Trivelpiece-Gould Modes Research Articles

RF antenna helicity dependent particle heating in a helicon source

Experiments have demonstrated that ion phenomena, such as the lower hybrid resonance, play an important role in helicon source operation. Damping of the slow branch of the bounded whistler wave at the edge of a helicon source (i.e. the Trivelpiece-Gould mode) has been correlated with the creation of energetic electrons, heating of ions at the plasma edge, and anisotropic ion heating. Here we present ion velocity distribution function measurements, electron density and temperature measurements, and magnetic fluctuation measurements on both sides of an m=|1| helical antenna in a helicon source as a function of the driving frequency, magnetic field strength, and magnetic field orientation relative to the antenna helicity. Significant electron and ion heating (up to two times larger) occurs on the side of the antenna consistent with the launch of the m=+1 mode. The electron and ion heating occurs within one electron skin depth of the plasma edge, where slow wave damping is expected. The source parameters for enhanced particle heating are also consistent with lower hybrid resonance effects, which can only occur for Trivelpiece-Gould wave excitation.

Trivelpiece–Gould modes and low-frequency electron–ion instability of non-neutral plasma

The frequency spectra of the Trivelpiece–Gould modes of a waveguide partially filled with non-neutral plasma are determined numerically by solving the dispersion equation. The modes having azimuthal number $m = 1$ are considered. The results are presented for the entire acceptable range of electron densities, magnetic field strengths, for different values of the charge neutralization coefficient. The Cherenkov resonance condition of an ion with a diocotron mode having a finite value of the longitudinal wave vector was studied. The characteristics of resonant low-frequency electron–ion instability caused by relative azimuth motion of electrons and ions in crossed fields and by the anisotropy of the distribution function of ions are discussed. Ions are created by ionization of residual gas in the plasma volume. Due to the anisotropy, instability occurs not only in the vicinity of the resonance, but also outside it. For typical values of plasma parameters in experiments, estimations of the frequency growth rate are given. A conclusion is drawn that this instability can be the cause of the low-frequency oscillations observed in linear devices with non-neutral plasma produced in an electron beam channel.

A simulation study of electron-scale turbulence generation by nonlinear wave–wave interaction in laser-produced plasmas

A laboratory simulation of a model based on the nonlinear interaction of plasma waves is presented to study the electron-scale magnetic turbulence in the magnetized plasma. In this perspective, the model equations of the Trivelpiece–Gould (TG) mode and extraordinary mode (pump wave) are developed, taking the relativistic change in electron mass and the ponderomotive force into account. Laboratory simulations utilizing the pseudo-spectral method along with the predictor–corrector scheme and finite difference method are performed to solve the formulated coupled model equations. The propagation angle (θ) of the TG mode from the magnetic field affects the dispersive properties of the dynamics (TG mode), which, in turn, impact the density perturbation, scale size of the filamentary structures, density harmonics, magnetic field enhancement, and the spectral index of the turbulence generation. The simulation results reveal that the observed turbulent spectra resemble the magnetic turbulence reported in various studies of the interaction of the intense laser with plasma at the laboratory astrophysics scale relevant to astrophysical events. A simplified model in the paraxial limit is also given to understand the effect of the propagation direction (angle of propagation θ) of the TG mode on the localized structures of the pump laser beam in the magnetized plasma.

Spontaneously excited longitudinal backward waves in dusty plasmas

A spontaneously excited longitudinal backward dust wave was experimentally examined. Dust particles were confined in a vertical glass cylinder. The spontaneously excited wave propagated downward. A dispersion relation was obtained by measuring the wavenumber and phase velocity. The dispersion relation showed that the wave consisted of two different modes, including a backward wave. The Trivelpiece–Gould (TG) mode was invoked as a dispersion relation to explain the backward wave, although the cyclotron angular frequency, ωc, appearing in the TG mode was replaced with ωa ∼ ωpd, where ωpd is the dust angular frequency. The experimental results were compared with the calculated dispersion relation, including the TG mode. The group velocity of the spontaneously excited backward wave is larger than the value predicted with the TG mode. It is necessary to explain the backward wave to modify the TG mode model or to construct a new model.

Squeezed state of an electron cloud as a “quasi-neutral” one-component plasma

We present a one-dimensional model, which gives a novel physical interpretation to the specific state of an ensemble of electrons continuously injected into an electrostatic potential well immersed in a strong applied magnetic field preventing radial expansion. When the space-charge field of the electrons accumulated in the potential well compensates the external electrostatic field, a force-free steady-state of the electron cloud forms. This state of equilibrium is known in another context as a squeezed state of an electron beam. It is shown that the spatial distribution of the electron number density in this steady-state correlates with the shape of the potential well. Perturbations of the steady-state propagate along the electron cloud in the form of Trivelpiece–Gould modes.

Excitation of Trivelpiece–Gould mode in a magnetized plasma having dust by relativistic electron beam?

Excitation of various modes by a beam propagation through a partially or fully ionized plasma has been a subject of great interest. Trivelpiece–Gould (TG) mode are the electrostatic waves having frequency in between the ion plasma frequency and electron cyclotron frequency. In the investigations, scientists have included dust grains in the plasma. We have noticed that in a theoretical investigation, the speed of electron beam propagating through a dusty plasma has been taken relativistic. Since the inclusion of dust grains in plasma and consideration of relativistic speed of electron beam could be a challenging task for theoretical treatment, we have looked into the details of such work and have found that for such situation, the dispersion relation is so complicated that its analysis is not feasible. We have found that earlier work has several discrepancies.

Theory of mode conversion in helicon plasmas

The helicon wave can be converted into the Trivelpiece–Gould (TG) mode and vice versa in nonuniform helicon plasmas. A theory is presented to study this phenomena using a full wave analysis with the stratified model and spectral collocation methods. The properties of the power flux are analyzed, and the reflection and conversion coefficients are obtained by constructing appropriate solutions representing the incident helicon or TG mode propagating toward the mode conversion layer (MCL). It is found that these coefficients are independent of the incident mode type in the collisionless limit, and that conversion of power from the helicon to the TG mode is the same as that from the TG to the helicon mode, regardless of collisions. It is also found that the radiation resistance peaks approximately where the conversion peaks when there is an MCL, or the reflection reaches its local minimum when there is no MCL. Finally, the absorbed power of the waves excited by an antenna is decomposed into those of the incident helicon and TG modes, which shows that their partitions are affected by the existence of mode conversion as well as the collision frequency.

The Effect of Key Parameters on Power Absorption in Helicon Plasma Sources

In recent years, helicon plasma sources have attracted much attention from scientific centers and industry. In this regard, the power coupling is of particular importance for the production of dense plasma in these systems. Therefore, in order to achieve maximum efficiency in these types of plasma sources, it is necessary to analyze the key parameters affecting plasma power absorption such as magnetic field (50–700 G), gas pressure (1–20 mTorr), radiofrequency power (800–2200 W), and RF frequency (6.28, 13.56, 20.34, and 27.12 MHz) applied to Nagoya antenna. Finding the optimal range of these parameters is one of the key parameters in the design and construction of helicon plasma sources. Therefore, in this article, we investigate the parameters affecting the power absorption mechanisms in these types of plasma sources using a 3-D simulation of the helicon plasma source with the finite-element method. Our findings indicate that absorption mechanisms have different effects on the absorption power due to the option of the most optimal plasma parameters’ value. The results show that changes in absorption power in terms of plasma density are nonlinear and show changes in the mode from E to H and eventually to W. Because of the domination of collisionless mechanism, the absorbed power increases in the pressures lower than 2 mTorr. In addition, at pressures above 2 mTorr, it increases due to the collisional mechanism. Plasma density increases linearly as the magnetic field intensity increases. The density peak is observed at the vacuum chamber edge in low magnetic fields (in the range of 65 G) which represents the Trivelpiece-Gould (TG) mode formation.

Modified plasma waves described by a logarithmic electrodynamics

The propagation of plasma waves in a new nonlinear, logarithmic electrodynamics model is performed. A cold, uniform, collisionless fluid plasma model is applied. Electrostatic waves in magnetized plasma are shown to correspond to modified Trivelpiece-Gould modes, together with changes in the plasma and upper-hybrid frequencies, driven by the logarithmic electrodynamics effects. Electromagnetic waves are described by a generalized Appleton-Hartree dispersion relation. The cases of propagation parallel or perpendicular to the equilibrium magnetic field are analyzed in detail. In particular, generalized ordinary and extraordinary modes are obtained. We determine the changes, due to logarithmic electrodynamics, in the allowable and forbidden frequency bands of the new extraordinary mode. Estimates are provided about the strength of the ambient magnetic field so that the nonlinear electrodynamics effects become decisive.

Generation of THz radiation by parametric coupling of laser and Trivelpiece–Gould mode

A scheme to generate terahertz radiation is proposed, wherein the high power, perpendicular-extraordinary laser pump (X-mode) (ω0) parametrically decays into the Trivelpiece-Gould (TG) (ω1) mode and kinetic Alfvén wave (ω2) at an oblique angle with the external magnetic field. The nonlinear interaction amongst the three interacting waves leads to the existence of quasi-static ponderomotive force. Furthermore, under the influence of this force, electrons acquire nonlinear oscillatory velocity. A strong transient current is generated by the nonlinear coupling of laser velocity and TG density perturbation. This nonlinear current drives a wave, whose frequency is obtained in the terahertz range (ω2=ω0−ω1) after optimizing various laser-plasma parameters. For the resonant excitation of the wave, the requisite phase matching condition is shown by modelling the parallelogram, which follows the principle of conservation of energy and momentum simultaneously. The coupling coefficients for the three-wave interaction process along with the growth rate of decay instability are investigated.

Effect of an External Magnetic Field on the Absorption Efficiency of the RF Power in a Spatially Bounded Inductive Plasma Source

The measured dependences of the equivalent plasma resistance on the external magnetic field (0–50 G) in a 46-cm-diameter RF inductive plasma source operating at frequencies of 2, 4, and 13.56 MHz and a power of 100–500 W are presented. The experiments were carried out in argon at pressures of 0.1–30 mTorr. The presence of the external magnetic field leads to the appearance of resonance domains of efficient RF power absorption corresponding to the conditions of resonance excitation of helicons coupled with Trivelpiece–Gould modes. It is shown that RF power absorption at frequencies of 2 MHz can be optimized by applying an external magnetic field corresponding to the domains of resonance absorption. The effect is enhanced with increasing operating frequency.

Study of the operating parameters of a helicon plasma discharge source using PIC-MCC simulation technique

In this work, a Particle in Cell-Monte Carlo Collision simulation technique is used to study the operating parameters of a typical helicon plasma source. These parameters mainly include the gas pressure, externally applied static magnetic field, the length and radius of the helicon antenna, and the frequency and voltage amplitude of the applied RF power on the helicon antenna. It is shown that, while the strong radial gradient of the formed plasma density in the proximity of the plasma surface is substantially proportional to the energy absorption from the existing Trivelpiece-Gould (TG) modes, the observed high electron temperature in the helicon source at lower static magnetic fields is significant evidence for the energy absorption from the helicon modes. Furthermore, it is found that, at higher gas pressures, both the plasma electron density and temperature are reduced. Besides, it is shown that, at higher static magnetic fields, owing to the enhancement of the energy absorption by the plasma charged species, the plasma electron density is linearly increased. Moreover, it is seen that, at the higher spatial dimensions of the antenna, both the plasma electron density and temperature are reduced. Additionally, while, for the applied frequencies of 13.56 MHz and 27.12 MHz on the helicon antenna, the TG modes appear, for the applied frequency of 18.12 MHz on the helicon antenna, the existence of helicon modes is proved. Moreover, by increasing the applied voltage amplitude on the antenna, the generation of mono-energetic electrons is more probable.

Directional power absorption in helicon plasma sources excited by a half-helix antenna

This paper deals with the investigation of the power absorption in helicon plasma excited through a half-helix antenna driven at The simulations were carried out by means of a code, HELIC. They were carried out by taking into account different inhomogeneous radial density profiles and for a wide range of plasma densities, from to The magnetic field was and A three-parameter function was used for generating various density profiles with different volume gradients, edge gradients and density widths. The density profile had a large effect on the efficient Trivelpiece–Gould (TG) and helicon mode excitation and antenna coupling to the plasma. The fraction of power deposition via the TG mode was extremely dependent on the plasma density near the plasma boundary. Interestingly, the obtained efficient parallel helicon wavelength was close to the anticipated value for Gaussian radial density profile. Power deposition was considerably asymmetric when the ratio was more than a specific value for a determined density width. The longitudinal power absorption was symmetric at approximately irrespective of the magnetic field supposed. The asymmetry became more pronounced when the plasma density was The ratio of density width to the magnetic field was an important parameter in the power coupling. At high magnetic fields, the maximum of the power absorption was reached at higher plasma density widths. There was at least one combination of the plasma density, magnetic field and density width for which the RF power deposition at both side of the tube reached its maximum value.

Neutrino-driven electrostatic instabilities in a magnetized plasma

The destabilizing role of neutrino beams on the Trivelpiece-Gould modes is considered, assuming electrostatic perturbations in a magnetized plasma composed by electrons in a neutralizing ionic background, coupled to a neutrino species by means of an effective neutrino force arising from the electroweak interaction. The magnetic field is found to significantly improve the linear instability growth rate, as calculated for supernova type II environments. On the formal level, for wave vectors parallel or perpendicular to the magnetic field, the instability growth rate is found from the unmagnetized case replacing the plasma frequency by the appropriated Trivelpiece-Gould frequency. The growth rate associated with oblique propagation is also obtained.

Nonlinear helicons bearing multi-scale structures

The helicon waves exhibit varying characters depending on plasma parameters, geometry, and wave numbers. Here, we elucidate an intrinsic multi-scale property embodied by the combination of the dispersive effect and nonlinearity. The extended magnetohydrodynamics model (exMHD) is capable of describing a wide range of parameter space. By using the underlying Hamiltonian structure of exMHD, we construct an exact nonlinear solution, which turns out to be a combination of two distinct modes, the helicon and Trivelpiece-Gould (TG) waves. In the regime of relatively low frequency or high density, however, the combination is made of the TG mode and an ion cyclotron wave (slow wave). The energy partition between these modes is determined by the helicities carried by the wave fields.

Effect of the radial plasma nonuniformity on the propagation of guided m=+1 and m=−1 modes in helicon discharges

Theoretical as well as numerical analyses of the full set of Maxwell's equations is carried out to study non-axisymmetric (m≠ 0) guided modes in radially nonuniform helicon (HE) discharges. Unlike the axisymmetric (m = 0) modes, these modes reveal a non-reciprocal behavior with respect to the azimuthal direction. We develop the conditions for propagation and non-propagation of the various modes in the helicon parameter range, thereby focussing on the important role of the radial density gradient. Three types of modes occurring in different parameter ranges are described, i.e., the helicon (HE) mode, the electrostatic (ES) or Trivelpiece-Gould mode, and the locally coupled (LC) mode that is characterized by mode coupling (MC) in a certain region of the plasma density profile. In contrast to m=+1 modes, the parameter range of m=−1 modes is much more restricted as rather high densities are needed for the propagation of the helicon and LC modes. An important issue of the investigations is the rf power coupling and absorption via the various modes. Computations based on a simple antenna-plasma model show that the axial wavenumber of the antenna determines decisively which type of mode is excited. In case of LC mode excitation, the dominant role of the MC layer for the absorption is demonstrated. Finally, the rf power coupling to helicon modes is studied. The density limit for m=−1 helicon mode propagation and the narrow magnetic field profiles of these modes are the main reasons why the rf power absorption in helicon discharges occurs via m=+1 helicon modes.

Trivelpiece-Gould modes in a uniform unbounded plasma

Trivelpiece-Gould (TG) modes originally described electrostatic surface waves on an axially magnetized cylindrical plasma column. Subsequent studies of electromagnetic waves in such plasma columns revealed two modes, a predominantly magnetic helicon mode (H) and the mixed magnetic and electrostatic Trivelpiece-Gould modes (TG). The latter are similar to whistler modes near the oblique cyclotron resonance in unbounded plasmas. The wave propagation in cylindrical geometry is assumed to be paraxial while the modes exhibit radial standing waves. The present work shows that TG modes also arise in a uniform plasma without radial standing waves. It is shown experimentally that oblique cyclotron resonance arises in large mode number helicons. Their azimuthal wave number far exceeds the axial wave number which creates whistlers near the oblique cyclotron resonance. Cyclotron damping absorbs the TG mode and can energize electrons in the center of a plasma column rather than the edge of conventional TG modes. The angular orbital field momentum can produce new perpendicular wave-particle interactions.

Effect of Radial Density Configuration on Wave Field and Energy Flow in Axially Uniform Helicon Plasma**supported by National Natural Science Foundation of China (No. 11405271)

The effect of the radial density configuration in terms of width, edge gradient and volume gradient on the wave field and energy flow in an axially uniform helicon plasma is studied in detail. A three-parameter function is employed to describe the density, covering uniform, parabolic, linear and Gaussian profiles. It finds that the fraction of power deposition near the plasma edge increases with density width and edge gradient, and decays in exponential and “bump-on-tail” profiles, respectively, away from the surface. The existence of a positive second-order derivative in the volume density configuration promotes the power deposition near the plasma core, which to our best knowledge has not been pointed out before. The transverse structures of wave field and current density remain almost the same during the variation of density width and gradient, confirming the robustness of the m=1 mode observed previously. However, the structure of the electric wave field changes significantly from a uniform density configuration, for which the coupling between the Trivelpiece-Gould (TG) mode and the helicon mode is very strong, to non-uniform ones. The energy flow in the cross section of helicon plasma is presented for the first time, and behaves sensitive to the density width and edge gradient but insensitive to the volume gradient. Interestingly, the radial distribution of power deposition resembles the radial profile of the axial component of current density, suggesting the control of the power deposition profile in the experiment by particularly designing the antenna geometry to excite a required axial current distribution.

Helicon waves in uniform plasmas. IV. Bessel beams, Gendrin beams, and helicons

Electromagnetic waves in the low frequency whistler mode regime are investigated experimentally and by digital data superposition. The radiation from a novel circular antenna array is shown to produce highly collimated helicon beams in a uniform unbounded plasma. The differences to Bessel beams in free space are remarked upon. Low divergence beams arise from the parallel group velocity of whistlers with phase velocity either along the guide field or at the Gendrin angle. Waves with angular momentum are produced by phasing the array in the circular direction. The differences in the field topologies for positive and negative modes numbers are shown. It is also shown that in uniform plasmas, the radial amplitude profile of the waves depends on the antenna field topology. Thus, there are no helicon “eigenmodes” with radial Bessel function profiles in uniform plasmas. It is pointed out that phase measurements in helicon devices indicate radial wave propagation which is inconsistent with helicon eigenmode theory based on paraxial wave propagation. Trivelpiece-Gould modes also exist in uniform unbounded plasmas.

On the density limit in the helicon plasma sources

Existence of the density limit in the helicon plasma sources is revisited. The low- and high-frequency regimes of a helicon plasma source operation are distinguished. In the low-frequency regime with ω<ωciωce, the density limit is deduced from the Golant-Stix criterion of the accessibility of the lower hybrid resonance. In the high-frequency case, ω>ωciωce, an appropriate limit is given by the Shamrai-Taranov criterion. Both these criteria are closely related to the phenomenon of the coalescence of the helicon wave with the Trivelpiece-Gould mode. We draw a conclusion that the derived density limits are not currently achieved in existing devices, perhaps, because of high energy cost of gas ionization.