Analysis and comparison of ion-acoustic wave reflection using laser-induced fluorescence and Langmuir probes

Since the first measurements of the Ion-Acoustic Wave (IAW),¹ the existence of wave reflection and the mechanisms behind it have been a topic of research in plasma physics. Experiments and theory have been developed to understand the behavior of reflection relative to the type of sheath^2–5 (ion-rich or electron-rich), surface (solid,^6–8 wire,⁹ mesh,^9–11 mesh + plate,^6,12–14 others), and wave frequency relative to velocity or density gradient scale lengths.^5,10,15–17

The phase-space behavior of ions is critical to IAW reflection. A common theme in the previous experiments and theory is the importance of ion flow in reflection. From ions flowing in the direction of the reflected wave to velocity gradient scale lengths, the ion phase-space was fundamental in the reflection mechanism. Most of the experiments cited before used Langmuir probes as the main diagnostic instrument. While probes are useful in measuring changes in densities and potentials, the amount of information they provide is limited—e.g., ion velocities and flows cannot be readily determined from them (except for Mach probes, which were not typically used in reflection experiments) —and the presence of the direct electronic signal (from fast electrons perturbed by the antenna) obscuring the incident and reflected wave information. In this work, we propose using laser-induced fluorescence (LIF) as a technique to avoid some of the pitfalls of probe diagnostics and to further the understanding of IAW reflection.

Laser-induced fluorescence is the resonant scattering of light from ions. It acts on the ion's internal quantum states, leaving the classical dynamics nearly unchanged. The fluorescence signal produced by the process is proportional to the density of ions in the metastable state that is surveyed. A narrow linewidth laser can be used to make Doppler resolved measurements of the ions at different velocities, resulting in the ability to measure the ion velocity distribution function (IVDF). With this, an observation of reflection in terms of ion velocities and flows can be performed, and new analysis methods become available.^18–20 

The goal of this paper is to show how the non-perturbative nature of laser-induced fluorescence can help shed light on IAW reflection without the complications inherent to probes. The experiment presented here is for the case of reflection from a solid conducting surface. Different frequencies and reflector bias voltages are tested. A comparison between Langmuir probe and LIF measurements is made for similar techniques, and observables unique to fluorescence diagnostics are displayed.

This paper is organized as follows. Section II describes the experimental setup, while Sec. III presents the diagnostic techniques used for the present experiment. In Sec. IV, graphs and tables with the experimental results are displayed, and an analysis of the results is given. Finally, Sec. V presents the conclusions and potential future work.

The experiment is performed in a multi-pole, cylindrical chamber. The chamber is 73 cm long, and it has a diameter of 49 cm. Singly ionized Argon (ArII) plasma is produced by means of impact ionization by the electrons emitted from a hot, lanthanum-hexaboride (LaB6) cathode. The plasma is contained by 16 rows of magnets with alternating polarities. The rows are set around the plasma periphery. The magnetic field is ∼1000 G close to the magnets but falls off rapidly to <1 G. The plasma is very uniform up until the cusp fields. The Larmor radius of ions in the region of interest (center of the chamber) is ρ_Li ∼ 1 m, leading to unmagnetized ions. Plasma parameters can be adjusted using two separate power supplies that control the heating and the emission of the cathode, respectively. A mass flow controller is used to keep the neutral pressure in the chamber constant. Typical plasma parameters for the experiment are T_e ∼ 2.1 eV, n ∼ 10⁸ cm⁻³, and a neutral pressure of 7.8 × 10⁻⁴Torr. The ion-neutral collision is ν_i−n < 1 kHz; the electron-neutral collision is ν_e−n ∼ 8 MHz, which is much smaller than the electron plasma frequency f_pe ∼ 90 MHz.

Figure 1 shows a diagram of the chamber. A circular, mesh-grid antenna with 81% transparency and a diameter of 5.6 cm is used to launch sinusoidal IAWs with a typical amplitude of δn/n < 0.1 and frequencies of 45, 100, 150, and 200 kHz. A hole of 2 cm diameter at the center of the antenna is used to allow laser light to go through without scattering. The antenna can be moved vertically with the use of a translation stage with a precision of 1 mm. A conducting electrode is used as the reflecting boundary for the experiment. The electrode is made out of a stack of razor blades with the razor edge facing the laser light, thus acting as a beam dump and reducing the amount of light scattered from the electrode to a minimum. The electrode can be moved vertically by a stepper-motor mounted on a translation stage with a precision of 1 mm and can be biased to different voltages using an external power supply. In this experiment, the bias was set to either +10 V or –40 V relative to ground to achieve electron-rich and ion-rich sheaths, respectively, at the reflector. Detailed measurements have been made of the ion flows at these bias voltages.²¹ 

The diagnostic instruments utilized for this experiment are a Langmuir probe, a double-tipped electric field probe, and LIF. The Langmuir probe has a circular tip with a diameter of ∼6.4 mm and can be moved by rotation as seen in Fig. 1. The probe is biased at –18 V to measure the ion current for a more direct comparison with LIF data. The electric field (E-field) probe has two tips constructed from two opposite-sided, S-shaped wires to prevent shadowing. The tips have a length of ∼5 mm and a separation of d = 0.53 mm. The tips are unbiased, and the probe is not absolute calibrated, as we are only interested in the relative amplitudes of the incident and reflected waves. The E-field probe is set such that the tips lay flat and parallel to the electrode. The probe is mounted on a translation stage that allows it to move horizontally, with a precision of 1 mm. For the fluorescence setup, a Sirah Matisse DR, tunable dye laser with a wavelength of 611.6 nm is used to employ the 3d²G_9∕2 → 4p²F_7∕2 → 4s²D_5∕2 transitions resulting in fluorescence with a wavelength of 461 nm. The laser beam travels through an Acousto-optic Modulator (AOM) used to “chop” the laser amplitude. The beam then goes into a fiber optic coupler that conveys the beam to the chamber. Laser power is monitored with a power meter for normalization purposes.

The laser beam enters the chamber through a window at the top. The laser propagates through the hole in the antenna, and its path is parallel to the axis between the antenna and the electrode. The viewing volume is positioned on that same axis, as shown in Fig. 1. Light from the plasma is collected through a large window on the side of the chamber, which has a Fresnel lens optically fused to it. The lens images the light onto a 3 mm tall aperture (which defines the size of the viewing volume) on the front of an aluminum can. Inside the can are optics for spectrally filtering the light (linewidth of 3 nm @461 nm) and focusing it onto a single channel photomultiplier tube (PMT) that is connected to an amplifying circuit.

The Langmuir probe, E-field probe, and PMT outputs are connected to a Zürich Instruments' MFLI lock-in amplifier, which is used to measure the linear response of the plasma due to the wave excited by the antenna. To achieve this, the “synch” output of the signal generator used to drive the antenna is used as the reference.

In past experiments, several different techniques have been used to observe IAW reflection. One method for detecting incident and reflected waves is by using an interferometer technique. In the Conventional Interferometer (CM) technique, the distance between the exciter and the reflector is kept constant. The probe position is scanned between the two, and the output is sent through a frequency mixer (a lock-in amplifier in the case of our experiment) along with the exciter's reference signal in order to obtain the change in the linear response function between the exciter and the reflector. The procedure used in this experiment follows the interferometer method devised by Schott.^9–11 In Schott's Method (SM), the distance between the exciter and the probe remains constant, while the separation between the probe and the reflector is changed. As the reflector is moved, the incident wave contribution remains constant as the distance between the exciter and the probe is fixed; so, any significant changes in the amplitude and phase of the signal are due to the reflected wave.

Schott's method holds some advantages over CM for observing IAW reflection. In CM, both incident and reflected wave signals are overlapped, and their phases change with respect to the probe's relative position between the exciter and the reflector. In cases where the incident amplitude is much greater than the reflected, it is difficult to separate the signals. In SM, this is not the case as the contribution of the incident wave to the interferometer pattern is constant. SM also holds some advantages over other typical techniques to measure IAW reflection, such as the time-of-flight (or sine-burst) technique. In the sine-burst method, it is difficult to separate the incident and reflected pulses for low frequency waves in short probe-reflector distances. This technique also suffers from dispersion and damping effects that complicate the analysis. Schott's interferometer technique works well for small, frequency-dependent reflections.

An additional complication arises in reflection experiments. Under some conditions, a wave can be “induced” by the Direct Electronic (DE) signal between the antenna and the reflector^22–24 produced by fast electrons. This wave is created at the reflector at nearly the same time as the incident wave is emitted and propagates back towards the antenna. This “induced” wave can be erroneously interpreted as a reflected wave. However, with SM, it can easily be separated from the reflected wave. As the reflector distance from the antenna and probe increases, the phase change is doubled as it accounts for the increase in the distance between the exciter and the reflecting surface for the incident wave and the distance between the reflector and the probe for the reflected wave. For the “induced” wave, as the reflector is moved away, the change in phase is affected by the increase in separation between the electrode and the probe only. Since the phase changes of the reflected wave and the “induced wave” are different, their respective contributions to the pattern will be as well. The waves can then be separated one from the other, and a reflected wave amplitude can be measured.

While SM is good at extracting the reflected wave amplitude, it suffers from the inability to obtain the incident wave amplitude. This is because for electric probes, there is yet another contribution due to the DE signal between the exciter and the probe that cannot be separated with this method. For this reason, Schott calculates “reflection efficiencies”^10,11 as opposed to true reflection coefficients. LIF on the other hand allows us to calculate reflection coefficients as the observed signal is mostly unaffected by the DE signal between the exciter and “diagnostic volume” (the metastable ion population can potentially change due to the fast electrons of the DE signals, but this can be isolated). Figure 2 shows the main signals affecting probe and LIF measurements.

First, for comparison, SM is applied using both a negatively biased Langmuir probe and single-wavelength LIF (with the wavelength set at the maximum of the first order velocity distribution function, f₁). The results are compared, and an analysis is carried out for both the ion-rich and electron-rich sheath cases. Finally, LIF's Doppler resolving capabilities are used to obtain new reflection measurements.

A reduction in the phase velocity is expected due to the value of the excited wave frequencies (45, 100, 150, and 200 kHz) relative to the ion plasma frequency of f_pi = 332 kHz. This produces a modification of the wave number k²⁵ as shown in the following equation:

where k is the wave number, ω is the excited wave frequency, ω_pi ≈ 2.1 × 10⁶ s⁻¹ is the ion plasma frequency, and $Cs=Te/mi$ is the ion-acoustic speed. Equation (1) is for the case of hot electrons and cold ions (⁠$κTi/mi≪ω/k≪κTe/me$⁠). The effect of this on the nominal phase velocity C_s = 2300 m/s is shown in Fig. 3. In the plot, each vertical line represents the position of each of the four frequencies used in this experiment. This dispersion is experimentally observed.

For the case of the positively biased electrode, IAW reflection is expected.² Figure 4 shows the real part of an interferometer scan obtained with the lock-in amplifier for a frequency of 200 kHz. The periodic variation in the signal suggests the existence of reflection. Similar patterns are observed at the other frequencies.

Fourier Transforms (FFT) of the patterns' complex vectors are computed to better extract information about the different modes present in the pattern. Figure 5 shows the FFT for a 200 kHz. In Fig. 5, the x-axis has been multiplied by 0.5 to account for the phase change of the reflected wave as described in Sec. III. The peak labeled A, C corresponds to the sum of the incident wave and the DE signal between the antenna and probe because the contribution of both is constant during a scan. To understand what peaks B and D represent, their respective phase velocities are calculated. The velocity for the case of f = 45 kHz was not obtained as peaks B and D could not be resolved separately.

Table I shows the wave numbers k = 2π/λ measured for peaks B and D and their respective phase velocities u = ω/k. Peak D corresponds to the reflected wave as the calculated velocities are in general agreement with those shown in Fig. 3. For peak B, which represents the “induced” wave, the “*” notation is used to denote a corrected k value. The interferometer phase for this component depends only on the distance between the probe and the electrode and not on the distance between the antenna and the electrode.

The signals referenced above were confirmed by time-of-flight (TOF) measurements made with an electric-field probe connected to an oscilloscope. A signal generator is used to launch a 1 Vpp sinusoidal burst from the antenna that is 6.5 cm away from the probe. The same signal is used to trigger the oscilloscope measurement. The time-of-flight method allows us to chronologically separate the signals that reach the electric-field probe. Figure 6 shows TOF traces for a single-period, 45 kHz pulse. Figure 6 also shows how the change in the signals measured by the electric-field probe with respect to electrode separation. Figure 6 is “observed at an angle,” with the higher lines representing the larger distances.

In Fig. 6, peak A represents the DE signal between the antenna and the probe as the peak's temporal position remains unchanged with respect to electrode separation, and it occurs close to the triggering time suggesting that it is not an acoustic wave. The incident wave is marked by peak C, as it remains almost constant as the reflector moves. The peak's temporal position matches the time it takes for a pulse to travel from the antenna to the probe at the group velocity. The position of peak D changes linearly as the distance between the electrode and the probe increases, and it occurs at a later time than the incident wave peak, suggesting that it represents the reflected pulse. A calculation of the pulse's group velocity matches the expected velocity [for f = 45 kHz, Eq. (1) can be approximated to ω ≈ C_sk]. Finally, there is a slight linear change in the position of peak B with respect to electrode separation, but it is detected before the incident wave. This is consistent with the “induced” wave. The pulse changes as the electrode is moved, and it is detected before the incident wave as the distance between the probe and the reflector is shorter than that between the antenna and the probe.

For the LIF interferometer scans, the wavelength is Doppler selected to match v ∼ v_ti (where $vti=kBTi/mi$ is the ion thermal speed) which is the maximum of the first order velocity distribution function. The electrode scan range was reduced to 8 cm as the signal-to-noise ratio decreased considerably at longer distances. Figure 4 shows a scan pattern while Fig. 5 shows an FFT for the 200 kHz wave for LIF measurements. The same as with the probe data, three distinct peaks are present. Table II shows the calculated phase velocities for the peaks. The wave numbers of the peaks match closely the values of those obtained for the Langmuir probe, and the presence of the “induced” wave is further confirmed.

The relative amplitudes of the LIF FFT peaks are different from those obtained by the probe. DE signals between the diagnostic and the exciter exist for the probe but not for LIF. For LIF, the peaks represent the actual amplitudes for incident and reflected waves, and so, it is possible to calculate a reflection coefficient.

Figure 7 shows the calculated reflection coefficients from the LIF measurements. The reflection coefficients were calculated by obtaining the Root-Mean-Square amplitude of the incident and reflected peaks (given their respective widths in k-space) and then dividing the latter by the former. Figure 7 shows a constant reflection coefficient value for the analyzed frequencies.

For the case of the negatively biased electrode, complete absorption of the wave is expected.² The Langmuir probe scans were obtained, but single-wavelength LIF scans were not as strong ion flows cause a constant laser wavelength to not always correspond to the same point in the distribution function. While single wavelength LIF scans are not presented, the Langmuir probe results in this subsection can be compared to those of LIF presented in Subsection IV D. For the ion-rich sheath, we separate the near-field from the far-field behavior. When looking at the effects of a negative surface bias on IAW reflection, only the far-field will be considered, x > 3 cm, where x is the separation between the diagnostic volume and the electrode.

Figure 8 shows the real part of scans for 200 kHz and 45 kHz. For 200 kHz, in the far-field, both amplitude and phase of the signal remain fairly constant, suggesting that no reflection of the wave exists. Waves at frequencies of 100 kHz and 150 kHz showed a very similar behavior, with no periodic patterns evident. For 45 kHz, a slight periodic modulation is present.

Assuming small amplitude fluctuations, the velocity distribution function of the ions can be approximated as

where f₀ is the equilibrium part (independent of time) of the velocity distribution function and f₁(t) is the time dependent part representing the fluctuations in the IVDF. In Eq. (2), f₁ ≪ f₀.

Taking advantage of LIF's Doppler resolving capabilities, the zeroth order IVDF (f₀) can be measured for the different cases. Using the lock-in amplifier with the excited wave signal as the reference, the first order IVDF (f₁) can be obtained as well.²⁶ Figure 9 shows the f₀ and f₁ for the positively biased electrode case, while Fig. 10 presents the negative electrode bias case.

Figure 9 was obtained by measuring the zero and first order velocity distribution functions for several electrode-viewing volume separations, following a method similar to SM. A 3D graph is then constructed with the IVDFs obtained, and top views of the interpolated versions of the graphs are presented. In Fig. 9, the x-axis depicts the separation between the electrode and the viewing volume. The y-axis shows the ion velocity, with the positive velocities representing the direction going from electrode to viewing volume. Finally, the colors show the amplitude of the signal for each phase-space location, with the color bar showing how the colors are to be interpreted.

For the positively biased electrode (Fig. 9), no substantial flows are detected. The positive velocity offset seen in Fig. 9(a) is within experimental and calibration errors. The lack of flow fulfills the requirement for reflection, which is the existence of ions traveling in the direction of that of the reflected wave.² This is further confirmed by the pattern shown in Fig. 9(b), where f₁ data were collected for different separations for a 200 kHz wave. The change in the phase and amplitude suggests the existence of reflection. Figure 9 also shows an interesting feature, where large peaks are followed by smaller peaks, repeating periodically as the electrode separation increases. This is consistent with the pattern obtained using SM in Sec. IV B, where the superposition of the “induced” and reflected waves resulted in a “double-peaked” pattern.

Figure 10 was put together in the same manner as Fig. 9. Figure 10(a) shows a significant drift in the ion flow near the electrode, with a flow close to the expected phase velocity. Looking at the far field in Fig. 10(b) (x > 3 cm), there are no visible patterns to suggest the existence of reflection or any other “backward” propagating waves, consistent with Bertotti's theory.² Wave frequencies of 45, 100, and 150 kHz present similar results, with no presence of periodic patterns in the far field.

In conclusion, laser-induced fluorescence provides a way to address the problems associated with reflection measurements using Langmuir probes, allowing us to obtain the incident and reflected amplitudes and calculate reflection coefficients. LIF's Doppler resolving capabilities make it possible to extract velocity space information about reflection that could further be studied by means of kinetic analysis. The technique illustrated here can be used to study IAW in other contexts.

We wish to thank S. Mattingly, F. Chu, and R. Hood for their help in setting up the experiment. This work was supported by the U.S. Department of Energy under Grant No. DE-SC0016473.

This research is a part of a Ph.D. dissertation to be submitted by J. Berumen to the Graduate College, University of Iowa, Iowa City, IA.