The co-excitation of electrostatic ion-cyclotron wave and ion-Bernstein-like wave launched from a grid

1.   Introduction

Bernstein waves are electrostatic waves in magnetized plasmas [1]. On the ion time scale, there are two kinds of electrostatic waves propagating vertically to the magnetic field. They are electrostatic ion-cyclotron waves (EICWs) [2], that propagate along the direction nearly perpendicular to the magnetic field, and pure ion-Bernstein waves (IBWs) [3], that propagate along the direction strictly perpendicular to the magnetic field. For the EICWs, there exists a small but finite parallel wave number $k_{\parallel}$ satisfying the condition $k_{\parallel} v_{\rm{ti}} \ll \omega \ll k_{\parallel} v_{\rm{te}}$, where $\omega$ is the angular frequency of the wave, $v_{{\rm{t}}\alpha} = (2T_{\alpha}/m_{\alpha})^{1/2}$ ($\alpha = { {\rm{e}}, {\rm{i}}}$) is the thermal velocity, and $m_{\alpha}$ and $T_{\alpha}$ are the mass and temperature, respectively, of the corresponding species. In this case, the electrons can move freely along the field lines to neutralize the charge-separation field. Thus, the EICWs were also termed the neutralized ion-Bernstein waves [3]. For the IBWs, on the other hand, the much more stringent condition, $\omega \gg k_{\parallel} v_{\rm{te}}$, must be satisfied. Therefore, it is usually difficult to excite the pure IBWs in experiments.

The EICWs and IBWs were extensively studied in the past, both theoretically and experimentally [4–9]. These waves have important applications in magnetic confinement and space plasmas. In magnetic fusion devices, radio-frequency (RF) waves in the ion cyclotron range of frequencies were often used to heat the plasmas [10–15]. The RF waves first converted into the EICWs or IBWs in the outer layer of the tokamak plasma. Then these waves propagated into the core region and were absorbed by the plasma via cyclotron damping or Landau damping processes. Furthermore, it was shown that the IBWs could be used to control the profiles and transport of tokamak plasmas [13]. In space plasmas, the EICWs were often invoked to explain satellite-observed phenomena such as the ion acceleration in auroral regions [16]. Besides, the parametric excitation of the IBWs was proposed in ionospheric modification experiments [17]. From the point of view of these applications, it is essential to investigate the excitation and propagation processes of the EICWs and IBWs.

For the EICWs, the majority of previous experimental investigations were conducted in current-carrying Q-machines [6, 18, 19], in which the waves were excited via an internal instability driven by the magnetic-field-aligned current (see, for example, reference [20] for a review). Through this, only the waves with maximum growth rates were detected, and the observed wave frequencies were usually near the ion cyclotron frequency $\omega_{\rm{ci}}$ or near the harmonics of $\omega_{\rm{ci}}$ [19], i.e., the frequencies of excited waves were uncontrollable. Only a few experiments were devoted to the external excitation of the EICWs, in which the waves were launched from antennas [21–26] or an excitation grid [7]. In these experiments, the excited-wave frequency was the same as the external driving frequency. The wave number $k$ was measured as a function of the frequency, thus enabling the direct measurement of the dispersion relations. However, the results indicate that the measured dispersion relation is only partially consistent with the kinetic theory of the EICW. Therefore, there is a need to further investigate the external driving of the EICWs and their propagation properties.

For the pure IBW, the first experimental demonstration was conducted by Schmitt [3] in a Q-machine, in which the wave was externally launched from a long, thin wire aligned parallel to the magnetic field. Another experiment on the external excitation of the IBW was conducted in a toroidal device by Ono and Wong [27], in which the wave was excited using two electrostatic antennas placed at the outer edge of the plasma. In these experiments, the spatial waveforms were detected and the wave number was directly measured for a given driving frequency, yielding the dispersion relations that were in good agreement with the results of the kinetic theory. Besides these early experiments on the IBW, almost all the later experimental studies were connected to the heating of fusion plasmas, where the IBWs were routinely launched from RF antennas. It was shown that the antenna-launched RF waves first excite fast magnetosonic waves in the plasma, and then the fast waves convert into the IBWs in the ion cyclotron resonance layer [28]. The mode-converted IBWs were inferred from far infrared laser scattering and microwave scattering signals [29–31] or from magnetic and RF probe data [29, 32]. In these experiments, the waveforms were not directly measured, and the experimental data on the dispersion relations were very few. In some other experiments conducted in steady-state toroidal plasmas [33, 34], the spatial wave patterns and the dispersion relations as well as the wave attenuation lengths were studied in nonuniform plasmas. However, the measured dispersion relations were only in a limited frequency range. In addition, the IBWs were also excited in an ion beam-created plasma column through an internal instability arising from the beam-plasma interaction [35, 36], but it was unable to study the dispersion relations because of the uncontrollable frequency of the instability-driven waves.

From the point of view of basic plasma physics, the wave propagation characteristics are crucial to the identification and coupling of different kinds of excited waves. In the previous experimental investigations on the EICWs and IBWs, the wave property evolutions have not been well studied. Furthermore, the EICW and pure IBW are only the ion mode in two limiting cases, i.e., $\omega / k_{\parallel} \ll v_{\rm{te}}$ and $\omega / k_{\parallel} \gg v_{\rm{te}}$. In practice, for the externally excited waves, there exists a small value of $k_{\parallel}$ because of the finite size of the antennas. The condition $\omega / k_{\parallel} \gg v_{\rm{te}}$ for the pure IBW is difficult to satisfy. Conversely, the excited wave may satisfy the condition $\omega / k_{\parallel} \sim v_{\rm{te}}$. The waves in this regime, as we call the IBW-like waves, are important to the wave-plasma energy coupling process because of the strong electron Landau damping. Besides, the EICWs and IBW-like waves may be excited at the same time in different frequency bands. The co-existence of the EICWs and IBW-like waves further complicates the propagation properties. These issues have not been investigated.

This work is devoted to the experimental study of the co-excitation of the EICWs and IBW-like waves. The waves are launched from an excitation grid. It will be shown that these waves can be simultaneously excited and a transition of the wave properties from the EICW to the IBW-like wave occurs in the frequency domain from low to high bands of the ion cyclotron frequency. The paper is organized as follows. Section 2 presents the experimental arrangements, the plasma parameters, and the details of the wave excitation and detection. Section 3 presents the wave propagation properties and the detailed data processing. Section 4 gives the results of the measured dispersion relations of the excited waves with a comparison to theoretical models and the identification of the modes. The main conclusions and discussions are summarized in section 5.

2.   Experimental arrangements

2.1   Experimental setup

The experiment was carried out in a magnetized plasma device, as schematically shown in figure 1. The main chamber is 89 cm in diameter and 40 cm in length, connected with two side chambers of 23.8 cm in diameter and 36 cm in length. The two side chambers are covered with two coils, which can provide an axial magnetic field up to 1200 G at the center of the main chamber. The plasma was generated in one of the side chambers by the hot-cathode discharge between a set of hot filaments (HF) and an anode mesh (AM) and diffused into the main chamber. The background pressure was $2.0\times10^{-4}$ Pa and the working gas pressures were $2.5\times10^{-2}$ Pa for argon (Ar) and $6.8\times10^{-2}$ Pa for helium (He). The discharge voltages and currents were 30 V and 0.53 A for Ar and 45 V and 0.66 A for He, respectively. The waves were launched from a stainless steel grid (32 cm long and 21 cm wide) and detected with two ring-shaped cylindrical probes (one naked and the other insulated, separated ~ 1 cm). The grid was placed in a vertical plane parallel to the magnetic field at the center of the device.

Figure  1.  The experimental device (top view). HF: hot filaments, AM: anode mesh, CP: ring-shaped cylindrical probe, EG: wave excitation grid. The origin of the coordinates $(z,r)$ is at the center of the grid.

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2.2   Plasma parameter distributions

The plasma parameters were diagnosed with one longitudinally movable probe and one transversely movable probe. The probes (made of tungsten wire of 0.02 cm in diameter) were bent in a ring shape of 1 cm in diameter. It was measured that the plasma parameters were quite uniform in the direction parallel to the magnetic field. However, the parameter distributions perpendicular to the magnetic field were nonuniform. Figures 2 and 3 show the electron density $n_{\rm{e}}$, temperature $T_{\rm{e}}$, and plasma potential $V_{\rm{p}}$ distributions with the distance from the excitation grid. As can be seen from the figures, the plasma parameters are uniform in the absence of the magnetic field. However, with the application of the magnetic field, the parameters become nonuniform. The electron density first decreases away from the grid and then shows a plateau region between 4 and 8 cm. The electron temperature exhibits apparent variations. The plasma potential increases slightly with the distance, but the increase is slower in the Ar than in the He case. The relative uniform region between 4 and 8 cm was chosen to perform the wave propagation experiments.

Figure  2.  The transverse distributions of the electron density $n_{\rm{e}}$ (a), temperature $T_{\rm{e}}$ (b), and plasma potential $V_{\rm{p}}$ (c) with the distance from the excitation grid under different magnetic strengths indicated in (a) in the Ar discharge.

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Figure  3.  The transverse distributions of the electron density $n_{\rm{e}}$ (a), temperature $T_{\rm{e}}$ (b), and plasma potential $V_{\rm{p}}$ (c) with the distance from the excitation grid under different magnetic strengths indicated in (a) in the He discharge.

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It can also be seen that the plasma density is lower in the He than in the Ar case (note the different vertical scales in figures 2 and 3 for $n_{\rm{e}}$). One reason for the lower discharge density of He is that its ionization energy (24.6 eV) is higher than that of Ar (15.7 eV), and thus it is more difficult to ionize He. Another reason is that the mass of He is ten times smaller than Ar and therefore, He has a larger transport coefficient than Ar, i.e., He’s loss on the wall is faster than Ar’s.

2.3   Excitation and detection of the waves

The waves were launched by applying sinusoidal signals to the excitation grid. In order to fix the grid strictly parallel to the magnetic field, it was rotated until its floating voltage reached a maximum value, at which the electron flux to the grid in the vertical direction is frozen by the magnetic field so that the ion flux charges the grid to the maximum positive potential with respect to the plasma. By this, the grid surface was regarded to be perfectly parallel to the magnetic field.

Figure 4 is a schematic diagram of the circuit used to launch and detect the waves. The voltage signals from a signal generator were first amplified and then applied through a capacitor to the grid. The peak-to-peak amplitude of the applied voltage signals was $10$ V. The use of the capacitor is to keep the grid at DC floating voltage (unbiased).

Figure  4.  The circuit of the wave excitation and detection. The input to the channel 1 (CH1) of the oscilloscope is the driving signal. The inputs to the channels 2 and 3 (CH2 and CH3) are the signals from the naked and insulated probes. The output of the oscilloscope is the difference signal between the channels 2 and 3. Because the two probes cannot be made perfectly the same, the channel 3 signal is multiplied by a coefficient of 0.97, which gives the best result of the difference (see figure 5).

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The excited wave signals were detected by the probes biased at a positive voltage. To eliminate the electromagnetic induction signal that unavoidably couples to the real wave signal, we used the method of difference between the signals from the naked and the insulated probes. The naked probe receives both signals, but the insulated one receives only the electromagnetic induction signal. Figure 5 is a plot of the signals from the naked (CH2) and the insulated (CH3) probes and the output of their difference (${ {\rm{CH2}}} - 0.97 \times {\rm{CH3}}$) in the absence of the plasma. Because the two probes cannot be made perfectly the same, the channel 3 signal is multiplied by a coefficient of 0.97, which gives the best result of the difference. The bottom panel in figure 5 shows that the difference signal is close to the background noise. Thus, the elimination of the electromagnetic induction signal is efficient.

Figure  5.  The electromagnetic induction signals from the naked probe (CH2, top), the insulated probe (CH3, middle), and their difference $\rm{CH2} - 0.97\times \rm{CH3}$ (bottom) in the absence of the plasma. The difference signal is close to the background noise.

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3.   Propagation properties and data processing

In the presence of the plasma, a typical output of the wave signal in Ar plasma when $B = 300$ G is shown in figure 6(a), which shows that the pattern of the excited wave signal deviates from the sinusoidal shape. Figures 6(b) and (c) are the fast-Fourier-transform (FFT) spectra of the amplitude and phase of the wave signal shown in figure 6(a), indicating that the received wave signal contains harmonic components in addition to the fundamental component at the frequency, $f_{\rm{drive}}$, of the driving voltage. The harmonic waves are the result of the nonlinear evolution of the sheath near the excitation grid. The application of the driving signal to the grid causes the oscillation of the sheath, which then propagates out into the plasma to form the wave. Since the sheath is a highly nonlinear region, its oscillation is a nonlinear process. Therefore, the wave contains harmonic components.

Figure  6.  The typical detected wave signal (a) and its fast-Fourier-transform spectra for amplitude (b) and phase (c) in Ar plasma with $B = 300$ G.

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Moving the probes to different positions from the grid and using the FFT spectra at each position, the variations of the wave amplitude and phase with the propagation distance $r$ are shown in figures 7 and 8 for $f_{\rm{drive}}$ and $2f_{\rm{drive}}$ signals (the two largest frequency components). The amplitude decreases rapidly with $r$ both in the Ar and He discharges. It can be seen that the amplitude of the excited waves is substantially larger in the Ar case than in the He case. However, the amplitude variation shows a region of slow increase from $r = 5$ cm to $r = 7$ cm for the second harmonic signal in the Ar case and from $r = 5$ cm to $r = 6$ cm for the fundamental signal in the He case. The complex variation of the amplitude may be related to the nonuniformity of the plasma parameters, which is out of the scope of the present study and needs future investigation.

Figure  7.  The variations of the amplitude (a) and phase (b) of the FFT spectral components at $f_{\rm{drive}}$ and $2f_{\rm{drive}}$ with respect to the distance from the grid for Ar discharge at $B = 300$ G and $f_{\rm{drive}} =$ 25 kHz. $k_1$ and $k_2$ are the linearly fitted values of $k_{\perp}$ for the $f_{\rm{drive}}$ and $2f_{\rm{drive}}$ signals.

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Figure  8.  The variations of the amplitude (a) and phase (b) of the FFT spectral components at $f_{\rm{drive}}$ and $2f_{\rm{drive}}$ with respect to the distance from the grid for He discharge at $B = 300$ G and $f_{\rm{drive}} = 240$ kHz. $k_1$ and $k_2$ are the linearly fitted values of $k_{\perp}$ for the $f_{\rm{drive}}$ and $2f_{\rm{drive}}$ signals.

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In contrast to the complex variation of the amplitude, the phase versus $r$ plot exhibits a quite linear variation both in the Ar and He cases within the range of measurement as shown in figures 7(b) and 8(b). Supposing, to the lowest order of approximation, the phase of the wave being given by $\psi = \omega t - k_{\perp} r$, one can obtain the perpendicular wave number $k_{\perp} = - \partial \psi / \partial r$. Using the linear fitting of the experimental data of the phase, we can obtain $k_{\perp}$ with uncertainties for $f_{\rm{drive}}$ and $2f_{\rm{drive}}$ signals as indicated in the figures. It is seen that the measured $k_{\perp}$ for $2f_{\rm{drive}}$ signal is larger than that for $f_{\rm{drive}}$ signal.

To determine the parallel wave number, $k_{||}$, we use the axially movable probe and the insulated probe to obtain the wave signal propagating along the direction parallel to the magnetic field. Using the same method as to obtain $k_{\perp}$, $k_{||}$ can be measured for each given driving frequency.

4.   Dispersion relations

4.1   Theoretical background

In the fluid description, the dispersion relation for the EICW when $k \simeq k_{\perp} \gg k_{\parallel}$ is [37]

where $\omega_{\rm{ci}} = e B / m_{\rm{i}}$, $c_{\rm{s}} = (T_{\rm{e}}/m_{\rm{i}})^{1/2}$ is the ion acoustic speed, $e$ is the electronic charge, and $B$ is the strength of the magnetic induction.

In the kinetic description, the dispersion relation of the electrostatic waves is governed by

where $\chi_{\alpha}$ is the susceptibility of the corresponding species [3]

where $\lambda_{{\rm{D}}\alpha} = (\epsilon_0 T_{\alpha} / q_{\alpha}^2 n_{\alpha 0})^{1/2}$ is the Debye length, $\epsilon_0$ is the vacuum dielectric constant, $q_{\alpha}$ and $n_{\alpha 0}$ are the charge and unperturbed density of the species $\alpha$, $\zeta_{\alpha {n}} = (\omega - {n}\omega_{{\rm{c}}\alpha}) / k_{\parallel} v_{{\rm{t}}\alpha}$ (${n} = 0, \pm1, \pm 2, ...$), ${Z}(\zeta_{\alpha {}}) = {\text{π}} ^{-1/2}\int {\rm{d}}x \exp({-x^2})/ (x - \zeta_{\alpha {n}})$ is the plasma dispersion function with the argument $\zeta_{\alpha {n}}$, ${A}_{n}(b_{\alpha}) = {I}_{n}(b_{\alpha}) \exp(-b_{\alpha})$, ${I}_{n}(b_{\alpha})$ is the $n$-th order and Bessel function of imaginary argument, $b_{\alpha} = k_{\perp}^2\rho_{\alpha}^2$, $\rho_{\alpha} = v_{{\rm{t}} \alpha}/ \sqrt{2} \omega_{{\rm{c}}{\alpha}}$ is the mean gyro-radius.

For both the EICW and IBW, it is satisfied that $\omega \gg k_{\parallel} v_{\rm{ti}}$, and thus we have $\zeta_{{\rm{i}}n} \gg 1$ and ${Z}(\zeta_{{\rm{i}}n}) \simeq -1/\zeta_{{\rm{i}}n}$. Therefore, the ion susceptibility can be simplified as

For the EICW, $v_{\rm{ti}} \ll \omega / k_{\parallel} \ll v_{\rm{te}}$. Thus, we have $\zeta_{\rm{e0}} \ll 1$ and $\zeta_{\rm{e0}} {Z}(\zeta_{{\rm{e}}n}) \ll 1$. Since the electrons are easily magnetized, it is usually that $b_{\rm{e}} \ll 1$ even though the magnetic field is weak. Thus, only the ${n} = 0$ term contributes to the electron susceptibility. In this case, the dispersion relation is determined by

where $\omega_{\rm{pi}} = v_{\rm{ti}} / \sqrt{2} \lambda_{\rm{Di}}$ is the ion plasma oscillation frequency.

For the IBW, $\omega / k_{\parallel} \gg v_{\rm{te}}$, $\zeta_{{\rm{e}}n} \gg 1$, and ${Z}(\zeta_{{\rm{e}}n}) \simeq -1/\zeta_{{\rm{e}}n}$. Thus, the expression for $\chi_{\rm{e}}$ is similar to that for $\chi_{\rm{i}}$. In this case, only ${n} = 1$ term is sufficient to retain in $\chi_{\rm{e}}$ since $b_{\rm{e}} \ll 1$. Using ${A}_1(b_{\rm{e}}) \simeq b_{\rm{e}}/2$, the dispersion relation for the IBW is determined by

4.2   Experimental results in Ar

For each driving frequency $f_{\rm{drive}}$, the excited wave contains harmonic components. The signals with $f_{\rm{drive}}$ and $2f_{\rm{drive}}$ were used in the measurement of $k_{\perp}$. The resulting dispersion relations, i.e., $k_{\perp}$ versus $\omega$ plots, are shown in figures 9 and 10 for the Ar plasma. As a comparison, the calculated dispersion relations from equations (1) and (5) are plotted in the same figures using the experimentally measured plasma parameters listed in table 1.

Figure  9.  The measured (data points) and calculated (lines) dispersion relations in the Ar plasma when $B = 300$ G. Triangles and circles: the data from the $f_{\rm{drive}}$ and $2f_{\rm{drive}}$ signals obtained from the FFT spectra. Solid lines: the dispersion relation calculated from the kinetic theory. Dashed line: the dispersion relation calculated from the fluid model.

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Figure  10.  (a) The same as figure 9 except for $B = 500$ G. (b) The dispersion relation of the pure IBW calculated from the kinetic theory.

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Table  1.  Plasma parameters based on the measured electron density and temperature and the magnetic induction in Ar gas. The ion temperature is taken to be $T_{\rm{i}} \sim 0.03$ eV. $f_{{\rm{c}}\alpha} = \omega_{ {\rm{c}} \alpha}/2 {\text{π}}$ and $f_{ {\rm{p}} \alpha} = \omega_{ {\rm{p}} \alpha}/2 {\text{π}}$.

$B$ (G)	$n_{\rm{e}}/10^8$ (cm$^{-3}$)	$T_{\rm{e}}$ (eV)	$v_{\rm{te}}/10^7$ (cm/s)	$v_{\rm{ti}} / 10^4$ (cm/s)	$f_{\rm{pe}} / 10^8$ (Hz)	$f_{\rm{pi}} / 10^5$ (Hz)	$f_{\rm{ce}} / 10^8$ (Hz)	$f_{\rm{ci}} / 10^4$ (Hz)	$\lambda_{\rm{De}} / 10^{-2}$ (cm)	$\rho_{\rm{e}} / 10^{-3}$ (cm)	$\rho_{\rm{i}} / 10^{-1}$ (cm)
300	7.05	2.1	8.59	3.80	2.37	8.91	8.44	1.15	4.11	11.5	3.74
500	5.65	2.88	10.1	3.80	2.13	8.07	14.1	1.92	5.37	8.13	2.24

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In the case of $B = 300$ G, figure 9 shows that the overall variation tendency of $\omega$ versus $k_{\perp}$ is similar to the results of the fluid theory of the EICW. In the first and second frequency bands of $\omega_{\rm{ci}}$, the measured $\omega$ versus $k_{\perp}$ data are close to the results of the kinetic theory of the EICW. In the higher frequency bands of $\omega_{\rm{ci}}$, the measured data shift toward the higher $k_{\perp}$ values.

In the case of $B = 500$ G, figure 10(a) shows that $\omega$ increases with the increase of $k_{\perp}$ only in the first frequency band, in which the agreement with the dispersion relation of the EICW is observed. In the second frequency band, the variation of $\omega$ with respect to $k_{\perp}$ undergoes a transition from the increase to the decrease. From the third frequency band on, however, $\omega$ decreases with the increase of $k_{\perp}$. This is contradictory to the characteristics of the EICW. On the other hand, it is in qualitative agreement with the IBW, as shown in figure 10(b). Thus, in the third frequency band and above, the excited wave property is between the EICW and IBW, i.e., it is the IBW-like wave. However, compared with the theoretical dispersion relation of the IBW, the experimentally measured $k_{\perp}$ is systematically too low.

Since the excited wave contains multiple harmonic components, the harmonic signals may be the IBW-like waves, while the fundamental and/or second harmonic signals are the EICWs. Therefore, the results in the Ar plasma indicate that the EICW and IBW-like signals can be co-excited in the experiment.

4.3   Experimental results in He

In the case of the He plasma, the measured $\omega$ versus $k_{\perp}$ data are shown in figures 11 and 12 for $B = 300$ and 500 G, respectively. The corresponding plasma parameters are listed in table 2.

Figure  11.  The dispersion relations in the He plasma when $B =$$300$ G. Triangles and circles: the measured data of the $f_{\rm{drive}}$ and $2f_{\rm{drive}}$ signals obtained from the FFT spectra. Solid lines: the calculated results of the pure IBW from the kinetic theory. Note that the lower horizontal scale is for the experimental data, while the upper horizontal scale is for the theoretical lines.

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Figure  12.  The dispersion relations in the He plasma when $B =$$500$ G. Triangles: the measured data of the $f_{\rm{drive}}$ signals obtained from the FFT spectra. Solid lines: the calculated results of the pure IBW from the kinetic theory. Note that the lower horizontal scale is for the experimental data, while the upper horizontal scale is for the theoretical lines.

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Table  2.  Plasma parameters based on the measured electron density and temperature and the magnetic induction in He gas. The ion temperature is taken to be $T_{\rm{i}} \sim 0.03$ eV. $f_{{\rm{c}}\alpha} = \omega_{{\rm{c}}\alpha}/2 {\text{π}}$ and $f_{{\rm{p}}\alpha} = \omega_{{\rm{p}}\alpha}/2 {\text{π}}$.

$B$ (G)	$n_{\rm{e}}/10^8$ (cm$^{-3}$)	$T_{\rm{e}}$ (eV)	$v_{\rm{te}}/10^7$ (cm/s)	$v_{\rm{ti}} / 10^5$ (cm/s)	$f_{\rm{pe}} / 10^8$ (Hz)	$f_{\rm{pi}} / 10^6$ (Hz)	$f_{\rm{ce}} / 10^8$ (Hz)	$f_{\rm{ci}} / 10^5$ (Hz)	$\lambda_{\rm{De}} / 10^{-2}$ (cm)	$\rho_{\rm{e}} / 10^{-3}$ (cm)	$\rho_{\rm{i}} / 10^{-2}$ (cm)
300	3.25	2.55	9.47	1.20	1.61	1.91	8.44	1.15	4.11	12.7	11.8
500	2.75	3.35	10.9	1.20	1.49	1.78	14.1	1.92	5.37	8.77	7.08

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Figure 11 shows that, in the He plasma, $\omega$ decreases with the increase of $k_{\perp}$ in the first and second frequency bands, which is entirely different from the EICW dispersion relation. On the other hand, it is qualitatively consistent with the kinetic results of the IBW, but the measured $k_{\perp}$ is much lower than the theoretical expectation. However, in the third and fourth frequency bands, the variation of the $\omega$ versus $k_{\perp}$ is reversed, which is contradictory to the kinetic results of the IBW.

In the case when $B = 500$ G, figure 12 shows that the measured dispersion relation is much more complicated. With the increase of $k_{\perp}$, $\omega$ decreases in the fourth and the upper part of the third frequency bands, then changes from decrease to increase in the lower part of the third and the second frequency bands, and finally changes to decrease again in the first frequency band.

These results indicate that in the He plasma, the measured dispersion relation is qualitatively consistent with the theoretical one for the IBW only in the first and second frequency bands when $B = 300$ G but only in the first frequency band when $B = 500$ G.

Note that in the He case the excited wave amplitude is much lower than in the Ar case (c.f. figures 7 and 8). This is partially because the wave frequency is ten times higher in the He case than in the Ar case and the wave is more difficult to be excited in the high-frequency case. In fact, in the He case with $B = 500$ G, the second harmonic signal is very weak to deduce the well-shaped dispersion relation (this also is why in figure 12, the dispersion relation from the $2f_{\rm{drive}}$ signal is not included). Therefore, the experimental data in the He case are not as convergent as those in the Ar case.

4.4   Parallel wave numbers and discussions

The conditions $v_{\rm{ti}} \ll \omega / k_{\parallel} \ll v_{\rm{te}}$ and $\omega / k_{\parallel} \gg v_{\rm{te}}$ must be satisfied for the EICW and pure IBW, respectively. To measure $k_{\parallel}$, the longitudinally movable probe together with the insulated probe were used to detect the wave signals propagating parallel to the magnetic field. Figure 13 shows the dependence of the measured $k_{\parallel}$ versus $\omega / \omega_{\rm{ci}}$ for the same experimental conditions when measuring the $k_{\perp}$ versus $\omega / \omega_{\rm{ci}}$. It is shown that both $k_{\parallel}$ value and its variation are small. The typical value of $k_{\parallel}$ is $\sim 0.005$ ${\rm cm}^{-1}$. Using the parameters in tables 1 and 2, it is estimated that $\omega_{\rm{ci}} / k_{\parallel} \sim 1.4 \times 10^7 - 2.4 \times 10^7$ cm/s for Ar and $\sim 1.4 \times 10^8 - 2.4 \times 10^8$ cm/s for He for the magnetic fields of $300-500$ G, respectively. These values are much greater than the ion thermal velocity, $v_{\rm{ti}} \sim 10^4 - 10^5$ cm/s. However, the electron thermal velocity is $v_{\rm{te}} \sim 10^8$ cm/s, which can be either much greater than or the same order of $\omega_{\rm{ci}} / k_{\parallel}$.

Figure  13.  The measured $k_{\parallel}$ versus $\omega$ data in the Ar (a) and He (b) plasmas and for $B = 300$ (triangles) and 500 (circles) G, respectively.

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For the Ar plasma in low-frequency bands of $\omega_{\rm{ci}}$, it is usually held that $\omega / k_{\parallel} \ll v_{\rm{te}}$ and the EICWs are excited, which explains that the measured dispersion relations are close to the kinetic theory of the EICW. In higher frequency bands and for stronger magnetic fields, the wave frequency is increased, but $k_{\parallel}$ does not change appreciably. In this case, $\omega / k_{\parallel}$ is approaching $v_{\rm{te}}$, and thus, the excited wave changes its characteristic from the EICW to the one between the EICW and pure IBW.

For the He plasma, the ion mass is reduced, and the $\omega_{\rm{ci}}$ and hence the wave frequency are raised. Thus, for He, the regime $\omega / k_{\parallel} \geqslant v_{\rm{te}}$ and, in some cases, $\omega / k_{\parallel} \gg v_{\rm{te}}$ can be reached. Therefore, the measured dispersion relations are consistent qualitatively with the kinetic results for the pure IBW in some frequency bands. However, as a small change in $k_{\parallel}$ will result in the breakdown of the condition $\omega / k_{\parallel} \gg v_{\rm{te}}$, the measured dispersion relation easily deviates from the kinetic theory of the pure IBW in other frequency bands. Therefore, the excited waves are the IBW-like waves for most cases in the He plasma.

5.   Summary and perspective

The EICWs and IBW-like waves propagating perpendicular to the magnetic field were co-excited in the magnetized plasma device by launching the waves from the excitation grid. The excited waves contain harmonic components at multiple times of the driving frequency, with the EICWs and IBW-like waves appearing in different frequency bands. Using the FFT phase spectrum versus the propagation distance, the wave numbers and hence the dispersion relations were measured. It is shown that the condition for the excitation of the EICWs can be easily satisfied, while it is much more stringent to excite the pure IBWs. For Ar plasma, the measured dispersion relation is consistent with the fluid and kinetic models of the EICW in a limited frequency range. In high-frequency bands and for relatively strong magnetic fields, the excited waves change their property from the EICW to the IBW-like wave. For He plasma, the measured dispersion relations are only qualitatively consistent with the kinetic results of the pure IBWs in a few frequency bands but deviates significantly from the model in other frequency bands. The excited waves in the He plasma are IBW-like ones.

The EICWs and pure IBWs are only the kinds of the IBWs at two well-modelled extremities. In the spectral domain between these two extremities, the IBW-like waves, or the electron-Landau-damped IBWs as termed in the reference [3], have not been studied. It is important to study the IBW-like waves because they occur in a wide range of parameter regimes and are also easier to excite in experiments compared to the pure IBWs. Furthermore, they may have potential application in plasma heating since they have better energy coupling to the plasma via electron Landau damping due to the condition $\omega / k_{\parallel} \sim v_{\rm{te}}$. The dispersion relations of the IBW-like waves may be significantly different from the pure IBWs. In the present study, we were only able to qualitatively compare the measured dispersion relations with the results of the pure IBWs rather than those of the IBW-like waves. Therefore, it is necessary to further investigate the propagation properties of the IBW-like waves.

The reason why only the IBW-like waves rather than the pure IBWs were excited in the present experiment may be that the size of the excitation grid is finite, and the surface of the grid cannot be perfectly planar, which unavoidably results in a finite $k_{\parallel}$ value, though small it is. Moreover, the plasma and magnetic field nonuniformities may further complicate the experimental results, contributing to the discrepancy between the measured and theoretical dispersion relations.

It should be pointed out that the plasma is nonuniform in the present experiment. The relative uniform region in the transverse direction is only about several centimeters, which is comparable to (or even shorter than) the wavelength. From figures 7 and 8, it can be seen that the nonuniformity has a significant effect on the wave amplitude. It has minor effects on the phase compared to the amplitude, since the linearity of the phase versus distance is approximately maintained within the measurement distance, which enables us to obtain the wave numbers and identify the wave kinds to the lowest order of approximation. To the next order of approximation, the nonuniformity will also cause the phase shift in the propagation, resulting in the variation of the wave number versus distance that is correlated with the amplitude variation. From figures 11 and 12, it can be seen that the measured $k_{\perp}$ also has negative data points in the He plasma case, indicating the existence of backward-propagating waves. The backward-propagating waves may also be connected to the plasma nonuniformity. Thus, the plasma nonuniformity has multiple effects on the wave propagation. These will be the topics of future investigation.