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A design method of the non-periodic meander line slow-wave structure (ML-SWS) and folded-waveguide slow-wave structure (FW-SWS) for traveling-wave tubes (TWTs) is proposed based on effective phase velocity synchronization. For the non-periodic ML-SWS and FW-SWS, the unified effective axial phase velocity seen by electrons, the dispersion equation, and the ideal beam-wave synchronization voltage are derived. The theoretical results demonstrate that the non-periodic ML-SWS and FW-SWS reported to date are consistent with the proposed method. Furthermore, two examples are presented as applications for the proposed method to the non-periodic SWS design, including the concentric arc FW-SWS and the zigzag ML-SWS. In addition, the simulated results of the two examples are in good agreement with the theoretical ones.

The theory of electromagnetic eigenwaves of a cylindrical plasma waveguide located in a constant coaxial magnetic field is developed. The dispersion equation for a waveguide with high electronic conductivity is obtained. It is shown that the waveguide supports the propagation of bulk and bulk-surface hybrid waves of helicon origin. The nonreciprocity effect of waveguide eigenwaves with the identical field distributions, but differing in the direction of propagation along the azimuthal coordinate and also when the direction of the external magnetic field is reversed, is discovered.

In this article, the electromagnetic (or RF) behavior of the azimuthally partitioned axially periodic metal disk-loaded coaxial structure was investigated using an equivalent circuit approach in the presence of the electron beam. In the analysis, the linearized Maxwell’s fluid equation (also known as Vlasov–Maxwell’s equation) was used to analyze the region in which the electron beam was present. The expression for the equivalent series inductance and shunt capacitance per unit length for the equivalent transmission line was derived in the electron beam present and absent case. The derived expressions include all the harmonics effects present inside the structure. Furthermore, the expression of the dispersion relation and temporal growth rate (TGR) was derived using the calculated equivalent series inductance and equivalent shunt capacitance. The output RF power and energy were also calculated analytically and validated through the PIC simulation. The results were compared with the already reported literature, and the relative error between them was below 5%. In addition, the effect of different beam parameters on the TGR behavior was also analyzed.

In this article, by substituting the dielectric tensor in Maxwell’s equations and using the boundary conditions in a longitudinally magnetized plasma-filled coaxial elliptical waveguide, the dispersion relation in the considered configuration is derived. We study the energy and dynamic of an injected external electron with initial energy using radiation of an electromagnetic source in the considered waveguide having the considered hybrid mode. The trajectory and kinetic energy of an injected electron in the considered waveguide along the propagating axis and hybrid field components are plotted. The obtained differential equations are numerically solved by the fourth-order Runge–Kutta method, and the results are graphically presented. The effects of different parameters on the trajectory and kinetic energy of injected electrons are investigated.

The theory of electrostatic (or, more accurately, slow electric or quasi-electrostatic) waves propagation in 1-D magnetized plasma periodic structures, at frequencies below both the plasma and cyclotron frequencies, is presented using the transfer matrix method. It assumes that the applied static magnetic field is perpendicular to the surface of the plasma layers. The dispersion relation of the waves is obtained and it is shown to break up into bands of allowed and forbidden propagation, which are conspicuous in periodic structures. Furthermore, the transmittance behavior of the electrostatic waves as a function of frequency is studied. At a fixed value of the wavenumber, the frequency range of forbidden band gaps is found to become smaller by increasing the plasma width, while the forbidden band gaps become broader as external magnetic field increases.

The metallic grating structure is usually used in the Cherenkov maser. In order to further improve the gain and bandwidth of the sheet beam metallic grating waveguide system, the beam can be with initial transverse and axial velocity instead of pure axial velocity. In this situation, the sheet helical beam will interact with several space harmonics waves simultaneously, and then, the combined cyclotron and Cherenkov resonances can be utilized. Based on the linear theory, the “hot” dispersion equation of the sheet beam metallic grating cyclotron maser is derived and calculated. Through the numerical calculations, the contribution of each resonance, and the influence of beam voltage, the magnetic field and transverse to axial velocity ratio on the gain and bandwidth are analyzed. Besides, the effects of the beam voltage and magnetic field on the growth rate are also simulated, which can further verify the accuracy of the theoretical calculations. This analysis shows the potential of the sheet helical beam with a metallic grating waveguide system, which can be further studied and applied in the future.

In this article, an equivalent circuit approach has been used to study the bifrequency generation by the azimuthally partitioned axially periodic metal disk-loaded coaxial structure. Considering the effect of all harmonics present within the structure and using an equivalent circuit approach, the expression of the equivalent series inductance per unit length and equivalent shunt capacitance per unit length has been obtained for this equivalent transmission line. In addition, with the help of computed equivalent series induction and equivalent shunt capacitance, expressions for the dispersion relation, phase velocity, and characteristic impedance have been obtained. The maximum possible frequency difference (i.e., bandwidth) between the modes (i.e., resonating frequency) and its region of operation have also been analytically computed. Finally, the parametric studies have been performed to find out the effect of structural parameter variation on dispersion behavior. In order to validate the numerical analysis, the structure has been first simulated with the help of CST microwave studio, and then, the analytically obtained results (such as dispersion behavior and phase velocity) have been compared with the results obtained from simulation. The comparison of results shows that the relative error is below than 5%, which shows a good agreement between them.

The parallel propagating waves (Langmuir, R and L waves) are studies in different plasma environments by choosing the different values of $\eta =({mc^{2}}/{k_{B}T})$ and $({\omega ^{2}_{\mathrm{ pe}}}/{\omega ^{2}_{\mathrm{ ce}}})$ . The dispersion relations of these waves are derived under a weak magnetic field limit for a fully relativistic plasma environment. The analytical solution of the integrals in the dispersion relations is not possible, so we use a numerical quadrature approach for their solution. We observe that in both relativistic and weakly relativistic plasmas, these modes shifted to lower values of frequency. The cutoff points of R- and L-waves shifted to further lower values in high-density plasma compared to low density.

The dispersion relation of the reltron modulation section has been derived using the lumped-element equivalent circuit model of a biperiodic side-coupled cavity resonator. Simultaneous equations involving voltage–current have been structured as eigenvalue problems and solved by the matrix method. The solutions, i.e., the eigenvalues and the corresponding eigenvectors, give the frequencies and electromagnetic (EM) field patterns of the three normal modes and, subsequently, the dispersion relation. The analysis shows that the frequency differences among the three modes are not equal, in general, because of the nonlinear dependence. A reltron modulation section has been designed in the S-band and modeled using MAGIC and CST Microwave Studio (MWS). The eigenvalues and the corresponding eigenvectors have been obtained at different coupling depths, and the corresponding coupling coefficients ( $k$ ) are calculated. The effect of the coupling depth on the device efficiency has also been studied through the particle-in-cell (PIC) solver.

This article presents a small-signal theory of a tape helix traveling-wave tube (TWT) with distributed cold tube loss, including the recently discovered space charge effects on the circuit mode that was obtained from the exact dispersion relation of the corresponding lossless tube. The classical values of Pierce’s parameter ${C}$ and ${Q}$ are modified in this new formulation, while the cold tube loss will enter in a similar way as in Pierce. Realistic examples are given, including tubes with uniform circuit loss and with a spatial sever, and compared with the Pierce’s classical theory.

Based on a kinetic model, a linear theory of the 2-D sheet beam metallic grating cyclotron maser has been derived and analyzed numerically. The TE mode dispersion relation and the periodicity of the normalized amplitudes of spatial harmonics of the cyclotron maser with rectangular gratings are obtained. Through numerical calculations, the dependence of the growth rate on the gap distance between the metallic grating surface and the beam, the external magnetic field strength, the beam current, the transverse-to-axial velocity ratio, the axial velocity spread, and the beam voltage are analyzed. This article presents an analysis of interaction using the fundamental TE mode in the 2-D metallic grating sheet beam system. The fundamental TE mode in the 2-D metallic rectangular waveguide grating structure is the fundamental TMx mode in the 3-D rectangular waveguide grating structure and this method can be extended to the practical 3-D interaction system that operates at other TMx modes and has a good potential for the study of the combined resonances in cyclotron masers with grating structures.

Over the past 25 years, the dust acoustic wave has been a subject of intense study and the impact that thermal effects have on the dispersion relation has been a subject of investigation. Recently, a theory of dust correlations and the effect that these correlations have on the propagation of compressional modes has been developed that incorporates the effects of the confinement of the dust cloud in a finite volume within the larger plasma volume. This paper applies this model to a systemic study of the dispersion relation over a wide range of experimental conditions in a radio frequency (RF) discharge plasma. It is found the dust thermal pressure is larger than the dust electrostatic pressure and that both pressures are the square of the dust charge calculated using the orbit-motion-limited (OML) theory. It is also observed that the dust temperature and thermal pressure decreases with increasing neutral gas pressure.

The nonlocal theoretical model developed to observe current-driven low-frequency electrostatic modes in a collisional magnetized strongly coupled dusty plasma, in the strongly coupled kinetic regime $\omega \tau _{m}\gg ~1$ (i.e., wave frequency is much larger than the dust particle relaxation time) led to a new set of equations for dispersion relation, frequency, and the growth rate. The theoretical investigations on a cylindrical magnetized strongly coupled dusty plasma led to a current driven ion dust hybrid like longitudinal dust wave propagating nearly along radial direction and a peculiar transverse shear dust wave mode along axial direction. Dispersion properties of both longitudinal dust acoustic modes and transverse shear modes are modified in the presence of ion and dust currents and the effective perpendicular wave number provided by finite geometry of strongly coupled magnetized dusty plasma. The magnetic field has a destabilizing effect on the growth of longitudinal wave modes, while the dependence of transverse modes on the strength of magnetic field is not prominent. Transverse shear modes are characterized as the short wavelength modes with a long wavelength cutoffs. A comparison of nonlocal effects with local effects shows that in bounded magnetized strongly coupled dusty plasmas, the longitudinal wave modes need a large magnetic field and weak collisional regime to support the excitation of dust wave modes, whereas the transverse modes are a consequence of viscous and collisional damping.

The linear theory of the interaction of electromagnetic waves in a helix slow wave structure filled with thermal plasma driven by an electron beam is investigated. The dispersion relation is found by the application of boundary conditions. This dispersion equation determines the propagation and the gain of the modes in the plasma loaded helix-type traveling-wave tube. The complex transcendental method is used for the solution of the dispersion relation. The role of plasma temperature and the trend of change for the electron beam velocity on the gain were also evaluated. The results show that the growth rate is enhanced with plasma temperature, and the strong gain maximums are for the hybrid mode frequencies. It has also been shown that if the electron beams velocity decrease in the system, the growth rate will increase.

The amplitude modulation of weakly nonlinear electrostatic solitary waves is investigated using standard quantum hydrodynamic model with multiple scale perturbation technique by deriving a nonlinear Schrodinger equation for ultrarelativistic degenerate semiconductor quantum plasma. Due to nonlinear development of amplitude of electrostatic solitary waves, stable and unstable wave packets are generating and it is numerically analyzed that critical wave number separating stable and unstable wave packets is significantly dependent upon ultrarelativistic parameter, quantum Bohm potential, degenerate pressure, hole-electron mass ratio, and hole-electron temperature ratio. It is seen that critical wave number is varying in a different semiconductor material. Further instability growth characteristic is also examined for different semiconductors. In linear analysis, a dispersion relation has been evaluated analytically and the nature of two wave modes (slow frequency and fast frequency) is studied by varying different plasma parameters.

An analytical model is developed with the aim to investigate the propagation of dust-acoustic (DA) waves in a multicomponent thermal dusty plasma system including ion species as well. Dust particles are considered positively charged by electron and ion currents, thermionic and UV photo-ionic processes. The model is applied to laboratory plasma as well as Martian atmospheric plasma to explain the nature of dust potential variation as well as characteristics of DA propagation modes within the linear and nonlinear regimes for given plasma parameters. In the nonlinear regime, the KdV equation is derived, and the equation is seen to support the propagation of compressive solitons in laboratory plasma system and rarefactive solitons in the case of Martian atmospheric plasma. The inclusion of ion species in the system is seen to have an important role in determining the propagation speed, the amplitude, and the corresponding width of solitons. The thermionic work function W, photon yield factor Y, and low value of flux density of UV photon beam J do not have an appreciable impact on dispersion relation and DA soliton structures. On the contrary, noticeable changes occur in the case of the dust surface potential obtained in both laboratory and Martian plasmas.

Electron cyclotron resonance (ECR) plasma sources have been widely used for ultra-large-scale integration manufacturing processes because of their wide operating range and excellent uniformity over 300- or 450-mm diameter wafer. However, microwave electromagnetic fields in magnetized plasma-filled reactors are not well understood because of the complexity of electromagnetic wave field analysis. We conducted microwave electromagnetic field analysis of an ECR plasma etcher by the finite-element method using a simulation platform. Complicated electric field patterns were clearly observed under some conditions. We investigated the patterns in detail to determine whether they were spurious or not. The patterns proved to be Trivelpiece–Gould waves according to their characteristics imposed by the dispersion relation of waves. This analysis approach is a powerful method because very fine-wave structures can be identified.

As it is well known that the simplified classical Pierce’s three-wave theory can provide scaling insights into traveling-wave tube (TWT) operation for its close-form solutions, and thus becomes a useful guide for TWT design. However, the classical Pierce’s three-wave theory shows poor agreement with Lagrangian theory, so determining how to achieve a more accurately Pierce’s three-wave theory from Pierce’s four-wave theory becomes an open problem. In this paper, a more accurately revised Pierce’s three-wave dispersion relation is deduced by using an approximate treatment on the nonlinear term of the dispersion relation of Pierce’s four-wave beam-wave interaction theory. Meanwhile, the boundary conditions of classical Pierce’s three-wave theory are revised by theoretical analysis. Thus, the revised Pierce’s three-wave theory is established. The Pierce small-signal theories are compared to each other and Lagrangian theory on a set of TWT parameters which are based on a single pitch section of C- and Ku- bands TWTs. It is found that the revised Pierce’s three-wave theory agrees extremely well with Lagrangian theory and gains more accuracy than the classical Pierce’s three-wave theory in the small-signal beam-wave interaction region. Finally, the phase difference between Pierce’s four-wave theory and revised Pierce’s three-wave theory is discussed.

Dispersion is one of the important characteristics of dusty plasma. In this paper, the real and imaginary parts of the complex refractive index of dusty plasma are first deduced and verified by the Kramers–Kronig dispersion relation. And then, the dependence of dispersion ability on the temperature, pressure, angular frequency of incident electromagnetic wave, as well as the number density, radius, and charge of dust particles is given. Finally, the theoretical results indicate that: 1) the dispersion ability decreases with the increase of angular frequency; 2) the dispersion ability monotonically varies with the pressure approximately, but nonmonotonically varies with the temperature; and 3) the dispersion ability monotonically increases with the increase of dust particle’s charge, but nonmonotonically changes with the number density and radius of dust particles. In particular, the involved physical explanations are proposed.

The effects of polarized Debye sheath (i.e., polarization force) and trapped ions on the nonlinear propagation of dust-acoustic (DA) waves are investigated, including the dust inhomogeneity in the strongly coupled dusty plasma. Dispersion relation and propagation of possible modes are calculated from the underlying usual hydrodynamic model. Then, a variable coefficient’s modified Korteweg-de Vries equation describing the dynamics of DA solitary waves is derived from a weakly nonlinear analysis, and is solved to get the stationary solutions using a sine–cosine method. It is found that for experimental conditions under which the interaction of resonant ions and highly negatively charged dust grains is important, only soliton with negative potential can propagate in such a strongly coupled inhomogeneous dusty plasma. In the presence of trapped ions, the amplitude of solitary waves is found to be smaller compared with that for the Boltzmann distribution. Moreover, it is argued that the soliton profile is highly sensitive to the effects of modified strength of the polarization force, dust temperature, and dust density inhomogeneity.

In this paper, the dispersion relation of the beam wave interaction of the confocal waveguide gyrotron-traveling-wave tube (confocal gyro-TWT) is derived from kinetic theory for the first time; according to this dispersion relation, we can make the linear analysis of confocal gyro-TWT as well as the cylindrical waveguide gyro-TWT. At the same time, a numerical code based on the self-consistent nonlinear theory is written to simulate the behavior of electron beam and electromagnetic wave during beam wave interaction. When neglecting the spread of velocity and guiding center, the gain and growth rate calculated by the nonlinear theory code makes great agreement with that of linear theory in the small-signal region.

In this paper, a novel waveguide made of a hollow long metallic cylindrical column with a transverse anisotropy dielectric rod (TADR) is studied. The dispersion relation of hybrid electromagnetic waves is obtained in this structure, and its diagram variations will be investigated. The effects of dielectric permittivity coefficients of TADR on the dispersion relation of slow waves is studied. The mechanism of wave generation and its amplification phenomena by a thin annular relativistic electron beam are investigated for hybrid modes. Also, the graphs of operating frequency and the graphs of growth rate will be presented for the waves during the electron beam injection. The computations have done for terahertz hybrid waves.

The ladder multigap resonant cavity finds wide application in the terahertz vacuum electric devices. It consists of a single slotted plate (called ladder) enclosed by a rectangular waveguide, which offers advantages in mechanical simplicity as well as the capability of high gain per circuit length and frequency tuning. The theoretical model for the high-frequency characteristics of such circuit was developed. The derivation of the characteristic impedance ( $R/Q$ ) and the intrinsic quality factor ( $Q_{0}$ ) in combination of the established dispersion relation enables a complete analysis for this circuit. Examples at the terahertz frequencies were used to demonstrate the usefulness of the theoretical model with the 3-D finite-element electromagnetic simulation as a benchmark. The formula system is described and the computational results are presented.

The propagation of ionic perturbations in a dusty plasma is considered through a three-species kinetic simulation approach, in which the temporal evolution of all three elements, i.e., electrons, ions, and dust particles are followed based on the Vlasov equation coupled with the Poisson equation. Two cases are focused upon: first, a fully electron-depleted dusty plasma, i.e., a plasma consisting of ions and dust particles. The second case includes dusty plasmas with large electron-to-ion temperature ratios. The main features of the ionic waves in these two settings, including the dispersion relation and the Landau damping rate, are studied. It is shown that the dispersion relation of the ionic waves perfectly matches the dispersion relation of Langmuir waves and hence is called Langmuir-like ionic waves and can be considered as Langmuir-like ionic waves. These waves can be theoretically predicted by the dispersion relation of the dust-ion-acoustic waves. The transition of ionic waves from dust-ion-acoustic to Langmuir-like waves is shown to be sharp/smooth in first/second case. The Landau damping rates based on simulation results are presented and compared with theoretical predictions wherever possible.

Spoof surface plasmons (SSP) has become an active research topic in microwave and terahertz (THz) spectrum since its extraordinary optical and physical properties. The strong near field of SSP mode on the corrugated metal surfaces makes it especially attractive for developing a THz electronic source. A THz electronic source based on the efficient generation of SSP modes on the doubly corrugated metallic waveguide is proposed and studied in this paper. The analytical dispersion relations of SSP modes are obtained based on a modal expansion method and the field profiles of SSP modes are also presented by the finite integration method. Besides, the interaction between SSP and injected electron beam is modeled and implemented by particle-in-cell (PIC) simulation based on finite difference time domain algorithm. The gap size between the doubly corrugated metal surfaces can significantly influence the output power and PIC simulation results reveal that output power can be increased from 272 mW to 36.5 W when the gap size decreases from 90 to $40~\mu \text{m}$ at the frequency near 1 THz by the 19.55 kV, 1 A injected electron beam within 4.5-mm interaction length. The dependencies of the output performance on electron beam parameters are also investigated and we find that there is an optimized beam voltage for the given operation frequency. Various electron beams of pulse and direct current electron beam are studied and we find that half pulsewidth of periodical electron beam is more preferable than other emissive shape of injected electron beam for the given structure. Our studies on the efficient generation of SSP modes on the doubly corrugated metallic waveguide may provide a new way to develop THz electronic sources.

The dispersion relation of terahertz hybrid modes in a plasma waveguide made of dielectric tube surrounded by long cylindrical metallic wall with elliptic cross section is obtained. The plasma region has been placed on outer surface of dielectric tube. The waveguide mentioned earlier is considered instable by the injection of two electron beams with opposite velocities as source energy. The dispersion relation graph and its characteristics, such as its dependence to geometrical dimensions and characteristics of the electron beam, are presented. In addition, effects of relative permittivity constant of dielectric tube, geometrical dimensions, accelerating voltage and current density of electron beams on growth rate, and frequency spectra of waveguide will be investigated.

The quantum hydrodynamic model is considered to investigate the Jeans instability of three species ultracold inhomogeneous dusty plasma. The formulation of this paper is based on the assumption that the electron and ion inertial force is negligible in comparison with the electron and ion pressure and force associated with the electron and ion correlations at the quantum scale. The gravitational force acting on the negatively charged dust grains is considered in the presence of both the electrostatic and polarization forces taking into account the quantum diffraction and statistical contributions of dust particles. The modified dispersion relation is derived and analyzed to obtain the instability criterion by including the dust polarization force and the quantum corrections. Numerical results are presented to show the effect of these modifications on the growth rate of Jeans instability. The present investigation is relevant to the collapse of ultracold astrophysical objects like the inner core of white dwarf and neutron star.

The linear theory of a plasma-loaded gyrotron amplifier is studied in the fast- and mixed-wave modes. The dispersion relation is derived using the Vlasov-Maxwell equations. The effects of beam location, plasma column radius, electron beam parameters, and azimuthal harmonic number on the growth rate for the fast- and mixed-wave modes are investigated. In addition, the radial behavior of the electric field is studied. The results show that the growth rate and bandwidth of the mixed-wave mode are larger than those of the fast-wave mode. It is shown that the bandwidth of this structure is largest for a small value of the axial momentum spread. Moreover, it is indicated that for a lower harmonic number, the axial electric field has its maximum value at the center of the plasma.

In this paper, a sector-step approximation (SSA) model is introduced. With step admittance recurrence and field matching relation, the dispersion equation of rising-sun magnetron (RSM) with arbitrary resonators is derived. By this model, resonant frequencies of a 10-resonator sector and hole-and-slot resonator type RSM (SHSRSM) are theoretically calculated. In addition, computer simulation is performed to verify the accuracy of the theory. Besides, the effects of the hole radius on magnetron’s main characteristic parameters are analyzed, such as $\pi$ -mode resonant frequency, mode separation, and quality factor. The optimized SHSRSM is compared with two common kinds of RSMs, such as sector resonator type RSM (SRSM) and sector-and-slot resonator type RSM (SSRSM). The results show that the theoretical results are well consistent with the simulation results, and the maximum relative error is less than 3%, which verifies the accuracy and feasibility of the SSA model. Compared with SRSM and SSRSM, SHSRSM has greater mode separation and quality factor, which improves the stability and the energy storage of RSM.

The theory of the propagation of the magnetized hollow relativistic electron beam with a dielectric rod at its axis through the sheath helix that inserted into the magnetron-type conductor is presented. The dispersion relation is obtained with the aid of the proper boundary conditions. The effects of an electron beam, inner dielectric, helix, and magnetron-type conductor parameters on the growth rate are studied numerically. The numerical results show that the presence of the cavity considerably increases the growth rate. In the fast wave mode, we have the maximum value of the growth rate when the dielectric rod and the electron beam are close to each other. The maximum growth rates for the fast and slow wave modes occur to the special value of the helix radius. The results show that the growth rate increases with the decreasing of the axial momentum spread. Comparison of the two modes shows that the growth rate of the slow wave mode is higher than that of the fast wave mode, but the bandwidth of the fast wave mode is wider than that of the slow wave mode.

A backward-wave amplifier topology based on a coaxial negative refractive index transmission-line (NRI-TL) metamaterial is introduced. The dispersion relation of the isolated coaxial metamaterial is derived rigorously using a mode-matching technique. The structure is then homogenized to derive its effective material parameters. The electron beam is subsequently introduced by means of an anisotropic, spatially dispersive permittivity. With the electron beam present, the homogenized metamaterial is analyzed to obtain a coupled-mode dispersion relation that accounts for electron beam–metamaterial interaction. Solutions to this dispersion relation yield complex coupled modes. The analytical solutions are compared with full-wave simulation of the NRI-TL with electron beam present, and show close agreement.

Spokes are azimuthally propagating perturbations in the plasma discharge of Hall effect thrusters (HETs) that travel in the $E \times B$ direction. The mechanisms for spoke formation are unknown, but their presence has been associated with improved thruster performance in some thrusters motivating a detailed investigation. The propagation of azimuthal spokes are investigated in a 6 kW HET by using high-speed imaging and azimuthally spaced probes. The spoke velocity is determined from high-speed image analysis using three methods with similar results. The spoke velocity for three discharge voltages (300, 400, and 450 V) and three anode mass flow rates (14.7, 19.5, and 25.2 mg/s) are between 1500 and 2200 m/s across a range of magnetic field settings. The spoke velocity is inversely dependent on magnetic field strength for lower B-fields and asymptotes at higher B-fields. Spoke velocities calculated from the probes are consistently higher by 30% or more. An empirically approximated dispersion relation of $\omega ^\alpha = {v}_{\rm ch}^\alpha k_{\theta }^\alpha - \omega _{\rm ch}^\alpha $ where $\alpha \ge 1$ yields a characteristic velocity that matches the ion acoustic speed for $\sim 5$ eV electrons which exist in the near-anode and near-field plume regions of the discharge.

A linear theory of a nonstaggered bilateral metal-grating periodic slow-wave structure with a sheet electron beam is developed using the Borgnis potential function and field-matching method. The field-matching method has been used at the upper and bottom waveguide boundaries where the metal-grating structure is used to slow down the electromagnetic phase velocity. The derived dispersion equation has been solved through the iterative technique. Numerical calculations show the effects of some parameters on the linear growth rate. Moreover, the influences of the structural asymmetry on the linear growth rate have been analyzed as well.

In this paper, the electromagnetic properties of a beam free trapezoidally corrugated slow wave structure (TCSWS) are numerically and experimentally investigated. The dimensions of the TCSWS are selected so that it can be deployed in $X$ -band backward wave oscillators (BWOs). The theoretical dispersion characteristics for the axisymmetric ${\rm TM}_{01}\hbox{--}{\rm TM}_{03}$ modes are obtained by solving the dispersion equation numerically. In order to verify the numerical calculation, a cold test is conducted using a 6-period TCSWS cavity and a vector network analyzer (VNA). The cavity is excited by a small needle-type antenna and frequency of the VNA is swept over the passbands. The dispersion characteristics are obtained from frequency values of the discrete resonant peaks of reflection measurement of each TM mode. Experimental results show an excellent agreement with numerical ones. The numerical and cold test results reveal that TCSWS can be used as an alternative to sinusoidally corrugated slow wave structure in BWO experiments with an advantage of ease of fabrication.