# Uncertainty analysis of the plasma impedance probe

J. W. Brooks<EMAIL_ADDRESS>U.S. Naval Research Laboratory,
Washington, D.C., USA  M. C. Paliwoda  U.S. Naval Research Laboratory,
Washington, D.C., USA, Karle Fellow

###### Abstract

A plasma impedance probe (PIP) is a type of in-situ, radio-frequency (RF)
probe that is traditionally used to measure plasma properties (e.g. density)
in low-density environments such as the Earth’s ionosphere. We believe that
PIPs are underrepresented in laboratory settings, in part because PIP
operation and analysis has not been optimized for signal-to-noise ratio (SNR),
reducing the probe’s accuracy, upper density limit, and acquisition rate. This
work presents our efforts in streamlining and simplifying the PIP design,
model, calibration, and analysis for unmagnetized laboratory plasmas, in both
continuous and pulsed PIP operation. The focus of this work is a Monte Carlo
uncertainty analysis, which identifies operational and analysis procedures
that improve SNR by multiple orders of magnitude. Additionally, this analysis
provides evidence that the sheath resonance (and not the plasma frequency as
previously believed) sets the PIP’s upper density limit, which likely provides
an additional method for extending the PIP’s density limit.

Plasma impedance probe, RF probe, time-resolved, high-speed, plasma density,
plasma sheath, electron damping, electron collisions, Monte Carlo uncertainty
analysis

##  I Introduction

The Plasma Impedance Probe (PIP)[1, 2, 3, 4] is an in-situ plasma diagnostic
that is underutilized when compared with the Langmuir Probe (LP)[5, 6]. Both
probes share much in common: both are metal electrodes, often have similar
geometries (e.g. planar and spherical), and form sheaths when placed in a
plasma. When the sheath is large with respect to the probe size, both probes
struggle with accuracy [7, 8]. When electrically biased, both provide a
measurement of the coupled sheath-plasma impedance, $Z$, around the probes,
and fitting sheath-plasma models to these measurements provide many of the
same plasma properties. These properties include: the electron plasma
frequency,

$\begin{split}\omega_{p}=\sqrt{\frac{ne^{2}}{m_{e}\epsilon_{0}}},\end{split}$
(1)

electron density ($n$)[4], temperature ($T$)[9, 10], Debye length
($\lambda_{D}$), sheath thickness ($t_{sh}$)[7, 11], and plasma potential
($V_{p}$)[12]. Historically, PIPs are primarily used to measure density, but
have also measured electron damping ($\nu$). However, isolating the species-
dependent collisional and collisionless terms within $\nu$ is still an active
research topic in the PIP community[13, 14, 15].

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 ($V_{exc}>k_{b}T/e$) 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% to 50% [8]. In contrast, PIPs [4, 7] are
excited across a range of frequencies ($\omega_{exc}$) above the ion-plasma
frequency and near the electron-plasma frequency ($\omega_{p}$); therefore
PIPs measure the complex impedance spectra, $Z(\omega)$, and ignore ion
contributions. Additionally, PIPs are excited at relatively lower 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.

Physically, PIPs operate by measuring the frequency response of the plasma [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, $\epsilon_{sh}(\omega)$ and $\epsilon_{p}(\omega)$, respectively,
which are dependent on their respective plasma properties. These dielectrics
electrically couple with the PIP and modify the PIP’s electrical impedance,
$Z_{PIP}(\omega)$. An example of this coupling is it introduces two resonances
($\omega_{\pm}$) into the PIP’s impedance. Measuring $Z_{PIP}(\omega)$ 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, 16, 14, 15], 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 setup/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 [17, 18, 7], 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 under-representation 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 $Z_{PIP}(\omega)$ 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
$\omega_{p}^{2}$ (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[19, 20,
21, 22, 23, 24, 17], satellites[25, 26], and on the International Space
Station [27, 28, 29]. To a lesser extent, PIPs have also been used in lower-
density laboratory plasmas, including DC discharges [30, 7], Hall thruster
plumes [31, 16], and plasma processing applications [32] 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 Section II, we discuss our updated PIP monopole design and accompanying
analytical model. In Section III, we discuss our streamlined calibration,
operation, and analysis for both the continuous and pulsed PIP methods. In
Section IV, we perform a Monte Carlo uncertainty analysis that both quantifies
the improvement in SNR from the previous two sections and provides
recommendations for optimized PIP operation and analysis.

##  II Design and models

To extract meaningful results from a measurement of $Z_{PIP}$, we require a
physical model. Below, we A) introduce our PIP-monopole design which allows
for easier modeling and calibration, B) derive the accompanying PIP models for
unmagnetized plasmas, and C) discuss their assumptions and limitations.

###  II.1 Design

For unmagnetized plasmas, we recommend the spherical monopole design as it is
non-directional (unlike a dipole [33]) and is simpler to both construct and
model. Figure 1 shows a 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. The
head was a two-piece, machined SS316 hollow sphere, and the inner conductor
was attached with set screws. The head for our previous design [7] was a
drilled aluminum sphere that was press fit onto the inner conductor.

Figure 1:  a) PIP-monopole. b) CAD drawing. Calibration planes 2 and 3 are
discussed in Sec. III.2.

###  II.2 Models

The model presented below is a reformulation of Blackwell’s model [4] and
closely follows our previous derivation [7].

Similar to previous work [34, 35, 4, 36, 7], we begin our model by using a
lumped-element circuit framework and modeling the PIP’s head and surrounding
environment as one or more 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 $Z(\omega)=1/j\omega C$, and because of
the spherical geometries, we use spherical-shell capacitor models, which have
capacitance $C=4\pi\epsilon(1/r_{1}-1/r_{2})^{-1}$ with $r_{1}$ and $r_{2}$
being the radii of the inner and outer electrodes, respectively.

When no plasma is present (Figure 2a), we model the vacuum around the PIP as a
single capacitor, where the inner electrode is the PIP-monopole’s head with
radius $r_{m}$, the outer electrode is the grounded vacuum chamber wall with
approximate radius $r_{vc}$, and dielectric $\epsilon=\epsilon_{0}$.
Recognizing that $r_{m}/r_{vc}\approx 0$, the PIP’s vacuum impedance is
therefore

$\displaystyle Z_{vac}=$ $\displaystyle\frac{1}{4\pi
j\omega\epsilon_{0}r_{m}}.$ (2a) We further simplify this expression to
$\displaystyle Z^{\prime}_{vac}$ $\displaystyle=\frac{1}{j\omega^{\prime}}$
(2b)

by introducing a normalized impedance, $Z^{\prime}\equiv Z/Z_{m}$, a
characteristic impedance of the monopole,

$\begin{split}Z_{m}(r_{m},\omega_{p})\equiv\frac{1}{4\pi\epsilon_{0}r_{m}\omega_{p}},\end{split}$
(3)

and a normalized frequency, $\omega^{\prime}\equiv\omega/\omega_{p}$.

Figure 2:  Two circuit models of the PIP-monopole. a) The vacuum model uses a
single vacuum-filled capacitor. b) The plasma model uses two capacitors in
series: a vacuum dielectric for the sheath and a plasma dielectric for the
plasma ($\epsilon=\epsilon_{0}\epsilon_{p}$).

When plasma is present (Figure 2b), 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, $r_{sh}$.

Because the sheath’s density is much less than the bulk plasma, we approximate
its dielectric as a homogeneous vacuum (e.g. $\epsilon\approx\epsilon_{0}$)
with a normalized sheath thickness,

$\begin{split}t^{\prime}_{sh}\equiv\frac{r_{sh}-r_{m}}{r_{sh}}=\frac{t_{sh}}{r_{sh}}.\end{split}$
(4)

The sheath’s impedance is therefore

$\begin{split}Z^{\prime}_{sh}&=\frac{1}{j\omega^{\prime}}t^{\prime}_{sh}\end{split}$
(5)

where we have similarly normalized the impedance.

For the bulk plasma, we model its dielectric as a plasma that is homogeneous,
cold, collisional, and unmagnetized. Specifically,
$\epsilon=\epsilon_{0}\epsilon_{p}$, where

$\begin{split}\epsilon_{p}&=\left(1-\frac{1}{\omega^{\prime}\left(\omega^{\prime}-j\nu^{\prime}\right)}\right),\end{split}$
(6)

and $\nu^{\prime}\equiv\nu/\omega_{p}$ is a normalized electron damping rate.
The inclusion of $j\nu^{\prime}\omega^{\prime}$ effectively introduces a
damped, inductive term into this dielectric [4]. The bulk plasma’s impedance
is

$\begin{split}Z^{\prime}_{pl}&=\frac{1}{j\omega^{\prime}}\frac{(1-t^{\prime}_{sh})}{\epsilon_{p}}.\end{split}$
(7)

The total PIP impedance is the sum of the sheath and bulk plasma impedances,

$\begin{split}Z^{\prime}_{pip}&=Z^{\prime}_{sh}+Z^{\prime}_{pl}\\\
&=\frac{1}{j\omega^{\prime}}\left(t^{\prime}_{sh}+\frac{\left(1-t_{sh}^{\prime}\right)}{\epsilon_{p}}\right)\\\
&=\frac{\nu^{\prime}\omega^{\prime}(1-t^{\prime}_{sh})-j(\omega^{\prime
2}-\omega^{\prime 2}_{+})(\omega^{\prime 2}-\omega^{\prime
2}_{-})}{\omega^{\prime}\left(\nu^{\prime 2}\omega^{\prime 2}+(\omega^{\prime
2}-1)^{2}\right)}.\end{split}$ (8)

In electrical systems, resonances occur where Im($Z$)=0, and Eq. 8 has two
positive resonances [4, 7],

$\begin{split}\omega^{\prime}_{\pm}&=\sqrt{\frac{1+t^{\prime}_{sh}-\nu^{\prime
2}\pm\sqrt{\left(1+t^{\prime}_{sh}-\nu^{\prime
2}\right)^{2}-4t^{\prime}_{sh}}}{2}}.\end{split}$ (9)

We refer to the lower resonance ($\omega_{-}$) as the sheath resonance because
it disappears as the sheath vanishes,

$\begin{split}\lim_{t^{\prime}_{sh}\rightarrow 0}\omega_{-}=0,\end{split}$
(10)

and we refer to $\omega_{+}$ as the damped-plasma resonance because it
converges to $\omega_{p}$ as damping vanishes,

$\begin{split}\lim_{\nu^{\prime}\rightarrow
0}\omega_{+}=\omega_{p}.\end{split}$ (11)

Further solving Eq. 9, we find that both resonances merge and disappear with
large damping [37, 7],

$\begin{split}\nu^{\prime}\geq 1-\sqrt{t^{\prime}_{sh}}.\end{split}$ (12)

We also create a second PIP model by subtracting the vacuum impedance from Eq.
8,

$\begin{split}Z^{\prime}_{diff}&=Z^{\prime}_{pip}-Z^{\prime}_{vac}\\\
&=\frac{1}{j\omega^{\prime}}\left(\frac{1}{\epsilon_{p}}-1\right)\left(1-t_{sh}^{\prime}\right)\\\
&=\frac{\nu^{\prime}\omega^{\prime}(1-t^{\prime}_{sh})-j(1-t^{\prime}_{sh})(\omega^{\prime
2}-1)}{\omega^{\prime}\left(\nu^{\prime 2}\omega^{\prime 2}+(\omega^{\prime
2}-1)^{2}\right)}.\end{split}$ (13)

Both Eqs. 8 and 13 have the same real component. However, Eq. 13 has a
resonance at

$\begin{split}\omega_{diff}=\omega_{p},\end{split}$ (14)

which isolates $\omega_{p}$ from $\nu^{\prime}$ and $t^{\prime}_{sh}$.

To better understand $Z^{\prime}_{pip}$ and $Z^{\prime}_{diff}$, we plot both
in Figure 3 for $t_{sh}^{\prime}=0.1$ and $\nu^{\prime}=0.4$. We choose these
values because i) they are values we have encountered experimentally and ii)
the three resonances ($\omega_{\pm}$ and $\omega_{diff}$) 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].

Figure 3:  The real and imaginary components of both models
($Z^{\prime}_{pip}$ and $Z^{\prime}_{diff}$) are shown for
$t_{sh}^{\prime}=0.1$ and $\nu^{\prime}=0.4$. The three resonances are
indicated.

Above, we 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 $\nu^{\prime}$ and is
therefore its measurements are less susceptible to noise when damping is high
($\nu^{\prime}$ $\scriptstyle\gtrsim$ 1). This dependency on $\nu^{\prime}$ is
apparent when comparing Eq. 8 and Eq. 13, while considering Eq. 9. The
resonant frequencies ($\omega_{\pm}$ ) are heavily dependent upon
$\nu^{\prime}$, which are present in $Z_{pip}$ but not in $Z_{diff}$. At high
damping, $\omega_{+}$ and $\omega_{-}$ merge and disappear per Eq. 12, while
$\omega_{p}$ is isolated from $\nu^{\prime}$ and $t^{\prime}_{sh}$ per Eq. 14.
Second, we resolved $Z_{diff}$ using planar and cylindrical capacitors (in
addition to spherical), and all three results provide the same
$\omega_{diff}=\omega_{p}$ 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 $\omega_{p}$ by locating Im$(Z_{diff})=0$ in a measurement alone
(i.e. without the need of a model).

Despite building our models ($Z_{vac}$, $Z_{pip}$, and $Z_{diff}$) in
electrical impedance ($Z$), we instead prefer doing analysis with the
reflection coefficient [38, 39],

$\begin{split}\Gamma(\omega)=\frac{Z(\omega)-Z_{0}}{Z(\omega)+Z_{0}}.\end{split}$
(15)

In this expression, $Z_{0}$ is the characteristic impedance of the
transmission lines, typically 50 $\Omega$. We convert $Z$ to $\Gamma$ using
Eq. 15 for both PIP measurements (vacuum and plasma) and our three PIP models,
which results in $\Gamma_{vac}$, $\Gamma_{pip}$, and $\Gamma_{diff}$. We
prefer $\Gamma$ because it results in lower uncertainty (see Sec. IV).
Converting from $Z$ to $\Gamma$ does not change the location of the three
resonances: $\omega_{\pm}$ and $\omega_{diff}$.

###  II.3 PIP limitations and errors

##  III Methods

With the models established in Sec. II, we next turn to PIP setup,
calibration, and analysis for both swept and pulsed methods. Both setups,
diagrammed in Figure 4, have much in common despite apparent differences, and
we discuss both below.

Figure 4:  Diagram of the PIP setup for: a) Swept PIP operation using a VNA,
and b) pulsed PIP operation using an AWG and oscilloscope.

###  III.1 Swept setup

The most notable feature of the swept PIP setup (Figure 4a) is that it uses a
VNA as both the stimulus and the measuring device, which simplifies PIP
operation. Typical VNA acquisition rates are relatively slow and are
determined by several factors, including hardware limits and the amount of
averaging required to achieve a reasonable SNR. As a rough approximation, a
higher-end VNA will have a maximum sweep rate of order 100 Hz assuming no
averaging and measuring $10^{3}$ samples [40]. A VNA operates by transmitting
low-voltage frequency sweeps to the PIP through a series of transmission lines
(TLs). Upon arriving at the PIP, the voltage is attenuated, partially
reflected, and phase shifted due, in part, to the dielectric properties of the
sheath and plasma regions. The VNA measures both the reflected ($V_{refl}$) to
transmitted ($V_{trans}$) voltages, and the ratio is the definition of the
reflection coefficient [38, 39],

$\begin{split}\Gamma(\omega)\equiv\frac{V_{refl}(\omega)}{V_{trans}(\omega)}.\end{split}$
(16)

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 [41].

###  III.2 Swept calibration

The TL’s 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 Figures 1 and 4.

To assist with calibration, we intentionally built our PIP-monopole (Sec.
II.1) 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
[42, 7] 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[43].

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 [38, 7] 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 App. A.

App. B shows an example calibration using both steps.

###  III.3 Swept analysis

After acquiring and calibrating a $\Gamma_{pip}$ measurement, the next step is
to fit the model to the measurement in order to solve for the model’s three
unknowns: $\omega_{p}$, $\nu$, and $t_{sh}$. This requires several steps.
First, we normalize $\omega_{p}$ in Eq. 8 so that its value is roughly the
same order as $\nu^{\prime}$ and $t^{\prime}_{sh}$, and therefore the fitting
routine weights each parameter more equally. Second, we convert our $Z_{pip}$
model (Eq. 8) into $\Gamma_{pip}$ 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 $\Gamma_{pip}$ model to our
$\Gamma_{pip}$ 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 $\Gamma$. We use the
same procedure above when fitting with $Z_{diff}$.

Figure 5 shows an example of the $\Gamma_{pip}$ model fit to a calibrated,
swept $\Gamma_{pip}$ 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.

Figure 5:  An example swept measurement is fit with the $\Gamma_{pip}$ model.
The calculated plasma parameters are $\omega_{p}/2\pi=373$ MHz, $n=1.73\cdot
10^{15}$ m-3, $\nu^{\prime}=0.151$ ($\nu=353\cdot 10^{6}$ $s^{-1}$), and
$t^{\prime}_{sh}=0.126$ ($t_{sh}=0.91$ mm).

###  III.4 Pulsed setup

The pulsed PIP setup, shown in Figure. 4b, 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.

For our work, the AWG transmits a Gaussian monopulse,

$\begin{split}g_{mp}(t;a,\sigma_{mp})=\frac{at}{\sigma_{mp}^{2}}\text{exp}\left(-\frac{1}{2}\left(\frac{t}{\sigma_{mp}}\right)^{2}\right),\end{split}$
(17)

which is the time derivative of a Gaussian distribution. Here, $a$ and
$\sigma_{mp}$ are the amplitude and width of the Gaussian distribution,
respectively. Figure 6a shows an example measurement of both measured outbound
and reflected monopulses for both $V_{RF}$ and $I_{RF}$. The measured
impedance of a single pulse at the scope is

$\begin{split}Z(\omega)=\frac{\text{FFT}\left\\{V_{RF}(t)\right\\}}{\text{FFT}\left\\{I_{RF}(t)\right\\}}\end{split}$
(18)

where FFT$\left\\{\,\right\\}$ is the fast Fourier transform. $Z(\omega)$ is
then converted to $\Gamma(\omega)$ with Eq. 15. Figure 6b shows the calibrated
$\Gamma_{pip}$ measurement and subsequent fit (discussed in Sec. III.6).

Figure 6:  An example pulsed measurement and fit using the $\Gamma_{pip}$
model. a) Shows the raw oscilloscope measurements of $I_{RF}$ and $V_{RF}$.
The outbound pulses occur at 0 ns, and the reflected pulses, due to line
length, occur 25 ns later. b) Shows the processed and calibrated
$\Gamma_{pip}$ measurement and fit. Fit results: $\omega_{p}/2\pi=194$ MHz,
$n=4.67\cdot 10^{14}$ m-3, $\nu^{\prime}=0.185$ ($\nu=225\cdot 10^{6}$ s-1),
and $t^{\prime}_{sh}=0.148$ ($t_{sh}=1.60$ mm).

The monopulse has two important parameters. The first is the inverse of the
width, $\omega_{mp}\equiv 1/\sigma_{mp}$, which sets both the center frequency
and frequency resolution of the pulse. This concept is most easily understood
by plotting the monopulse’s magnitude spectrum,

$\begin{split}\left|\mathcal{F}\left\\{g_{mp}\right\\}\right|=a\frac{\omega}{\omega_{mp}}\,\text{exp}\left(\text{-}\frac{1}{2}\left(\frac{\omega}{\omega_{mp}}\right)^{2}\right),\end{split}$
(19)

in Figure 7. Here, $\mathcal{F}\left\\{\,\right\\}$ is the Fourier transform.
This plot shows that Eq. 19 is a distribution with a peak at
$\omega=\omega_{mp}$ and a finite width. SNR is roughly defined as the ratio
of this distribution to a noise floor, and SNR $\gtrsim 1$ is roughly required
to resolve the monopulse. If the noise floor were, for example, 10% of the
spectrum’s peak, then the resolvable frequency range would be approximately
$0.06<\omega/\omega_{mp}<2.8$ (shown in Figure 7) with the highest SNR at
$\omega/\omega_{mp}=1$. By default, we typically choose $\omega_{mp}$
$\scriptstyle\lessapprox$ $\omega_{p}$ as discussed further in Sec. IV.

Figure 7:  Magnitude spectrum of the Gaussian monopulse, normalized to its
peak amplitude. The monopulse’s resolvable frequency range is roughly defined
as the frequencies where this distribution is greater than the noise floor.
The highest SNR occurs at the peak ($\omega/\omega_{mp}=1$) and falls off on
either side.

The second important parameter is $\tau$, which is the time between sequential
pulses. The inverse of $\tau$ is the acquisition rate of the pulsed PIP
system. The primary factor limiting the lower bound of $\tau$ is the time
between the pulse and the settled reflected signal [18, 7]. 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.

###  III.5 Pulsed calibration

The calibration process for the swept and pulsed PIP approaches effectively
use the same two calibration steps but with some additional nuance. The first
is that the 1st cal. plane is now at the oscilloscope (Figure 4b). The second
is that the measurements of the 3 calibration standards are done differently:
the “truth” measurements are made with a VNA but the 2nd calibration plane
measurements are made with the pulsed hardware. Interpolation is likely
required to ensure that the frequency bases for the two measurements match.
When measuring the RF calibration standards, SNR can be improved by averaging
over multiple pulses. As the precise value of $\omega_{p}$ is not typically
known in advance, we often mitigate this unknown by calibrating and then
operating with multiple values of $\omega_{mp}$.

###  III.6 Pulsed analysis

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

The first is the careful selection of $\tau$ and $\omega_{mp}$ as discussed in
Secs. III.4 and III.5. The second is using windowing functions. Before
applying Eq. 18 to the raw measurements of $V_{RF}(t)$ and $I_{RF}(t)$, we
typically apply a Hann window with a width on order of $10^{2}\,\sigma_{mp}$
to $10^{3}\,\sigma_{mp}$ to each measured pulse to suppress the noise floor
between the sequential pulses. Optionally, sequential pulses can also be
averaged.

Figure 6b shows the calibrated $\Gamma_{pip}$ measurement and fit resulting
from the raw data in Figure 6a.

##  IV Uncertainty analysis

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 analysis [44] 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.

###  IV.1 Methodology

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. $10^{4}$), 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. $\alpha^{\prime}$,
$\omega_{mp}$, etc). Below we detail each step.

In developing our model for a noisy PIP measurement, we start with the
definition of the reflection coefficient (Eq. 16) which is the ratio of the is
the reflected signal ($V_{refl}$) from the PIP to the transmitted
($V_{trans}$) signal to the PIP. We assume that only $V_{refl}$ is susceptible
to noise as $V_{trans}$ is measured at the VNA before being transmitted.
Therefore, a noisy measurement of $\Gamma$ has the form,

$\begin{split}\Gamma_{meas}(\omega)&=\frac{V_{refl}(\omega)+V_{noise}(\omega)}{V_{trans}(\omega)}\\\
&=\Gamma_{ideal}(\omega)+\Gamma_{noise}(\omega),\end{split}$ (20)

which includes the ideal PIP measurement ($\Gamma_{ideal}\equiv
V_{refl}/V_{trans}$) plus noise ($\Gamma_{noise}\equiv V_{noise}/V_{trans}$).
For $\Gamma_{ideal}$, this analysis uses Eqs. 8 or 13. This analysis models
$V_{noise}$, the numerator of $\Gamma_{noise}$, as a complex noise floor with
the form,

$\begin{split}V_{noise}(\omega;\alpha)&=\mathcal{N}(0,\alpha)e^{jU(0,2\pi)}.\end{split}$
(21)

This expression’s amplitude is a Gaussian or normal distribution,
$\mathcal{N}(0,\alpha)$, with a zero mean and standard deviation, $\alpha$.
Eq. 21 has a uniformly random phase shift, $U(0,2\pi)$, between 0 and $2\pi$
with respect to $V_{trans}$. For the swept method, we assume
$V_{trans}(\omega)=a$, i.e. the transmitted signal has a constant amplitude,
and therefore the noisy swept PIP model is

$\begin{split}\Gamma_{noise}(\omega;\alpha^{\prime})=\mathcal{N}(0,\alpha^{\prime})e^{jU(0,2\pi)},\end{split}$
(22)

where $\alpha^{\prime}\equiv\alpha/a$ is our adjustable noise parameter and
$1/\alpha^{\prime}$ is effectively SNR. For the pulsed method, we assume that
$V_{trans}(\omega)$ is a Gaussian monopulse and therefore equal to Eq. 19. The
noisy pulsed PIP model is therefore

$\begin{split}\Gamma_{noise}(\omega;\alpha^{\prime},\omega_{mp})=\frac{\mathcal{N}(0,\alpha^{\prime})e^{jU(0,2\pi)}}{\frac{\omega}{\omega_{mp}}\text{exp}\left(-\frac{1}{2}\left(\frac{\omega}{\omega_{mp}}\right)^{2}\right)}.\end{split}$
(23)

Note that Eq. 22 is dependent on a single parameter, $\alpha^{\prime}$, and
Eq. 23 is dependent on two: $\alpha^{\prime}$ and $\omega_{mp}$.

With the noisy PIP models established, the next step is to simulate them. For
both swept and pulsed approaches, we define frequency, $\omega$, as a uniform
set of $N$ discrete frequencies with step size,
$\Delta\omega=(\omega_{f}-\omega_{i})/(N-1)$, where the $\omega_{i}$ and
$\omega_{f}$ are the initial and final frequencies of the sweep, respectively.
Unless otherwise specified, we use the following as default values for
calculating $\Gamma_{meas}$ in the MC analysis: $\omega_{p}/2\pi=100$ MHz,
$\nu^{\prime}=0.10$, $t_{s}^{\prime}=0.10$, $r_{m}=0.5$ inch (12.7 mm),
$\omega_{f}=\omega_{p}$, $\omega_{i}=\Delta\omega=\omega_{f}/N$, $N=10^{3}$,
$\alpha^{\prime}=10^{-4}$, and $\omega_{mp}=\omega_{p}$. We choose these
values because they are similar to our previous experimental measurements.

Next, we fit our simulated $\Gamma_{meas}$ with our models ($\Gamma_{pip}$ or
$\Gamma_{diff}$) to provide measurements of density ($n$), electron damping
($\nu$), and the sheath thickness ($t_{sh}$). For fitting to $Z_{pip}$, we
convert $\Gamma_{meas}$ to $Z$ with Eq. 15. Figure 8 shows an example
simulated measurement and fit.

Figure 8:  Example of a single simulation of a noisy, swept PIP measurement
($\Gamma_{meas}$) with fit. $N=316$ and $\alpha^{\prime}=10^{-1}$.

Next, we repeat the previous two steps $M=10^{4}$ times to generate $M$ fit
values for $n$, $\nu$, and $t_{sh}$. For each, we calculate the standard
deviation ($\sigma$) and divide by their ideal value to get the normalized
uncertainty, $\sigma^{\prime}$. For density, this takes the form,

$\begin{split}\sigma^{\prime}_{n}=\frac{\sigma_{n}}{n_{ideal}},\end{split}$
(24)

where $n_{ideal}$ is the ideal density value used to generate
$\Gamma_{ideal}$. In addition to uncertainty, we also calculate a normalized
bias error, $\beta^{\prime}$, for each plasma parameter. For density, this
would be

$\begin{split}\beta^{\prime}_{n}=\frac{\left<n\right>}{n_{ideal}}\end{split}$
(25)

where $\left<n\right>$ is the average density from the fit results. For our
work, $\beta^{\prime}$ is consistently multiple orders of magnitude smaller
than $\sigma^{\prime}$, and we therefore focus our analysis exclusively on
$\sigma^{\prime}$.

###  IV.2 Comparing analysis methods

In this and our previous work [7], we have alluded to several methods for
extracting plasma properties from PIP measurements, which include: fitting
with $\Gamma_{pip}$ (our recommended method), fitting with $Z_{pip}$, fitting
with $\Gamma_{diff}$, and identifying the resonance at Im$(\Gamma_{diff})=0$
(for density only). Our MC analysis allows us to compare the uncertainties
associated with each method, and Figure 9 shows the results. Note that for the
Im$(\Gamma_{diff})=0$ method, we identified the zero intercept by iteratively
applying a low-pass, forward-backward, Butterworth filter to our noisy
$\Gamma_{diff}$ measurement with a decreasing corner frequency until a single
zero intercept remained. For this analysis, $\omega_{f}=1.5\omega_{p}$.

Figure 9:  Uncertainty results for four analysis methods, as a function of
$\alpha^{\prime}$. Fitting with $\Gamma_{pip}$ results in the lowest
uncertainty in all three plasma properties ($n$, $\nu^{\prime}$, and
$t^{\prime}$) compared with the other three methods.

Figure 9 shows that, of the four methods, fitting with $\Gamma_{pip}$
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 $\alpha^{\prime}$
$\scriptstyle{\gtrsim}$ $10^{-3}$.

In the case of higher damping ($\nu^{\prime}$ $\scriptstyle{\gtrsim}$ $1.0$),
fitting with $\Gamma_{diff}$ results in lower uncertainty than $\Gamma_{pip}$
as shown in Figure 10. This is because the magnitude of Im($Z_{diff}$) is less
dependent on $\nu^{\prime}$ than Im($Z_{pip}$) (see the discussion in Sec.
II).

Figure 10:  When damping is large ($\nu^{\prime}\gtrsim 1$), fitting with
$\Gamma_{diff}$ results in lower uncertainties than fitting with
$\Gamma_{pip}$.

###  IV.3 Improving SNR in $\mathbf{\Gamma_{pip}}$ analysis.

When fitting either swept and pulsed measurements with $\Gamma_{pip}$, there
are a number of notable experimental and post-processing hyperparameter that
impact measurement uncertainty: $N$, $\alpha^{\prime}$, $\omega_{mp}$,
$\omega_{i}$, and $\omega_{f}$. To investigate their impact, we perform the MC
uncertainty analysis across a range of values for each hyperparameter.

Figure 11 shows the swept MC analysis for $N$ and $\alpha^{\prime}$ with
several notable results. First, the magnitude of the three uncertainties are
consistently ordered: $\sigma^{\prime}_{n}$ $\scriptstyle{\lesssim}$
$\sigma^{\prime}_{t}$ $\scriptstyle{\lesssim}$ $\sigma^{\prime}_{\nu}$.
Second, the uncertainties are linear in the log-log plot, meaning that each
takes the form of a power function ($\sigma^{\prime}$ $\propto$ $x^{b}$) where
$b$ is a scalar and $x$ is either $\alpha^{\prime}$ or $N$. We performed fits
for all three plasma parameters and found that the exponent, $b$, is 1.00 and
-0.50 for $\alpha^{\prime}$ and $N^{\prime}$, respectively. We also determined
that these dependencies were independent of each other, and therefore the
uncertainties for $\sigma_{n}^{\prime}$, $\sigma_{\nu}^{\prime}$, and
$\sigma_{t}^{\prime}$ are all proportional to

$\begin{split}\sigma^{\prime}\propto\frac{\alpha^{\prime}}{\sqrt{N}}.\end{split}$
(26)

The $\alpha^{\prime}$ and $N$ dependencies are intuitive because uncertainty
should decrease with lower $\alpha^{\prime}$ (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.

Figure 11:  MC uncertainty analysis provides the uncertainty results of $n$,
$\nu^{\prime}$, and $t^{\prime}$ for variable $\alpha^{\prime}$, and $N$,
respectively, while holding the other values constant. Power function fits
show good agreement for most of the domains.

The pulsed method introduces an additional parameter, $\omega_{mp}$, that sets
the frequency resolution of the Gaussian monopulse. Figure 12 shows the
uncertainty results across a range of $\omega_{mp}/\omega_{p}$ and reveals
that the uncertainties have a minimum at $\omega_{mp}\approx 2\,\omega_{p}$
but also that there is a steep increase in uncertainty at $\omega_{mp}\gtrsim
3\,\omega_{p}$. As $\omega_{p}$ is generally not precisely known before a
taking a measurement, we recommend choosing $\omega_{mp}\lesssim\omega_{p}$.
These results are consistent across other values of $\nu^{\prime}$ and
$t^{\prime}_{sh}$.

Figure 12:  Pulse peak uncertainty analysis. For the pulsed method,
uncertainties have a minimum around $\omega_{mp}\approx 2\,\omega_{p}$ but
also a steep rise around $\omega_{mp}\gtrsim 3\,\omega_{p}$

When using a VNA (i.e. the swept approach), we have independent control of
$\omega_{i}$ and $\omega_{f}$. Figure 13 shows the uncertainty analysis of
$\sigma^{\prime}_{n}$ while varying both frequencies, and the two resonant
frequencies, $(\omega^{\prime}_{-},\,\omega^{\prime}_{+})$ = (0.318, 0.994),
are indicated. The results reveal two minima in $\sigma^{\prime}_{n}$.

Figure 13:  Uncertainty of $n$ over a range of $\omega_{i}$ and $\omega_{f}$
for a fixed $N$ and $\alpha$. The lowest minima is the rectangular region
($0<\omega^{\prime}_{i}<\omega^{\prime}_{-}$,
$\omega^{\prime}_{-}<\omega^{\prime}_{f}<\omega^{\prime}_{p}$) and suggests
that fitting only around $\omega^{\prime}_{-}$ is required.

The less prominent (higher) minimum occurs around ($\omega^{\prime}_{i}$,
$\omega^{\prime}_{f}$) $\approx$ (1.0, 1.0) but for $\sigma_{n}$ only. This is
somewhat intuitive because $\omega^{\prime}_{+}$ (Eq. 11) has a strong
dependence on $\omega_{p}$, which relates to density (Eq. 1).

The more prominent (lower) minimum is roughly bounded by the rectangular
region ($\omega^{\prime}_{i}<\omega^{\prime}_{-}$ and
$\omega^{\prime}_{f}>\omega^{\prime}_{-}$) for $\sigma^{\prime}_{n}$. Analysis
of $\sigma^{\prime}_{\nu}$ and $\sigma^{\prime}_{t}$ show this same minima.
The location and depth of this minima indicates that the best fit occurs when
the measured sweep straddles $\omega_{-}$. This strongly suggests that
$\omega_{-}$ sets the frequency resolution of the PIP rather than $\omega_{p}$
as previously believed. We believe that this is likely due to the fact that
$\omega_{-}$ is the location of Im($\Gamma_{pip}$)=0, the minimum of
Re($\Gamma_{pip}$), and is surrounded by several inflection points and the
largest magnitude of curvature (see Figure 8). Arguably, it is fitting to
these features, in addition to normalizing with $\Gamma$, 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 $\omega_{-}$ (and not up to $\omega_{p}$) may effectively extend the
PIP’s upper density limit. In this example, $\omega_{-}$ is roughly half an
order-of-magnitude less than $\omega_{p}$, which relates to a full order-of-
magnitude in density resolution (Eq. 1). Note that $\omega_{-}$ is strongly
dependent on $t^{\prime}_{sh}$ (Eq. 10), and therefore this potential benefit
only applies when the sheath is thin. ii) Further reducing $\omega_{-}$ by
deliberately decreasing $t^{\prime}_{sh}$ (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.

###  IV.4 Uncertainty analysis summary

In summary, we recommend the following best practices to maximize PIP SNR. We
recommend fitting $\Gamma_{pip}$ to measurements if $\nu^{\prime}<1$ but
fitting with $\Gamma_{diff}$ if $\nu^{\prime}>1$. For $N$ with the swept
method, we recommend using a reasonably large number (e.g. $10^{3}$ or
$10^{4}$) that still provides an adequate acquisition rate. While we do not
have direct control over noise, $\alpha^{\prime}$, we can use techniques such
as averaging to reduce it. For $\omega_{mp}$, we nominally recommend
$\omega_{mp}=\omega_{p}$, but also recommend using multiple values to
safeguard against an uncertain $\omega_{p}$.

An interesting result from this section is that $\omega_{-}$ appears to be the
frequency that sets the upper density limit of the PIP, and not $\omega_{p}$
as previously believed. This effectively extends the PIP’s density limit,
particularly when the sheath is “thin”. Therefore when choosing $\omega_{i}$
and $\omega_{f}$, we recommend values that straddle $\omega_{-}$ while also
adjusting these values appropriately to safeguard against variability in
$\omega_{-}$.

##  V Conclusions

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. 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.

## Acknowledgments

This work was supported by NRL base funding. JWB wishes to thank his mentors,
Mike McDonald and Erik Tejero, for their assistance, without which this work
would not be possible. JWB wishes to thank Jeffrey Bynum for his assistance
with technical drawings. JWB also wishes to thank his late teacher, Dr. Hugh
W. Coleman [44], for teaching him the importance of quantifying measurement
uncertainty.

## Data Availability

The data that supports the findings of this study are available on Zenodo
[45].

##  Appendix A The two-port network model for the PIP’s stem

In order to calibrate the PIP’s stem, we construct a two-port model of the
stem using an ideal, lossless transmission line (TL) model [38, 39], which are
expressed in S (scattering) parameters in the following form,

$\begin{bmatrix}S_{11}(\omega)&S_{12}(\omega)\\\
S_{21}(\omega)&S_{22}(\omega)\end{bmatrix}=\begin{bmatrix}0&e^{-j\omega
L/v_{p}}\\\ e^{-j\omega L/v_{p}}&0\end{bmatrix}.$ (27)

Here $L$ is the length of the TL, $j$ is the imaginary number, $\omega$ is the
angular frequency, $v_{p}=c/\sqrt{\epsilon_{r}}$ is the velocity of
propagation of the transmission line, $\epsilon_{r}$ is the published relative
dielectric constant of the TL, and $c$ is the speed of light in vacuum. The
resulting matrix can then be passed to _scikit-rf_ ’s _Network_ function [43].
Note that Eq. 27 is only accurate when $L$ is much smaller than the
attenuation scale-length of the TL’s dielectric. If this condition is
violated, an attenuation (i.e. loss) term could be inserted into Eq. 27.

##  Appendix B Example calibration

Figure 14 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 the 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 ($<2\pi$). Cal. plane 3 shows the fully calibrated measurement, where
both $\omega_{\pm}$ are apparent. The calibration process is detailed in Sec.
III.2.

Figure 14:  Example measured signals at each calibration plane (cal. plane):
the uncalibrated PIP measurement (cal. plane 1), partially calibrated
measurement (cal. plane 2) and fully calibrated measurement (cal. plane 3).

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