Uncertainty analysis of the plasma impedance probe Open Access

Arguably the largest differences between PIPs and LPs are in how they are electrically operated and what they subsequently measure. LPs are biased across a range of large excitation voltages (⁠ $V exc > k b T e / e$⁠) and swept at low rates compared with the ion and electron plasma frequencies. They therefore only measure electrical resistance, Re(Z). Due to the large voltage sweeps, LP measurements encounter three distinct sheath regimes: unsaturated, ion-saturated, and electron-saturated. The complexity of the underlying models and subsequent challenges in numerical analysis often lead to total probe errors between 10% and 50%.⁸ In contrast, PIPs^4,7 are excited across a range of frequencies (ω_exc) above the ion-plasma frequency and near the electron plasma frequency (ω_p); therefore PIPs measure the complex impedance spectra, $Z ( ω )$⁠, and ignore ion contributions. Additionally, PIPs are excited at relatively low voltages, which avoids saturated sheaths and allows for linearization of the Boltzmann relation. We argue that these factors lead to a simpler PIP model, which results in improved measurement accuracy over the LP.

The PIP is one of many types of in situ radio frequency (RF) probes and is advantageous in that it is well suited for lower-density plasmas.² Compared with other RF probes,³ the PIP is consistently more accurate with errors around 10%–20%. It is worth noting that the analysis by Kim et al.³ is based heavily on prior PIP work⁴ and does not appear to include the corrections we have identified in our current and previous work,⁷ which will further reduce the PIP's error. These corrections include but are not limited to calibrating the PIP's stem, using numerical fitting instead of locating the PIP's second resonance to calculate density, and optimizing the PIP's geometry for the thin-sheath limit.

Physically, PIPs operate by measuring the frequency response of the plasma and sheath around the probe.^4,7 When placed in plasma, a sheath forms between the PIP and the bulk plasma. Both the sheath and plasma regions have their own frequency-dependent dielectrics, $ε s h ( ω )$ and $ε p ( ω )$⁠, respectively, which are also dependent on their respective plasma properties. These dielectrics electrically couple with the PIP and modify the PIP's electrical impedance, $Z pip ( ω )$⁠. An example of this coupling is it introduces two resonances (⁠ $ω ±$⁠) into the PIP's impedance. Measuring $Z pip ( ω )$ allows us to infer the two dielectrics and their plasma properties. Experimentalists typically perform this measurement by transmitting one of two types of waveforms to the PIP: swept waveforms and pulsed waveforms. Swept waveforms,^4,14–16 the traditional method, use a vector network analyzer (VNA) to transmit continuous frequency sweeps and measure both the outbound and reflected waves. This approach is generally recommended over the pulsed approach because it is simpler to set up/operate and provides a higher signal-to-noise ratio (SNR). However, it has a slower acquisition rate, typically no faster than 100 Hz. The pulsed method,^7,17,18 a more modern approach, uses a signal generator to transmit a series of single, broad-spectrum pulses separated by a finite time delay and an oscilloscope to measure the voltages and currents associated with each pulse. This approach allows for >1 MHz acquisition rates but at the cost of lower SNR and higher experimental complexity.⁷ Although SNR has greater impact on the pulsed waveform method, it is a limiting factor for both methods.

We attribute the PIP's underrepresentation in the laboratory to several factors:^2,7 PIP's higher cost, complexity, relatively low upper density limit (⁠ $< 10 16$ m⁻³), and the long heritage of LPs. This upper density limit is set by several factors, including the underlying assumptions of the PIP's model,⁷ but it is often attributed to the maximum resolvable frequency in a $Z pip ( ω )$ measurement. Historically, the community has believed that PIP measurements must resolve frequencies up to the plasma frequency for unmagnetized plasmas and up to the upper hybrid frequency for magnetized plasmas. Measuring higher densities, which scale with $ω p 2$ [Eq. (1)], becomes increasingly complicated as RF instrumentation cost, complexity, and sensitivity to calibration errors all increase appreciably with frequency. This, in part, is why PIPs have been historically developed for lower-density environments, specifically the ionosphere where $n < 10 12$ m⁻³. Examples include sounding rockets,^17,19–24 satellites,^25,26 and on the International Space Station.^27–29 To a lesser extent, PIPs have also been used in lower-density laboratory plasmas, including DC discharges,^7,30 Hall thruster plumes,^16,31 and plasma processing applications³² where $n < 10 16$ m⁻³.

The goal of this work is to increase SNR in order to (i) improve overall measurement quality, (ii) extend the upper density limit, (iii) increase acquisition rates, and (iv) allow the unit to be more accessible in the laboratory environment. To achieve this, we present here our work in modernizing and streamlining the PIP's model, design, operation, and analysis. In Sec. II, we discuss our updated PIP-monopole design and accompanying analytical model. In Sec. III, we discuss our streamlined calibration, operation, and analysis for both the continuous and pulsed PIP methods. In Sec. IV, we perform a Monte Carlo (MC) uncertainty analysis that both quantifies the improvement in SNR from the previous two sections and provides recommendations for optimized PIP operation and analysis.

To extract meaningful results from a measurement of Z_pip, we require a physical model. Below, we derive the PIP-monopole models for unmagnetized plasmas and discuss their assumptions and limitations.

The PIP-monopole model presented below is a reformulation of Blackwell's model⁴ and closely follows our previous derivation.⁷ 

Similar to previous work,^4,7,33–35 we begin our model by using a lumped-element circuit framework and modeling the PIP's head and surrounding environment as one or more spherical capacitors in series. To clarify, we ignore inductance within the PIP's head but not in the surrounding environment. The impedance spectra of any capacitor is $Z ( ω ) = 1 / j ω C$⁠, and because of the spherical geometries, we use spherical-shell capacitor models, which have a capacitance $C = 4 π ε ( 1 / r 1 − 1 / r 2 ) − 1$ with r₁ and r₂ being the radii of the inner and outer electrodes, respectively.

When plasma is present [Fig. 1(b)], we model the sheath and plasma regions as two concentric capacitors in series. The inner conductor is the monopole's head, the outer conductor is the vacuum vessel, and the middle conductor is the sheath–plasma boundary with radius, r_sh.

To better understand $Z ′ pip$ and $Z ′ diff$⁠, we plot both in Fig. 2 for $t s h ′ = 0.1$ and $ν ′ = 0.4$⁠. We choose these values because (i) they are values we have encountered experimentally and (ii) the three resonances (⁠ $ω ±$ and ω_diff) are distinctly visible in the imaginary components. The real components have a peak near the plasma frequency and a width that broadens with higher damping.¹⁶ 

So far, we have presented two models of the PIP when plasma is present (Z_pip and Z_diff), and we recommend using Z_pip for most cases (discussed further in Sec. IV). However, there are two cases where Z_diff may be preferred. First, Z_diff is less dependent on $ν ′$ and therefore its measurements are less susceptible to noise when damping is high (⁠ $ν ′ ≳$ 1). This dependency on $ν ′$ is apparent when comparing Eqs. (8) and (13), while considering Eq. (9). The resonant frequencies (⁠ $ω ±$⁠) are heavily dependent upon $ν ′$⁠, which are present in Z_pip but not in Z_diff. At high damping, $ω +$ and $ω −$ merge and disappear per Eq. (12), while ω_p is isolated from $ν ′$ and $t ′ s h$ per Eq. (14). Second, we resolved Z_diff using planar and cylindrical capacitors (in addition to spherical), and all three results provide the same $ω diff = ω p$ resonance. This suggests that this result may be independent of probe geometry (i.e., by subtracting the vacuum measurement, we may be partially calibrating out the probe's geometry.) This implies that in cases where the probe's geometry is difficult to model, it may be possible to identify ω_p by locating Im $( Z diff ) = 0$ in a measurement alone (i.e., without the need for a model).

The physical models (Sec. II A) include a number of assumptions, which both limit its applicability and introduce potential modeling errors: (i) The probe and sheath have a spherical geometry, without consideration of the protruding stem from the PIP's head. (ii) The plasma is cold (T_e = 0), non-magnetized, and homogenous. (iii) The derivation of Eq. (6) assumes small amplitude signals of V_exc, oscillating around the probe potential (⁠ $| V probe − V exc | / | V probe | ≪ 1$⁠).³⁹ The small amplitude signal is also necessary to maintain a consistent sheath thickness. (iv) The monopole probe is modeled as a lumped element, where the probe scale is far smaller than the wavelength (⁠ $L / λ ≪ 1$⁠). (v) A quasi-electrostatic assumption is made for the monopole probe to model it as a capacitor, ignoring its inductive components, so that it does not radiate and only measures the near-field. (vi) The sheath is modeled as a homogeneous vacuum and the transition to the plasma region is discontinuous. Some of the errors associated with these assumptions have been quantified in our previous work.⁷ 

The models above are designed for a PIP-monopole antenna, which is intended for non-magnetized plasmas. Although the PIP-monopole has previously been implemented in a magnetized plasma,⁴⁰ we recommend using a PIP-dipole antenna in magnetized plasmas and taking measurements both along and across the magnetic field line. Blackwell has previously created a dipole circuit model⁴¹ that is based on work by Balmain.⁴² Please note that a balun transformer, a device used to balance the impedance between the dipole's two conductors, is required and will have to be correctly calibrated.

With the models established in Sec. II, we next turn to PIP setup, calibration, and analysis for both swept and pulsed methods. Pulsed setup and operation are discussed in our previous work.⁷ Both setups, diagramed in Fig. 3, have much in common despite apparent differences, and we discuss both below.

Figure 4 shows our recent monopole antenna design. This probe was fabricated from a Pasternack RG401 semi-rigid coax cable where the outer conductor and dielectric were trimmed back to expose the inner conductor. Constructing the PIP-monopole in this way makes its stem easier to calibrate, as discussed below. The head was a two-piece, machined SS316 hollow sphere, and the inner conductor was attached with set screws. The head for our previous design⁷ was a drilled aluminum sphere that was press fit onto the inner conductor. Finally, our previous work⁷ also provides recommendations for determining monopole dimensions, including staying within a thin-sheath limit.

In this setup, the TLs include a DC-blocking filter, which protects the VNA from DC voltages and allows the PIP's head to electrically float. The coaxial cabling is RG316 with SMA connectors, securely routed, and kept at a constant temperature. Small changes in cable routing and temperature can result in calibration drift and therefore measurement error, especially at higher frequencies.⁴⁴ 

The TLs between the VNA and PIP add unwanted attenuation and phase shift to the PIP measurement. Calibration is the process of removing the contribution of the TLs and isolating the contribution of the PIP and surrounding dielectrics. In the RF engineering community, this is referred to as “moving the calibration plane.” PIP calibration consists of two steps and therefore has three calibration planes (cal. planes) shown in Figs. 3 and 4.

To assist with calibration, we intentionally built our PIP-monopole (Sec. III A) from a commercial, semi-rigid, coaxial cable, which provides two benefits. (i) It provides a convenient connector (e.g., SMA) that makes calibrating from cal. plane 1 to cal. plane 2 relatively simple. (ii) The stem of the PIP (the intact portion of the semi-rigid cable) can be modeled as a lossless transmission line, using the cable's published dielectric properties for calibrating from cal. plane 2 to cal. plane 3.

The first step (cal. plane 1 to 2) calibrates the transmission lines between the VNA and PIP's SMA connector. In this work, we use a 1-port error model^7,45 which requires measurements of 3 RF standards at both cal. plane 1 (we refer to these as the “truth” measurements) and at cal. plane 2. Modern VNAs have built-in one-port error calibration procedures, but for our work, we perform the measurements manually and use scikit-rf 's OnePort calibration function.⁴⁶ 

The second step (cal. plane 2 to 3) calibrates the PIP's stem. Because there is no convenient connector (e.g., SMA) at cal. plane 3 to connect RF standards, we instead model the PIP's stem using a two-port, lossless transmission line model^7,37 and the published dielectric constant of the stem. For our work, we build the model in scikit-rf and invert the model to de-embed (calibrate) the stem as described in  Appendix A.

 Appendix B shows an example calibration using both steps.

After acquiring and calibrating a Γ_pip measurement, the next step is to fit the model to the measurement in order to solve for the model's three unknowns: ω_p, ν, and t_sh. This requires several steps. First, we normalize ω_p in Eq. (8) so that its value is roughly the same order as $ν ′$ and $t ′ s h$⁠, and therefore the fitting routine weights each parameter more equally. Second, we convert our Z_pip model [Eq. (8)] into Γ_pip by applying Eq. (15) to the model. Third, we trim our calibrated measurement to a finite frequency range, which is discussed in Sec. IV. Fourth, we fit our Γ_pip model to our Γ_pip measurement using a minimization function (specifically, scipy's minimize or least_squares functions) with a residual function that includes both the real and imaginary components of Γ. We use the same procedure above when fitting with Z_diff.

Figure 5 shows an example of the Γ_pip model fit to a calibrated, swept Γ_pip measurement. We attribute discrepancies between the model and the fit to the simplicity of model, which includes ignoring plasma gradients, ignoring nearby electrical conductors (i.e., the PIP's stem), and using a simplistic sheath model.

The pulsed PIP setup, shown in Fig. 3(b), uses an AWG (arbitrary waveform generator) to transmit sequential, preprogrammed waveform pulses to the PIP. A custom circuit board (RFIV board)⁷ isolates signals proportional to the pulsed voltage (V_RF) and currents (I_RF), which are measured by an oscilloscope in the time domain. Because we are using a single, broad-spectrum pulse to measure the PIP's impedance (instead of a frequency sweep), we are able to achieve significantly higher acquisition rates (>1 MHz) but at the cost of lower SNR.

The second important parameter is τ, which is the time between sequential pulses. The inverse of τ is the acquisition rate of the pulsed PIP system. The primary factor limiting the lower bound of τ is the time between the pulse and the settled reflected signal.^7,18 Without sufficient time between pulses, new outbound pulses would overlap with the reflection (or their ringdowns) from previous pulses. A method of addressing this is to install the RFIV board in close proximity to the PIP in order to reduce the time delay between transmitted and reflected signals.

The calibration process for the swept and pulsed PIP approaches effectively uses the same calibration steps but with some additional nuance. First, the truth measurements of the 3 RF standards are measured directly by the VNA. The standards then are connected to the SMA connector at cal. plane 2 and measured by the oscilloscope. The impedance measurement from the oscilloscope, using Eq. (18), and the impedance measurement from the VNA are fed into the python scikit-rf OnePort calibration library.⁴⁶ Using the apply_cal method from this library, we move the calibration plane from cal. plane 1 to cal. plane 2. Interpolation is likely required to ensure that the frequency bases for the two measurements match, as one measurement was made by the VNA and the other by the scope. The same approach described in Sec. III B is used to remove the effects of the probe stem and move the calibration plane from cal. plane 2 to cal. plane 3. When measuring the RF calibration standards, SNR can be improved by averaging over multiple pulses. As the precise value of ω_p is not typically known in advance, we often mitigate this unknown by calibrating and then operating with multiple values of ω_mp.

Because of the inherent trade-off between time resolution and SNR when using the pulsed method, careful analysis of the measurements is particularly important. There are several notable differences between the swept and pulsed methods.

The first is the careful selection of τ and ω_mp as discussed in Secs. III E and III F. The second is using windowing functions. Before applying Eq. (18) to the raw measurements of $V R F ( t )$ and $I R F ( t )$⁠, we typically apply a Hann window with a width on order of $10 2 σ m p$– $10 3 σ m p$ to each measured pulse to suppress the noise floor between the sequential pulses. Optionally, sequential pulses can also be averaged.

Figure 6(b) shows the calibrated Γ_pip measurement and fit resulting from the raw data in Fig. 6(a).

In this work, we are arguing that our historical PIP analysis has not been optimized for SNR, and this has artificially lowered the PIP's upper density limit, decreased measurement quality, and reduced acquisition rates. In this section, we use a Monte Carlo (MC) uncertainty analysis⁴⁷ to quantify improvements in SNR due to our updated model (Sec. II) and methods (Sec. III) and make recommendations for optimized PIP operation and analysis.

In summary, our MC uncertainty analysis takes the following form. (i) We develop a model for PIP measurements that includes Gaussian noise. (ii) We simulate a noisy PIP measurement for a fixed set of parameters and (iii) perform a fit to calculate its plasma properties. (iv) We repeat steps ii and iii a statistically significant number of times (e.g., 10⁴), which allows the Gaussian distribution in the noise to propagate to each plasma parameter. (v) We quantify the uncertainty in each plasma parameter by taking the standard deviation of each parameter across the multiple fits. We then repeat this analysis for each set of desired hyperparameters (e.g., $α ′$ and ω_mp). Throughout the remainder of this section, we detail.

To better justify these models, we next attempt to isolate an experimental, pulsed measurement of Γ_noise and compare it with our pulsed noise model [Eq. (23)]. Note that isolating a measurement of Γ_noise is difficult because Eq. (20) more realistically has the form $Γ meas = Γ model + Γ noise + Γ calibration _ error + Γ missing _ physics$⁠, where $Γ missing _ physics$ includes physics not included in the model (e.g., density gradients and a more realistic sheath model). To minimize $Γ missing _ physics$ and $Γ calibration _ error$⁠, we start with a freshly calibrated, pulsed measurement (Γ_meas) of the PIP in vacuum, instead of plasma. The measurement was performed with $ω m p / 2 π = 100$ MHz. We then subtract the fit of Γ_meas [using Eq. (2a)] from our measurement to get our experimental measurement of Γ_noise, which is shown in Fig. 8. To compare with the model, we replace the numerator in Eq. (23) with the fixed amplitude, $0.15 ( 1 + 1 j )$⁠, and the result is an envelope that appears to reasonably capture the noise measurement distribution, including the minimum of variance at ω_mp.

With the noisy PIP models established, the next step is to simulate them. For both swept and pulsed approaches, we define frequency, ω, as a uniform set of N discrete frequencies with step size $Δ ω = ( ω f − ω i ) / ( N − 1 )$⁠, where ω_i and ω_f are the initial and final frequencies of the sweep, respectively. Unless otherwise specified, we use the following as default values for calculating Γ_meas in the MC analysis: $ω p / 2 π = 100$ MHz, $ν ′ = 0.10 , t s ′ = 0.10 , r m = 0.5$ in. (12.7 mm), $ω f = ω p , ω i = Δ ω = ω f / N , N = 10 3 , α ′ = 10 − 4$⁠, and $ω m p = ω p$⁠. We choose these values because they are similar to our previous experimental measurements.

Next, we fit our simulated Γ_meas with our models (Γ_pip or Γ_diff) to provide measurements of density (n), electron damping (ν), and the sheath thickness (t_sh). For fitting to Z_pip, we convert Γ_meas to Z with Eq. (15). Figure 9 shows an example simulated measurement and fit.

In this and our previous work,⁷ we have alluded to several methods for extracting plasma properties from PIP measurements, which include fitting with Γ_pip (our recommended method), fitting with Z_pip, fitting with Γ_diff , and identifying the resonance at Im $( Γ diff ) = 0$ (for density only). Our MC analysis allows us to compare the uncertainties associated with each method, and Fig. 10 shows the results. Note that for the Im $( Γ diff ) = 0$ method, we identified the zero intercept by iteratively applying a low-pass, forward–backward, Butterworth filter to our noisy Γ_diff measurement with a decreasing corner frequency until a single zero intercept remained. For this analysis, $ω f = 1.5 ω p$⁠.

Figure 10 shows that, of the four methods, fitting with Γ_pip consistently provides the lowest uncertainty, by up to 2 orders of magnitude (for density and damping) and up to 4 orders of magnitude (for sheath thickness). Note that the Z_pip results are incomplete because low SNR caused fitting to become inconsistent above $α ′ ≳ 10 − 3$⁠.

In the case of higher damping (⁠ $ν ′ ≳$ 1.0), fitting with Γ_diff results in lower uncertainty than Γ_pip as shown in Fig. 11. This is because the magnitude of Im(Z_diff) is less dependent on $ν ′$ than Im(Z_pip) (see the discussion in Sec. II).

When fitting swept and pulsed measurements with Γ_pip, there are a number of notable experimental and post-processing hyperparameters that impact measurement uncertainty: N, $α ′$⁠, ω_mp, ω_i, and ω_f. To investigate their impact, we perform the MC uncertainty analysis across a range of values for each hyperparameter.

The $α ′$ and N dependencies are intuitive because uncertainty should decrease with lower $α ′$ (e.g., using more averaging) and with higher N (adding additional measurement points). Finally, we find the same relation in Eq. (26) for both swept and pulsed approaches.

The pulsed method introduces an additional parameter, ω_mp, that sets the frequency resolution of the Gaussian monopulse. Figure 13 shows the uncertainty results across a range of $ω m p / ω p$ and reveals that the uncertainties have a minimum at $ω m p ≈ 2 ω p$ and also that there is a steep increase in uncertainty at $ω m p ≳ 3 ω p$⁠. As ω_p is generally not precisely known before taking a measurement, we recommend choosing $ω m p ≲ ω p$⁠. These results are consistent across other values of $ν ′$ and $t ′ s h$⁠.

When using a VNA (i.e., the swept approach), we have independent control of ω_i and ω_f. Figure 14 shows the uncertainty analysis of $σ ′ n$ while varying both frequencies, and the two resonant frequencies, $( ω ′ − , ω ′ + )$ = (0.318, 0.994), are indicated. The results reveal two minima in $σ ′ n$⁠.

The less prominent (higher) minimum occurs around (⁠ $ω ′ i$⁠, $ω ′ f$⁠) $≈$ (1.0, 1.0) but for σ_n only. This is somewhat intuitive because $ω ′ +$ [Eq. (11)] has a strong dependence on ω_p, which relates to density [Eq. (1)].

The more prominent (lower) minimum is roughly bounded by the rectangular region (⁠ $ω ′ i < ω ′ −$ and $ω ′ f > ω ′ −$⁠) for $σ ′ n$⁠. Analysis of $σ ′ ν$ and $σ ′ t$ shows this same minima. The location and depth of this minima indicates that the best fit occurs when the measured sweep straddles $ω −$⁠. This strongly suggests that $ω −$ sets the frequency resolution of the PIP rather than ω_p as previously believed. We believe that this is likely due to the fact that $ω −$ is the location of Im(Γ_pip) = 0, the minimum of Re(Γ_pip), and is surrounded by several inflection points and the largest magnitude of curvature (see Fig. 9). Arguably, it is fitting to these features, in addition to normalizing with Γ, that leads to the most accurate results.

This finding has two interesting implications. (i) This suggests that the community has been spectrally over-resolving PIP measurements. Resolving only up to $ω −$ (and not up to ω_p) may effectively extend the PIP's upper density limit. In this example, $ω −$ is roughly half an order of magnitude less than ω_p, which relates to a full order of magnitude in density resolution [Eq. (1)]. Note that $ω −$ is strongly dependent on $t ′ s h$ [Eq. (10)], and therefore this potential benefit only applies when the sheath is thin. (ii) Further reducing $ω −$ by deliberately decreasing $t ′ s h$ [Eq. (4)] has the potential to increase the PIP's density limit. Possible methods to do this include using a larger probe radius and electrically DC biasing the PIP at or near the plasma potential.

In summary, we recommend the following best practices to maximize PIP SNR. We recommend fitting Γ_pip to measurements if $ν ′ < 1$ but fitting with Γ_diff if $ν ′ > 1$⁠. For N with the swept method, we recommend using a reasonably large number (e.g., 10³ or 10⁴) that still provides an adequate acquisition rate. While we do not have direct control over noise, $α ′$⁠, we can use techniques such as averaging to reduce it. For ω_mp, we nominally recommend $ω m p = ω p$ and also recommend using multiple values to safeguard against an uncertain ω_p.

An interesting result from this section is that $ω −$ appears to be the frequency that sets the upper density limit of the PIP and not ω_p as previously believed. This effectively extends the PIP's density limit, particularly when the sheath is “thin.” Therefore, when choosing ω_i and ω_f, we recommend values that straddle $ω −$ while also adjusting these values appropriately to safeguard against variability in $ω −$⁠.

The goal of this work has been to provide an approach to PIP analysis that increases measurement SNR in order to improve overall measurement quality, extend the upper density limit, increase acquisition rates, and allow the probe to be more accessible in the laboratory environment. To achieve this, we presented our PIP design and model, our approach to PIP operation, calibration, and analysis, and finally a Monte Carlo uncertainty analysis that identified a number of operational and analysis steps that collectively improve SNR by multiple orders of magnitude. We additionally provided evidence that the sheath resonance (and not the plasma frequency as previously believed) effectively sets the PIP's upper density limit, which has implications in additionally extending the PIP's upper density limit.

In this work, we have also identified several paths for future work. These include identifying the new value of the PIP's upper density limit and actively reducing the normalized sheath thickness to further increase the upper density limit. While this work focuses on DC discharges, the PIP is also applicable to RF plasmas, with sources operating at ω_RF. Specifically, we think that operating the PIP with the pulsed method at $ω m p ≫ ω R F$ in conjunction with high-pass filters (⁠ $ω R F < ω filter ≪ ω m p$⁠) would allow the PIP to investigate RF-oscillating sheaths without interference from the RF source. Alternatively, the PIP could be operated with continuous sweeps to capture time-averaged sheaths in an RF plasma. Finally, we believe that PIPs are an ideal candidate for educational plasma laboratories, particularly with the availability of commercial VNAs on the internet that are under $100 and capable of resolving frequencies >1 GHz.

This work was supported by NRL base funding and M.C.P was specifically funded by the 2023-2024 Karle Fellowship. J.W.B. wishes to thank his mentors, Mike McDonald and Erik Tejero, for their assistance, without which this work would not be possible. J.W.B. wishes to thank Jeffrey Bynum for his assistance with technical drawings. J.W.B. also wishes to thank his late teacher, Dr. Hugh W. Coleman,⁴⁷ for teaching him the importance of quantifying measurement uncertainty.

The authors have no conflicts to disclose.

J. W. Brooks: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). M. C. Paliwoda: Formal analysis (equal); Investigation (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are openly available in Zenodo, Ref. 48.

Figure 15 shows an uncalibrated PIP measurement and how the measurement visually changes as each calibration step is applied. Cal. plane 1 is the raw, uncalibrated PIP measurement. Its “oscillatory” appearance is due to the phase shift caused by the length of transmission line between the VNA and the PIP-monopole. Cal. plane 2 is the partially calibrated PIP measurement that still contains contributions from the PIP and its stem and shows a small phase shift (⁠ $< 2 π$⁠). Cal. plane 3 shows the fully calibrated measurement, where both $ω ±$ are apparent. The calibration process is detailed in Sec. III C.