A direct measurement of the particle balance and derivation of the underlying particle source rate distribution in a helicon plasma developed for wakefield particle accelerators is presented. Parallel and radial ion fluxes are measured using laser induced fluorescence on single ionized argon. We find that the radial contribution to the source rate is an order of magnitude larger than the axial contribution. We also find that the axial source rate profile closely matches the radial density gradient axial profile, thus indicating the importance of the radial density profile for the particle balance. Notably, the peak ion source rate is located off-axis, about halfway between the axis and the vacuum wall on both sides of the axial center.

Helicons are bounded whistler waves, typically at tens of megahertz for hundreds of Gauss of magnetic field, which are currently being considered for applications in space thrusters,1 fusion materials testing,2 and wakefield accelerators.3 Wakefield accelerators are capable of generating accelerating electric fields well in excess of 100 GV/m.4 The accelerating gradient in a plasma accelerator is limited by the wave breaking field E w b n e [ 10 20 m 3 ] [ GV / m ]. The AWAKE project5 at CERN aims to accelerate electrons in an accelerating field in excess of 1 GV/m and targets a background plasma density of n e 7 × 10 20 m−3. Furthermore, AWAKE requires minimal variations, δ n e, in the axial density profile such that δ n e / n e < 0.25 %.6 Any larger variations would lead to a degradation of the quality of the accelerated bunch or even to its loss along the plasma, if it is drifted into the defocusing phase of the wakefields.

Helicon plasmas can exceed 1020 m−3 at moderate input powers, e.g., 27 kW for a 1 m long cell with diameter 5 cm as described by Buttenschön et al.7 Densities in the low to mid 1019 m−3 range are achievable at much lower input power of only a few kilowatts.8 The mechanisms responsible for the formation of this very efficient plasma are not well understood. The power deposition has been modeled in other work,9 offering some meaningful insight into this formation, particularly into the directionality of helicon propagation.10 Previous work has indicated that RF power is initially absorbed by the mostly radial, electrostatic Trivelpiece–Gould mode,11,12 which propagates primarily in the plasma edge. The power is then transferred into the helicon mode, which propagates axially.12 

Particle source rates have been calculated from electron temperature, density, and ionization cross sections.13 A detailed measurement of the one dimensional ion source rate on the MARIA device has previously been presented, in a precursor to our current work.14 However, to our knowledge, no direct measurement of the complete two dimensional ion source rate profile has been conducted. The measurements on MARIA only included the axial contribution to the source, whereas we present both the radial and axial measurements. We can, therefore, resolve the fully 2D-axisymmetric ion source rate distribution in a 1.3 kW helicon plasma in a new experiment.

The helicon device used in this work is the Madison AWAKE Prototype (MAP) at the University of Wisconsin–Madison. An example of the helicon plasma in MAP is shown in Fig. 1, which shows the central 1 m of the vacuum chamber, the helicon antenna, and some of the electromagnets. The vacuum vessel is 2.1 m long and made from borosilicate glass with an internal diameter of 52 mm. MAP is pumped to a high vacuum (low 10 4 Pa) and filled with 1 Pa of argon gas. An array of 14 electromagnets creates a uniform magnetic field of 50 ± 0.5 mT in the inner 1 m of the chamber, across the whole 52 mm diameter. An Advanced Energy Apex RF generator delivers power, 1.3 kW in this work with a capacity up to 10 kW, through a matching network and a half-helical antenna. For this work, reflected power is typically 0 10 W as measured by the RF generator.

FIG. 1.

A helicon plasma in the Madison AWAKE Prototype (MAP).

FIG. 1.

A helicon plasma in the Madison AWAKE Prototype (MAP).

Close modal

MAP is equipped with a narrow bandwidth laser induced fluorescence (LIF) diagnostic.15 LIF is a versatile, noninvasive plasma diagnostic that can measure ion and neutral particle velocity distribution functions, temperatures, and densities, and the electric field in a plasma.16–20 A laser is injected into a plasma, exciting a specific transition, preferably from a metastable state. The excited state de-excites quickly, and the fluorescent light from the de-excitation is measured. The laser frequency is scanned across the excited particle's velocity distribution function (VDF) so that the whole VDF can be measured. The Doppler shift in the position of the VDF in frequency space can be used to extract the flow speed of the particles in the direction of laser injection.17 To measure flow speeds in radial and axial directions, we inject the laser either radially or axially, respectively.

A Toptica diode laser at 668.6 nm is used to excite the transition from the 3d4F7/2 argon ion state (1) to the 4p4D5/2 state (2), which then decays with an emission at 442.6 nm to the 4s4P3/2 state (3) within nanoseconds. The laser is amplified to 300 mW, monitored for power and wavelength, modulated at 47 kHz, and then injected into an optical fiber for delivery to the plasma. The polarization of the beam is easily adjusted with a linear polarizer and quarter waveplate at the fiber output. After selection of the beam polarization, the beam is injected into the plasma. Light is collected by a movable light collection telescope and coupled into an optical fiber. The light is filtered with a 1 nm optical bandpass filter and, when necessary, a neutral density filter and then injected into a photomultiplier tube. The desired component of this signal is then extracted by an SRS510 lock-in amplifier.

The collection optics telescope is focused to a 3 mm diameter spot on the MAP axis and is mounted to a Thorlabs high-precision rotation stage with a micrometer giving 2.4 arcminutes per division, resulting in a radial resolution of about 33 μm in the chamber. Typically, a radial step size of approximately 66 μm was used for fine radial steps. The radial injection unit and collection scope are both mounted to a carriage that slides along the axis, with 1 mm precision. The axial laser injection unit is mounted on a translation stage with 1 mm precision. The outer limit of the axial injector's radial line of sight is 1.5 cm, so the inner 60% of the plasma is accessible for measurement in the axial direction.

The laser frequency is calibrated absolutely against a well known peak in the molecular iodine absorption spectrum at 668.612 72 ±  2.56 × 10 5 nm.21 The high degree of precision in the location of this peak gives a low minimum uncertainty of 7.2 m/s in ion velocity measurements. LIF is used in this work to measure both radial and axial velocities. For radial velocity measurements, the beam is polarized in the direction of the magnetic field to prevent Zeeman splitting effects and then injected radially. For axial velocity measurements, the velocity distribution function is sampled both with the beam right-hand and left-hand circularly polarized to sample both σ + and σ transitions. The average velocity from these two velocity distribution functions is used, according to the method described in detail by Green et al.17 The circularly polarized beam is injected along the axis of the device.

The population of the excited ion state (1) is related to the total ion density through a power scaling law as derived by Green et al.14 Specifically, the LIF intensity I measured by the lock-in amplifier is related to the plasma density n in units of m−3 as
n = A I B ,
(1)
where A and B are empirically derived constants. The constants used in this work, A = 9.9 × 10 17 and B = 0.30, were derived through comparison to electron density measurements made by a Langmuir probe in the MARIA device.22 It was found that this scaling law holds for electron temperatures below 6 eV. Above 6 eV, the scaling law also depends on Te, which complicates the relation between I and n.23 The value of B depends principally on the temperature and density dependent state dynamics of the plasma, which dictate what fraction of the total ion population is in state (1). The value of A also depends on factors such as the laser intensity, the size of the sample volume, and other diagnostic setup parameters. While the values of A and B were derived on a different device than the data presented here, the state dynamics are expected to be similar between the two devices, so that B, in particular, is expected to be similar to what would be measured on MAP. Since B is the coefficient that dictates how the density changes with the LIF intensity, while A only provides a scalar multiplier, we use the coefficients from MARIA in this work to investigate profiles in MAP. While an absolute uncertainty in the density measurements is expected until a diagnostic to measure the absolute density is available, the relative uncertainty between measurements remains low.
The equation for a steady state particle balance including the source term S is given by
· ( n V ) = S ,
(2)
where n is the particle density and V is the velocity. While it is documented that helicons have an azimuthal flow, this flow does not have a significant azimuthal gradient.24 Thus, considering cylindrical symmetry in Eq. (2), the particle balance can be described as
( n V z ) z + 1 r ( r n V r ) r = S ( z , r ) ,
(3)
where z and r are the axial and radial coordinates, respectively, Vz is the axial component of V, and Vr is the radial component of V. The three parameters required for the calculation of the source rate are then the ion density and the radial and axial ion velocities. These values must be measured with sufficient precision to produce differentiable flux profiles in the radial and axial directions.

The layout of Figs. 2 and 3 mirrors the picture in Fig. 1, with the antenna centered about z = 0 cm. The peak density at z = 13 cm corresponds to the brightest section of the plasma in Fig. 1. The radial positions in Figs. 2 and 3 are shown on the y-axis, with the MAP axis shown at r = 0.

FIG. 2.

Plasma density (top), radial (upper middle), and axial (lower middle) ion velocities and ion flux (bottom) are shown. Shaded regions at r > 1.5 cm in the Vz and Γ plots indicate regions that depend on extrapolated Vz measurements. The flux magnitude is represented by color, and the direction and magnitude are represented by the arrows. The position of the antenna is indicated by the vertical dashed lines in all plots.

FIG. 2.

Plasma density (top), radial (upper middle), and axial (lower middle) ion velocities and ion flux (bottom) are shown. Shaded regions at r > 1.5 cm in the Vz and Γ plots indicate regions that depend on extrapolated Vz measurements. The flux magnitude is represented by color, and the direction and magnitude are represented by the arrows. The position of the antenna is indicated by the vertical dashed lines in all plots.

Close modal
FIG. 3.

Radial contribution to the source rate (top), axial contribution to the source rate (middle), and total source rate (bottom) are shown for all measured positions in the MAP chamber. Shaded regions at r > 1.5 cm in the Sz and S plots indicate regions that depend on extrapolated Vz measurements. The axial contribution is insignificant compared to the radial contribution to the total source rate. The position of the antenna is indicated by the vertical dashed lines in all plots.

FIG. 3.

Radial contribution to the source rate (top), axial contribution to the source rate (middle), and total source rate (bottom) are shown for all measured positions in the MAP chamber. Shaded regions at r > 1.5 cm in the Sz and S plots indicate regions that depend on extrapolated Vz measurements. The axial contribution is insignificant compared to the radial contribution to the total source rate. The position of the antenna is indicated by the vertical dashed lines in all plots.

Close modal

The ion density was measured in MAP at 86 locations on a 2D grid ranging from z = −15 to z = 41 cm, and from r = 2.5 to r = 2.5 cm and then fit to a least squares bivariate spline. The density is shown in the top panel of Fig. 2. Within the measurement error, which is generally 10%–20%, the values at positive r match the values at negative r. This supports the assumption of cylindrical symmetry, which we reasonably conclude to apply for the remainder of this work.

The radial velocity was measured at over 350 locations and then fit to a least squares bivariate spline. The results plotted in the upper middle panel of Fig. 2 show that the ions flow outward at all locations in the device, as expected from profiles modeled in other work.12 

The axial velocity was measured in MAP at 45 locations and is plotted in the lower middle panel of Fig. 2. The axial velocity was measured at r = 0, r = 0.5, and r = 1.5 cm, fit to a cubic spline and then interpolated radially. A window at the end of MAP allows for axial injection of the laser at | r | < 1.5 cm, so velocities at | r | > 1.5 cm are extrapolated. In Figs. 2 and 3, all quantities that depend on the extrapolated Vz data are shaded in light gray.

The ion flux is calculated from
Γ = n ( V z z ̂ + V r r ̂ )
(4)
and is shown in the bottom panel of Fig. 2.

The density in MAP peaks near 2.2 × 10 19 m 3, about 13 cm to the right from the center of the antenna. This peak is off-axis at about r = 0.6 cm, so that the density profile is slightly hollow, dropping on axis to about 2.1 × 10 19 m 3, or 95% of the maximum, and close to the radial boundary dropping to 7.1 × 10 18 m 3, or 32% of the maximum density. There is also an axial density gradient, resulting in a minimum measured on axis density of 9.8 × 10 18 m 3, constituting 45% of the maximum measured density. These minimum values are at the left end of Fig. 2, beyond which the density rapidly drops to values that are too low to measure accurately with LIF.

Figure 2 shows that the radial and axial velocities both have clear regions with zero velocity, surrounded by regions of outward directed velocities. Particles flow out from the axis slowly when they are in the core, but with a very steep gradient as they move further outward. Notably, the outward directed velocity gradient is steepest at the axial position of the antenna. The axial velocity has significantly more structure than the radial, as it contains separated regions of positive and negative velocity separated at z = 23 cm as well as local maxima, such as at z = 12 cm, which corresponds to the position of the density maximum. The flow reversal at z = 23 cm is of particular interest, as such a flow reversal immediately suggests the presence of a particle source, with particles flowing away in all directions.

The source rate calculated from Eq. (3) is shown in Fig. 3. The contributions from radial and axial flux are separated and shown in the top and middle panels of Fig. 3, respectively. While the radial and axial velocities are of the same order of magnitude, the length scales over which they change differ by an order of magnitude. Consequently, the maximum of the axial contribution is 6.7 × 10 22 m 3 s 1, or 7% of the maximum of the radial contribution, 9.6 × 10 23 m 3 s 1. At the radial boundary, there is a narrow region in which the source rate is negative, signifying an ion sink. This is thought to be caused by the proximity to the wall, where the neutral population increases significantly due to recombination at the wall.

The source rate map depicted in the bottom panel of Fig. 3 shows a clear off-axis peak at r / a 0.6, which supports power coupling in this area, presumably through the well documented Trivelpiece–Gould mode seen in other works.11,12 This high source rate is calculated from the steep radial velocity gradient. One possible reason for the relatively steep radial velocity gradient is the size of the MAP chamber. The only other helicon plasma experiment with a comparable ion source rate calculation is described by Green et al.14 and was conducted on MARIA, the predecessor of MAP. The helicon plasma in MAP is much larger radially than the plasma in MARIA, despite the smaller MAP chamber diameter of 5.2 cm, compared to 14 cm in MARIA. Nearly the whole 5 cm tube is filled with plasma compared to a 1 2 cm diameter core in MARIA. This smaller chamber diameter results in a very small radial distance that ions can travel before they are affected by pre-sheath-like acceleration toward the wall.12 As seen in Fig. 2, the outward acceleration is significant starting around 1 cm out from the axis. A typical ion gyroradius rg in MAP is 0.9 cm, so there is a region of about 2 r g in the center of MAP where the ions could remain confined without necessarily being caught in this outward acceleration region. In contrast, the main core plasma in MARIA is on the order a few cm or 10 % 20% of the 14 cm diameter chamber, while rg is again under 1 cm. This leaves a very large distance in a large chamber like MARIA's between the core plasma and the region where acceleration caused by proximity to the wall becomes significant.

The measurements presented in Figs. 2 and 3 are congruent with the intuitive strong link between the plasma density and the ion source rate. Both values peak near r = 1 and z = 13 cm. Notably, this is also a region with very high power deposition as presented in Ref. 10, where we have shown that the helicon plasma directionality is the consequence of radial density gradients. Based on the measurements presented here, we conclude that the radial density gradient is due to a non-uniform source rate. Separating Eq. (3) into individual terms, we can isolate the contribution from the radial density gradient,
S = V z n z + n V z z + n V r r + V r n r + n V r r .
(5)

These terms are plotted in Fig. 4, with each term averaged over all r at each z. The radially averaged density gradient contribution is negative, indicating an ion sink. The first two terms make up the axial contribution and are plotted in shades of purple. The first term is nearly negligible, with only a very small peak near z = 17 cm, but the second term contributes slightly to the directionality of the source rate profile. The remaining terms make up the radial contribution and are plotted in shades of green. The third term represents the radially scaled flux, and this contributes strongly to the directionality. The fourth term is negative, representing a particle sink contribution peaked near the antenna. The last term represents a source peaked very close to the peak of the fourth term but stronger than the fourth term. In general, the radial terms are so much stronger—for the presented MAP diameter and neutral source level—than the axial terms that the directionality of the source rate profile is due almost completely to the radial contribution. The axial contribution primarily provides minor structure to the overall profile, particularly at z = 0 and z = 17 cm. It is clear that the ion source location is defined primarily by the fifth term, so that the radial density profile and the radial velocity gradient profile are the most significant contributors to the overall source rate profile.

FIG. 4.

The terms in Eq. (5) are averaged over all r and plotted against z. Axial terms are shades of purple, radial terms are shades of green, and the total source rate is black.

FIG. 4.

The terms in Eq. (5) are averaged over all r and plotted against z. Axial terms are shades of purple, radial terms are shades of green, and the total source rate is black.

Close modal

We presented in this work a first time 2D measurement of the particle balance in an argon helicon plasma, considering both radial and axial contributions to the particle source rate. LIF measurements show that while the axial contribution is easily identified from the density and velocity profiles, it is very weak compared to the radial contribution in a helicon device with small diameter. The measurement of the source rate shows that the short distance ions travel before reaching the radial boundary significantly alters the overall fueling mechanism in such a plasma, rendering the axial flux divergence insignificant in comparison with the radial flux divergence. LIF measurements also revealed that particles are ionized off axis, near r / a 0.6, indicating the coupling of the Trivelpice–Gould mode. These measurements show for the first time that the contribution from radial flow to the overall source rate profile far exceeds the contribution from axial flow. With the goal of an axially extremely homogeneous plasma in mind, this work points toward a careful adjustment of the axial positioning of the radial density and velocity gradient profiles by correct antenna placement and fine-tuning of the neutral source distribution. The eventual density distribution is a direct result of the density formation itself, which determines helicon wave dispersion, hence the power deposition that causes the ionization. Disentangling the interaction of RF coupling and ionization mechanisms at high power densities in the AWAKE relevant regime is a key goal for future studies.

We would like to express our gratitude to Barret Elward for his outstanding work during the construction of the MAP experiment. The research presented here was funded by the National Science Foundation under Grant Nos. PHY-1903316 and PHY-2308846 and NSF-CAREER Award No. PHY-1455210 as well as the College of Engineering at the University of Wisconsin-Madison through the Thomas and Suzanne Werner Professorship of the Department of Nuclear Engineering and Engineering Physics.

The authors have no conflicts to disclose.

M. Zepp: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Project administration (equal); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). M. Granetzny: Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Software (equal); Visualization (equal); Writing – review & editing (supporting). O. Schmitz: Conceptualization (equal); Formal analysis (supporting); Funding acquisition (lead); Methodology (equal); Project administration (equal); Resources (lead); Supervision (lead); Writing – original draft (supporting); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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