuum electron devices.

3.2. Selection of the Input Parameters

The input parameters of the SWS are input RF power, beam power ($P_b$), and the axial magnetic field ($B_b$). For THz SWS, the input RF power is in the order of 1 mW to 10 mW [15,16,44]. The beam power is the product of the beam voltage ($V_b$) and beam current ($I_b$). Traveling-wave tubes are linear beam devices, where the electron beam surrounded by SWS is passing through the beam tunnel with an applied beam voltage ($V_b$) and beam current ($I_b$), and it is focused by an applied magnetic field ($B_b$) in Tesla. The applied magnetic field is calculated as follows:

$$B_b=0.83\times {10}^{-3}{\displaystyle \frac{I_b^{1/2}}{r_m{V}_b^{1/4}}}$$

(1)

where $r_m$ is the equilibrium radius under the Brillouin matching condition. An input electromagnetic (RF) signal with input average power propagates along selected SWS (either helix or coupled cavity or planar) with a phase velocity ($v_p$) [9]. The beam voltage ($V_b$) determines the velocity of the electrons in the beam. It is calculated based on the desired electron beam velocity ($v_e$), which must match the phase velocity of the RF wave in the SWS. For example, a staggered double vane (SDV) SWS of alternating metallic vanes on opposing waveguide walls is considered and a dispersion curve is drawn as shown in Figure 6. The forward wave of fundamental mode (mode 1) with higher-order harmonic (m = 1) is completely flat with the voltage beam line, which determines the broader bandwidth of 100 GHz. The phase velocity can be calculated by using Equation (2).

$$v_p=\beta c=c\sqrt{1-{\displaystyle \frac{1}{{(1+\frac{V_b}{5.11\times {10}^5})}^2}}}$$

(2)

where $\beta$ is the propagation constant, $V_b$ is the beam voltage, and c is the speed of light in m/s. The beam velocity can be calculated by Equation (3):

$$v_e=\sqrt{{\displaystyle \frac{2e{V}_b}{m_e}}}$$

(3)

where $e=1.6\times {10}^{-19}$ C is the electron charge and $m_e=9.109\times {10}^{-31}$ kg is the mass of the electron. From Figure 6, mode 1 (fundamental) dominates with the wide bandwidth, while mode 2 and mode 3 (higher-order) cause instabilities like backward-wave oscillations. The spatial harmonics (m = 0, 1, 2) arise from periodicity; m = 0 enables ideal beam coupling but requires impractical THz beam voltages. In this study, the operation uses Mode 1 with m = 1 harmonic. The second parameter of the beam power ($P_b$) is the beam current ($I_b$) which can be chosen based on the selected beam tunnel area and the beam current density. At 1 THz, the beam current density is in the order of 300 to 400 A/cm² to overcome the beam interception challenges.

3.3. Phase Velocity Mismatch

The phase velocity mismatch in the design of a slow-wave structure (SWS) for a traveling-wave tube (TWT) arises when the phase velocity of the electromagnetic wave ($v_p$) in the SWS does not align with the velocity of the electron beam ($v_e$). This mismatch can severely degrade TWT performance by reducing the efficiency of energy transfer from the electron beam to the RF signal. At high power levels, non-linear effects such as space charge forces or harmonic generation can cause deviations in the phase velocity, further exacerbating the issue. In broadband SWS designs, this problem is particularly common, and the improper selection of the beam voltage can worsen the mismatch. Consequences include increased beam interception, thermal loading, and potential device damage. To address this issue, the beam voltage must be carefully selected to satisfy the synchronization condition ($v_p=v_e$), which leads to amplification of the RF signal as illustrated in Figure 7.

3.4. Interaction or Coupling Impedance

The interaction impedance measures the strength of the interaction between the RF (or electromagnetic) wave and the electron beam. A higher interaction impedance typically results in greater slowing of the RF wave, which extends the interaction time between the electron beam and the wave. This prolonged interaction enhances energy transfer, leading to higher output power. It is mathematically defined as the ratio of the square of the electric field amplitude to the power flow per unit length in the structure, expressed as [15]

$$K_c={\displaystyle \frac{|E_z^2|}{2{\beta }_n^2{P}_w}}$$

(4)

where $E_z$ is the electric field amplitude in the axial direction, ${\beta }_n$ is the phase constant, and $P_w$ is the power flow:

$${\beta }_n={\beta }_0+{\displaystyle \frac{2n\pi }{L}},n=0,\pm 1,\pm 2,...$$

(5)

where ${\beta }_0=\varphi /L$ is the phase constant for the fundamental wave, $\varphi$ is the phase shift in one period, and L is the period length. The interaction is greater at the lower cutoff frequency of a dispersion as observed in Figure 6, which is not practically possible. This is influenced by the structure dimensions and the operating frequency. In general, the interaction impedance is calculated at the center of the electron beam, but in reality, it is the average as shown in Figure 8.

3.5. Design of Reflector, RF Coupler, Attenuator, and Transition Section

As the RF signal travels through the SWS, a portion of it may leak into the electron gun, which is an undesirable effect. These RF leaks can disrupt the electric fields, causing instability in the electron beam formation and resulting in a distorted beam. Additionally, the RF waves can induce currents in the electron gun, leading to localized heating. If the heating becomes severe, it can cause the electron gun to fail. So, the reflector at the two ends of SWS helps to reflect the RF signal at the desired frequency band and stop the RF leakage into the electron gun. A bad-impedance-matching section, the Bragg reflector is helpful in stopping (leaking) the RF signal. Single, double, and even triple Bragg reflectors are used [19]. In Figure 9a, a double Bragg reflector is presented.

The RF coupler (see Figure 9b) plays a crucial role in efficiently coupling electromagnetic energy into and out of the slow-wave structure (SWS). The input coupler transfers the input RF signal (THz wave) from an external source into the SWS with minimal reflection and maximum power transfer, enabling effective interaction with the electron beam. On the other hand, the output coupler extracts the amplified RF signal from the SWS and delivers it to the external load with minimal losses. These couplers are designed to match the impedance of the SWS with the external circuit, ensuring proper impedance matching to minimize reflections and standing waves. In the THz frequency range, mode conversion of the input signal may also be necessary to achieve efficient coupling, which is a critical consideration for optimal performance.

An attenuator is a passive device used to absorb or suppress unwanted electromagnetic waves, such as reflected signals or oscillations, within the slow-wave structure (SWS) [19]. Typically, it is a lossy structure positioned at specific locations (input, middle, or output) to absorb RF energy as shown in Figure 9d. It not only suppresses unwanted signals but also helps to maintain a flat gain response across the operating bandwidth by reducing signal variations and enhancing stability. However, the improper placement or alignment of the attenuator can lead to increased insertion loss and disruption of the electron beam.

In the THz frequency band (110 GHz to 1100 GHz), rectangular waveguides such as WR-10 to WR-1 are commonly used [45]. To connect the RF coupler to these rectangular waveguides, a transition section is required to carry the generated output power signal. Various types of transition sections, including stepped, ridge-based, and brick-type designs, are employed to efficiently transmit the signal. These transition sections should match the impedance for efficient power transfer with low return and insertion loss. Stepped and brick type transition sections are shown in Figure 9e.

The design of these devices often lacks a standardized methodology and frequently relies on a trial-and-error approach, as the optimal parameters can vary significantly depending on the specific type of structure being used. Design parameters are typically selected based on the analysis of S-parameters obtained from software simulations, which provide critical insights into the performance and behavior of the structure. This iterative process allows for the refinement of design to achieve the desired performance characteristics.

3.6. Filling Factor

The fill factor is a critical parameter that describes the extent to which the electron beam interacts with the RF wave in the SWS of TWTs. It is the ratio of the electron beam cross-sectional area to the beam tunnel area or beam–wave interaction area. When the fill factor is equal to 1, the electron beam occupies the entire region of the RF field, resulting in maximum energy transfer and higher gain. However, achieving a fill factor of 1 is not practical in reality because it would lead to significant beam interception, causing excessive heat and potential damage to the SWS. When the fill factor is less than 1, the electron beam interacts only weakly with the RF wave, leading to lower efficiency and reduced gain [46]. Therefore, the choice of the fill factor is a critical consideration in the design of SWSs, especially for applications at terahertz (THz) frequencies by adjusting the beam current, voltage, and radius to achieve a high fill factor without causing instability or excessive space charge effects. From Figure 10, the beam tunnel and electron beam for both the pencil and sheet electron beam is presented. The expression for filling factor is expressed as follows:

$$Filling factor={\displaystyle \frac{A_b}{A_T}}={\displaystyle \frac{\pi r_b^2}{\pi r_T^2}}={\displaystyle \frac{a_1\times b_1}{a_2\times b_2}}$$

(6)

3.7. Tapered Section

Tapering at both ends of the slow-wave structure (SWS) involves a gradual modification of the geometric dimensions of the structure. This modification can follow either a linear or non-linear profile, and the designer selects an appropriate tapering profile based on the specific shape and requirements of the SWS. In a non-tapered SWS, the phase velocity ($v_p$) remains constant, which can cause the electron beam velocity ($v_e$) to fall out of synchronization as the beam loses energy to the RF wave. In contrast, in a tapered SWS, the phase velocity is progressively adjusted to match the changing velocity of the electron beam as it loses energy. This synchronization enhances the interaction between the RF wave and the electron beam, resulting in higher output power, increased gain, and improved efficiency. Additionally, tapering helps reduce the localized heating and minimize damage to the SWS, thereby enhancing its reliability and longevity. It also supports broadband operation by enabling better performance across a wider frequency range. Hence, the emitter-end taper enables impedance matching (minimizing S11), mode control (suppressing higher-order modes), and optimized beam injection. The output taper eliminates discontinuities to maximize |S21| by reducing THz wave scattering and harmonic interference while dissipating heat from the beam–RF interaction. There are two types of tapering sections available which are the convergent [47] and divergent [48] type as shown in Figure 11. However, tapering increases the overall length of the SWS, which leads to higher metal losses at 1 THz due to the extended interaction region.

3.8. Material Selection

In the 1 THz band, conventional traveling-wave tubes (TWTs) face challenges like low efficiency, limited output power, and narrow bandwidth, which hinders high-frequency vacuum electron device (VED) development. To overcome these challenges, this study explores three material-based SWS innovations: 2D material coatings, dielectric loading, and metamaterial integration, demonstrating their potential for enhanced device performance. Graphene, a 2D carbon material of a honeycomb structure (see Figure 12a), called a semi-metal or zero gap semiconductor because this acts like a conductive metal, tunes like semiconductor, and manipulates light like a dielectric, offers a breakthrough solution. These unique characteristics enable surface plasmon excitation, allowing sub-wavelength periodic structures to achieve tuneable plasmon resonance which lead to high-efficiency THz modulators [49] and absorbers [50]. Graphene’s unique wave vectors enable efficient plasmon excitation by free electrons [51], offering a breakthrough approach for THz generation, detection, and modulation to overcome conventional VED limitations. In 2024, Fan Deng et al. [52] coated graphene to staggered double grating SWS for BWO (see Figure 12b) to improve the output power and shorten the starting oscillation time.

Dielectric loading refers to the strategic integration of dielectric materials like ceramics and quartz into the SWS to manipulate electromagnetic wave properties [53]. In THz SWS, it serves to slow the phase velocity to synchronize with the electron beam for efficient energy transfer, improves the bandwidth, reduces ohmic losses by shielding metal surfaces from high-frequency skin effects, confines EM fields to enhance beam–wave interaction, flattens dispersion for wider bandwidth, and improves thermal management. Simakov et al. [54], in 2019, incorporated the ceramic material into the Ka-band TWT as shown in Figure 12c to achieve the broader bandwidth.

Figure 12. (a) Graphene honey comb structure, (b) graphene-coated staggered double grating [52], (c) ceramic dielectric loading into SWS of TWT [54], and (d) metamaterial-based helical SWS with double sheet beam [55].

Figure 12. (a) Graphene honey comb structure, (b) graphene-coated staggered double grating [52], (c) ceramic dielectric loading into SWS of TWT [54], and (d) metamaterial-based helical SWS with double sheet beam [55].

Metamaterials (MTM) are artificially engineered materials with sub-wavelength structures that exhibit extraordinary electromagnetic properties not found in nature, such as negative permittivity ($ϵ<0$), negative permeability ($\mu <0$), or near-zero refractive index. MTM-based SWSs outperform conventional SWS designs by leveraging their strong resonant properties. The intensified electric fields in MTM based VEDs significantly enhance beam–wave interaction efficiency, leading to superior device performance. Jian et al. [55], in 2023, simulated a 0.22 THz metamaterial-based SWS for TWT with double sheet beam (see Figure 12d) with a higher interaction impedance of 5 ohm followed by higher output power, and 5.4 GHz, 3 dB bandwidth. In 2024, Thakur et al. [56] designed and simulated an MTM-based double negative medium (DNM) helical SWS for dual mode (forward and backward wave) operation. It was successfully verified with a theoretical approach, and the device is utilized for TWT and BWO.Leveraging these remarkable material characteristics, researchers have developed several metamaterial-based SWS configurations [57,58,59]. However, their practical application scope demands further extension to fully exploit their capabilities in vacuum electronic devices.

3.9. Fabrication Difficulties at THz

Fabricating slow-wave structures (SWSs) for THz frequencies presents significant challenges due to their extremely small dimensions, typically in the order of microns or nano-meters, and high precision is also required. Several techniques, such as NANO-CNC machining, 3D printing, and lithography are employed for the fabrication of these devices, but each comes with its own limitations. While 3D-printing is still in the developmental phase for such applications, lithography remains prohibitively expensive. Currently, NANO-CNC technology is the preferred method for fabricating these devices at the THz frequency. For instance, ref. [20] demonstrated a micro-fabricated double-staggered grating waveguide along with WR-3 flanged rectangular waveguide and S-parameter measurements using a vector network analyzer (VNA) (as shown in Figure 13).

Material selection (see Section 3.8) is based on properties such as high thermal conductivity, low loss tangent, and high heat capacity (e.g., copper, gold, or silicon carbide). During the fabrication process, meticulous attention must be paid to the dimensions, as even minor deviations can have a significant impact on performance. Additionally, issues such as voids, cracks, and surface deformations can arise, which must be minimized to ensure the metal surface maintains a low surface roughness value. Achieving this level of precision is critical for optimal functionality.

Skin depth and surface roughness are the two factors which are related to the material properties. Skin depth is a measure of how much the RF signal can go deeper into the conductive material which can be calculated as follows:

$$\delta =\sqrt{{\displaystyle \frac{2}{{\mu }_0\omega \sigma }}}$$

(7)

where ${\mu }_0$($=4\pi \times {10}^{-7}$) permeability in $H/m$, and $\sigma$($=5.8\times {10}^7$) conductivity of pure copper in S/m. When the RF signal penetrates deeper into the conductive material, it results in higher ohmic losses, reduced energy transfer efficiency, and increased temperature, which can lead to thermal stress and material degradation. Additionally, deeper penetration may cause increased dispersion, negatively impacting signal integrity. However, at higher frequencies, the skin depth becomes smaller, confining the RF signal closer to the conductive surface. This reduces losses and improves the overall performance of the device. Hence, the selection of higher electrical conductivity material, material with smooth surface, and surface coatings is helpful to reduce the losses.

The second property is the surface roughness, which is irregularities or voids forming on the surface during the fabrication, leading to severe metal losses. It can cause scattering of the RF signal, leading to signal distortion and reduced performance, and the RF signal may not travel a longer distance [60]. The effective conductivity (${\sigma }_{eff}$) of a material while considering the skin depth and surface roughness is calculated as follows:

$${\sigma }_{eff}={\displaystyle \frac{\sigma }{{\left(1+\frac{2}{\pi }arctan[1.4\Delta /\delta ]\right)}^2}}$$

(8)

where $\Delta$ is the surface roughness and $\delta$ is the skin depth. Applying a thin layer of high-conductivity material (e.g., gold or silver coating) can improve the surface smoothness and reduce metal losses.

3.10. Fabrication Tolerance Analysis

For THz TWTs, fabrication tolerance analysis quantifies how dimensional errors (e.g., ±100 nm in periodicity) affects SWS performance using simulations and sensitivity rankings [61]. This analysis is required due to the extreme miniaturization of the device, high sensitivity to small geometric deviations, and complex coupling between tolerances. At 1 THz, even sub-micron errors significantly degrade efficiency (around 15–20% drops) and bandwidth (5% shifts), making this analysis essential for choosing fabrication methods (lithography or NANO-CNC or 3D-printing) and setting realistic specifications. The analysis specifies the required manufacturing precision (tolerances) and verifies production capability, ensuring feasible fabrication.

3.11. Application of Slow-Wave Structures in Other Components

Beyond amplifiers like TWT, slow-wave structures (SWSs) play a crucial role in enhancing the performance of various high-frequency components. For backward-wave oscillators (BWOs), optimized SWS configurations allow tunable frequency generation, supporting applications in spectroscopy and imaging. Additionally, SWS-integrated millimeter and sub-millimeter wave antennas [62,63] benefit from miniaturization and improved radiation efficiency. Recent advances also explore SWS in filters [64], where controlled dispersion properties are essential, and in phase shifters [65] critical for beam forming in 5G/6G and aerospace systems. This section highlights the versatility of SWS technology across emerging RF and photonic systems.

4. Case Study: Design of THz Staggered Double Vane Slow-Wave Structure

In the THz frequency range, fabricating devices poses significant challenges due to their extremely smaller in dimensions. Among the potential SWS structures, the staggered double vane slow-wave structure (SDV-SWS) stands out for its ability to have higher interaction impedance, high output power and broader bandwidth as explained in the introduction section. This case study focuses on the SDV-SWS, presenting it as a practical example in a tutorial format. To improve the output power, a sheet beam is chosen instead of a pencil electron beam since the sheet beam covers a larger cross-sectional area, enabling it to carry a greater beam current with the same current density. This characteristic makes the sheet beam more advantageous for improving the efficiency and output power in traveling-wave tubes (TWTs).

4.1. Single Cell Design and Dispersion Characteristics

Figure 14 illustrates the staggered double vane structure, the sheet electron beam, and the vacuum model of a single cell, along with the associated design parameters. The primary objective of analyzing the dispersion characteristics is to determine whether the electron beam velocity can synchronize with the phase velocity of the specific SWS structure. Based on this analysis, the single cell is optimized. The optimized design parameters for the single cell are summarized in

Table 1. Design parameters of the staggered double vane single cell.

Parameter	Symbol	Value ($\mathsf{\mu }$m)
Period of the cell	P	98
Depth of the cell	d	150
Vane thickness	V	30
Width of the slot	W	68
Height of the slot	h	150
Height of the beam tunnel	$h_{BT}$	50

.

After identifying the design parameters, the operating voltage is determined by analyzing the dispersion characteristics along the fundamental mode. A beam line corresponding to a specific voltage is drawn to align with the forward wave, aiming to maximize bandwidth. However, even when the beam line appears relatively flat to the fundamental mode, the achieved bandwidth is often narrower than expected. From Figure 15, the 25.3 kV beam line intersects the forward wave near its lower cutoff frequency, where the interaction impedance is higher. The corresponding phase velocity and the interaction impedance are calculated from CST and shown in Figure 16. The dispersion analysis reveals a normalized phase velocity of 0.235 to 0.322 at the frequency band of 0.98 to 1.07 THz. The interaction impedance is calculated at the center and multiple locations of sheet beam which are found to be 0.4 $\Omega$ and 1.2 $\Omega$, respectively, as shown in Figure 16 across the entire bandwidth from 0.98 to 1.07 THz, indicating the presence of a strong interaction.

4.2. Transmission Characteristics of SDV-SWS

Figure 17a shows the simulated vacuum model of the complete structure, which is connected at both ends with a double Bragg reflector to prevent RF signal leakage. Port 1 and port 2 are input and output RF signal ports whereas port 3 and port 4 are the beam tunnel ports which emit and collect the electron beam, respectively. A RF coupler is used to connect the SWS and standard WR-1 waveguide with impedance matching. The number of periods are 200, and the length of complete structure is 21 mm. The background of the model is treated as a lossy metal (copper), accounting for copper with surface roughness. The conductivity of the copper metal is set to 2.5 × 10⁷ S/m.

From Figure 17b, it is observed that S11 is below −10 dB in the frequency range of 1.02 to 1.08 GHz (60 GHz) and reaches −55 dB at 1.04 THz, indicating excellent impedance matching. This means that most of the signal is transmitted forward with minimal reflection, leading to lower return loss. However, at higher frequencies, such as in the terahertz (THz) range, the transmission parameter S21 (see Figure 17c) shows a significantly lower value of −60 dB between 0.98 and 1.1 THz, primarily due to conductive losses. These losses stem from three key factors: the skin effect, which confines currents to the conductor’s surface, raising effective resistance at high frequencies; surface roughness that scatters EM waves and excites lossy higher-order modes; and higher sensitivity to minor variations in phase/group velocities, disrupting energy propagation. Together, these effects degrade performance in THz slow-wave structures. Additionally, the parameters S31 and S41 exhibit lower values, demonstrating that the structure acts as a perfect reflector with minimal electromagnetic wave leakage from the beam tunnel.

4.3. Beam–Wave Interaction Analysis

The complete structure of the staggered double vane slow-wave structure (SDV-SWS), consisting of 200 periods with a total length of 21 mm, is simulated using CST Particle Studio to analyze the beam–wave interaction. The beam tunnel has dimensions of 150 × 50 $\mathsf{\mu }$m, while the beam dimensions are 130 × 40 $\mathsf{\mu }$m, resulting in a filling factor of approximately 70%. A beam voltage of 25.3 kV and a beam current of 18.2 mA are applied, corresponding to a beam current density of 350 A/cm². To ensure beam stability over the entire length of the structure, an applied magnetic field of 0.7 T is used. The input RF power is set to 3 mW for the beam–wave interaction.

When the RF input power is applied to Port 1 and beam power is directed to Port 3 in the axial direction, electrons are emitted from the electron gun and travel through the cross-sectional area of the slow-wave structure (SWS), exhibiting perfect interaction as shown in Figure 18a. When the phase velocity of the RF signal matches the velocity of the electron beam, electron bunching occurs, which can be observed starting from the 100th period. The amplification of the RF signal becomes evident after the 170th period, and the amplified signal is collected at the output Port 2. Figure 18b also demonstrates how electrons gain and loose energy in the Z-direction. Most of the electrons decelerate, losing energy, while a smaller fraction accelerates, indicating that the amplification of the RF energy is clearly taking place.

The output power measured at Port 2 is 650 mW, achieving an electronic efficiency of 0.14% and a gain of 23.35 dB. The output power saturates after 0.75 ns, and the device operates at a frequency of 1 THz. A competing mode is observed at 2 THz, but it has no significant effect on the output performance as shown in Figure 19a. The maximum output power is obtained at 1 THz with the gain of 23.35 dB. There is a sweep of input power at a frequency of 997 to 1003 GHz, and the obtained 3dB bandwidth is 3 GHz which is exactly shown in Figure 19b. There is a parametric analysis of input RF power conducted from 1 to 10 mW. After 6 mW of input power, there is almost a saturation of output power, electronic efficiency, and gain as shown in Figure 20. The input and output parameters of the staggered double vane (SDV) compared with the deformed quasi sine wave-guide (D-QSWG) are tabulated in

Table 2. Comparison of the staggered double vane SWS with D-QSWG SWS.

Input/Output Parameters	Units	D-QSWG	SDV
Electric conductivity	S/m	$2.0\times {10}^7$	$2.5\times {10}^7$
Current density	A/cm2	400	350
Input current	mA	17	18.2
Beam voltage	kV	23.6	25.3
Magnetic field	T	0.65	0.7
Output Power	mW	320	650
Electronic Efficiency	%	0.08	0.14
RF Signal frequency	THz	1.05	1
Gain	dB	25.05	23.35
Number of periods	-	230	200
Length of SWS	mm	25	21

, and the SDV shows better results compared with D-QSWG.

Due to micro-fabrication challenges, a comparison was made between the simulated and fabricated tolerance values (±2 $\mathsf{\mu }$m and ±5 $\mathsf{\mu }$m as shown in

Table 3. Comparison of simulated and fabricated device tolerances.

Parameters	Simulation		Fabrication			Units
		−2 $\mathsf{\mu }$m	+2 $\mathsf{\mu }$m	−5 $\mathsf{\mu }$m	+5 $\mathsf{\mu }$m
Dispersion	1–1.08	1.01–1.095	0.98–1.065	1.03–1.115	0.96–1.045	THz
Normalized Phase velocity	0.32–0.23	0.32–0.23	0.32–0.23	0.32–0.23	0.33–0.24	-
Interaction impedance	0.4	0.35	0.35	0.35	0.35	$\Omega$
S11 (under −7.3 dB)	40	30	30	45	10	GHz

) for the key SWS parameters. The dispersion shifts downward when accounting for design parameter variations within the range of −5 $\mathsf{\mu }$m to +5 $\mathsf{\mu }$m. The interaction impedance drops consistently (0.4 $\Omega$ to 0.35 $\Omega$) across all fabrication cases. Notably, the S11 bandwidth degrades severely at 5 $\mathsf{\mu }$m (10 GHz vs. 40 GHz simulated), highlighting critical sensitivity to over-etching or inaccurate fabrication. The normalized phase velocity shows minimal variation (±0.01), suggesting robust tolerance to dimensional errors. Additionally, increased fabrication tolerance further impacts the dispersion characteristics. These results underscore the need for tight control (±2 $\mathsf{\mu }$m) to maintain performance, especially for impedance and bandwidth targets.

Hence, the step-wise design and analysis of the staggered double vane slow-wave structure (SDV-SWS) at a frequency band of 0.98 to 1.1 THz is presented in this section, which is helpful for the designers to obtain an overview of the design of any SWS.

Table 4. Comparison of various SWS configurations and performance characteristics.

Reference	SWS	Frequency (THz)	Beam	Input Parameters (RF Power, Beam Voltage, Beam Current)	Output Parameters (RF Power, Gain)	Efficiency (%)	Fabrication	Testing
2020 [68]	CDSGW	0.2	SEB	1 W, 18.8 kV, 100 mA	74 W, 24.75 dB	3.9	No	Simulation
2024 [66]	FGFW	0.2	SEB	0.8 W, 18.6 kV, 80 mA	126.7 W, 22 dB	8.5	CNC	Cold test
2024 [69]	NQ	0.2	SEB	0.2W, 23.7 kV, 250 mA	285 W, 31.5 dB	4.8	CNC	Cold test
2025 [70]	SUGSWG	0.2	PEB	25 mW, 23 kV, 53 mA	49 W, 32.9 dB	4.02	CNC	Hot test
2019 [71]	FWG	0.3	PEB	7.6 mW, 16.2 kV, 24.5 mA	3.17 W, 26.2 dB	0.8	CNC	Hot test
2025 [67]	QFGG	0.3	SEB	10 mW, 28.4 kV, 125 mA	177.3 W, 42.5 dB	1.25	CNC	Cold test
2023 [44]	MSSWG	0.34	PEB	7 mW, 23.8 kV, 30 mA	17.4 W, 33.9 dB	2.43	CNC	Cold test
2016 [72]	HELR	0.4	SEB	0.2 W, 17 kV, 20 mA	19.3 W, 19.5 dB	5.67	No	Simulation
2025 [73]	SWG	0.65	PEB	19.14 kV, 10 mA	647 mW	0.34	CNC, DRIE	Cold test
2022 [16]	DQSWG	1	SEB	1 mW, 23.6 kV, 17 mA	320 mW, 25 dB	0.08	CNC	Cold test
2021 [15]	FRSWG	1.03	SEB	1 mW, 22 kV, 17 mA	487 mW, 26.87 dB	0.13	DRIE, UV-LIGA	Cold test
Present case study	SDV	1	SEB	1 mw, 25.3 kV, 18.2 mA	650 mW, 23.35 dB	0.14	No	Simulation

presents a detailed comparison of various slow-wave structure (SWS) configurations and their performance metrics across studies published between 2016 and 2025. It includes parameters such as operating frequency (ranging from 0.2 THz to 1.03 THz), beam type (sheet electron beam (SEB) or pencil electron beam (PEB)), input power, voltage, and current, as well as output power, gain, and efficiency. Notable results include the 2024 study [66], achieving 126.7 W output power at 0.2 THz with 8.5% efficiency, and the 2025 study [67] showing a high gain of 42.5 dB at 0.3 THz. The case study presents a step-by-step THz slow-wave structure (SWS) design, achieving 650 mW output power, 23.35 dB gain, and 0.14% efficiency at 1 THz, demonstrating optimized high-frequency performance. Fabrication techniques like CNC machining, DRIE, and UV-LIGA are highlighted, alongside testing methods such as simulations, cold tests, and hot tests. Table 4 reveals a trend toward higher output power and gain in recent years, though efficiency varies significantly, with some designs prioritizing power or gain over efficiency. This comparison underscores the ongoing advancements and trade-offs in SWS design for terahertz applications.

5. Conclusions

Vacuum electron devices remain indispensable for high-power, broadband applications, driving continued demand for advanced traveling-wave tubes (TWTs) in the THz regime. This article presents a systematic design methodology for TWT slow-wave structures (SWS), covering parameter initialization, performance analysis, and prototype validation. A comprehensive approach is detailed—from initial parameter selection to final testing—while addressing critical design challenges such as phase velocity mismatch, dispersion optimization, structural components, fill factor, tapering, material selection, and fabrication constraints, along with practical solutions.

As a case study, a staggered double vane (SDV) SWS operating at 980–1080 GHz is designed, achieving 650 mW output power, 23.35 dB gain, and 0.14% electronic efficiency within a compact 21 mm interaction length. Performance comparisons with a deformed quasi-sine waveguide (D-QSWG) SWS validate the efficacy of the proposed design approach, demonstrating its utility for developing high-performance THz SWS structures.