# High-speed imaging of magnetized plasmas: when electron temperature matters

Simon Vincent  Vincent Dolique  Nicolas Plihon  Univ Lyon, ENS de Lyon, CNRS,
Laboratoire de Physique, F-69342 Lyon, France

###### Abstract

High speed camera imaging is a powerful tool to probe the spatiotemporal
features of unsteady processes in plasmas, usually assuming light fluctuations
to be a proxy for the plasma density fluctuations. In this article, we
systematically compare high speed camera imaging with simultaneous
measurements of the plasma parameters – plasma density, electron temperature,
floating potential – in a modestly magnetized Argon plasma column at low
pressure (1 mTorr, magnetic fields ranging from 160 to 640 G). The light
emission was filtered around $488\pm 5$ nm, $750\pm 5$ nm, $810\pm 5$ nm. We
show that the light intensity cannot be interpreted as a proxy for the plasma
density and that the electron temperature cannot be ignored when interpreting
high speed imaging, both for the time-averaged profiles and for the
fluctuations. The features of plasma parameter fluctuations are investigated,
with a focus on ion acoustic waves (at frequency around 70 kHz) at low
magnetic field and low-frequency azimuthal waves (around a few kHz) at larger
magnetic fields. An excellent match is found between the high speed images
fluctuations and an Arrhenius law functional form which incorporates
fluctuations of the plasma density and of the electron temperature. These
results explain the discrepancies between ion saturation current and narrow-
band imaging measurements previously reported in the literature.

††preprint: AIP/123-QED

##  I Introduction

High-speed camera imaging is a useful tool to deduce time and space-resolved
features of laboratory plasma dynamics from the light it naturally emits.
Following the recent technological evolutions of high-speed cameras over the
last two decades, a number of studies followed the path initiated decades ago
in the fusion community Carruthers and Davenport (1957); Medley  _et al._
(1985). For instance, recent investigations using high-speed visible imaging
of tokamak plasmas allowed to gain insight into the dynamics of avaloids or
blobs Fuchert  _et al._ (2014); Tynan  _et al._ (2009), filaments Farley  _et
al._ (2019), the plasma response to magnetic perturbations Angelini  _et al._
(2015) or edge plasma dynamics Cavalier  _et al._ (2019). Indirect
measurements of the local plasma dynamics Gas Puff Imaging Zweben  _et al._
(2017) also offered great insights into the spatio-temporal dynamics of fusion
plasmas. High speed imaging of lower temperature plasmas either dedicated to
basic plasma studies or applications has experienced a recent boom. In the
context of plasma propulsion, the transverse and streamwise dynamics of
breathing modes Désangles  _et al._ (2020) and rotating spokes Mazouffre  _et
al._ (2019) in Hall thrusters or MHD modes in hollow cathodes Becatti, Goebel,
and Zuin (2021) benefited from images acquired up to several hundreds of
thousands frames per seconds. In the context of basic plasma physics, the
characterization of the dynamics of magnetized plasma archs benefited from
high speed imaging Hsu and Bellan (2003); Soltwisch  _et al._ (2010);
Sklodowski, Tripathi, and Carter (2021). High speed imaging in linear plasma
devices allow to gain insight on the ion dynamics Claire  _et al._ (2018), the
transition to drift wave turbulence Thakur  _et al._ (2014a) or the generation
of zonal flows Xu  _et al._ (2019). It can also be used for the comparison
with simulations Russell  _et al._ (2011), or as a complement to other
diagnostics Thakur  _et al._ (2014b); Brandt, Thakur, and Tynan (2016).
However, since the light radiated from a plasma depends in a very non-trivial
way on the plasma parameters, extracting physical information from camera
imaging is seldom very quantitative. However, camera imaging can be used alone
as a conclusive diagnostic technique. In cylindrical geometry for instance,
where the spatial fluctuations of an entire azimuthal wave can be captured at
once, and if the propagation velocity is low enough, modern cameras allow the
realization of detailed mode decompositions Brandt  _et al._ (2011); Ohdachi
_et al._ (2017); Yamasaki, Fujisawa, and Nagashima (2017). High speed plasma
imaging has also been used for the extraction of experimental dispersion
relations Light  _et al._ (2013); Cui  _et al._ (2016) or the estimation of
turbulent transport Cui  _et al._ (2015). In the latter examples, the light
intensity is usually considered as a proxy for the ion saturation current, or
for the density, with correlations up to 0.75 Cui  _et al._ (2015). In low
temperature and weakly ionized plasma, this approximation is usually by the
Corona model Fantz (2006); de la Cal  _et al._ (2010); Gonzalez-Fernandez  _et
al._ (2020) which assumes that the main source for photons emission is
radiative decay from excited states that are populated by electronic
collisions from the ground state. As a consequence the intensity of the light
emitted by the radiative decay from the electronic level $p$ to level $k$ of a
species $\alpha$ satisfies:

$I_{pk}\propto n_{e}n_{\alpha}K_{ex}^{g\rightarrow p}(T_{e}),$ (1)

where $n_{e}$ is the electron density, $n_{\alpha}$ is the ground state
density of species $\alpha$, and $K_{ex}^{g\rightarrow p}$ the excitation rate
constant of state $p$ from the ground state of $\alpha$. The densities of the
neutrals and the electron temperature are usually assumed to be spatially
homogeneous, and their temporal fluctuations are neglected. The emitted light
then depends only on the plasma density: for light emitted by excited states
of neutral species, the light intensity $I_{\rm neutral}\propto n_{e}$ and for
light emitted by excited states of ions in a single ion species plasma, the
light intensity $I_{\rm ion}\propto n_{e}^{2}$. While extremely convenient for
the interpretation of camera images, this simplification is often crude, which
is the reason for camera imaging to be mostly used as a qualitative diagnostic
tool.

In the recent years great efforts were made to investigate the correlations
between high speed camera imaging and electrostatic probes measurements –
either the ion saturation current for an estimation of the density, or the
floating potential – in laboratory Argon plasmas Antar, Yu, and Tynan (2007);
Oldenbürger  _et al._ (2010); Light  _et al._ (2013); Light, Thakur, and Tynan
(2019). These studies usually showed a high correlation between the ion
saturation current and the broadband light fluctuations (typically spanning
from 0.45 Antar, Yu, and Tynan (2007) to 0.6 Oldenbürger  _et al._ (2010) or
even 0.75 Cui  _et al._ (2015)), while it was significantly reduced when
considering filtered light (typically reduced to the order of 0.4 Oldenbürger
_et al._ (2010); Light, Thakur, and Tynan (2019)).

In this article we present simultaneous measurements of plasma density,
electron temperature and floating potential using an electrostatic probe, with
high speed imaging of the plasma. To our knowledge, it is the first
experimental study taking into account a well resolved spatial and temporal
evolution of the electron temperature in order to interpret light intensity of
visible light spectral lines directly recorded by camera in a low pressure,
modestly magnetized plasma column. We show that, for typical conditions where
low frequency waves are excited, the correlation between the light
fluctuations and the ion saturation current is of the order of 0.7, and
increases up to 0.87 for the electron temperature. The correlation with a
model based on the Corona approximation and taking into account both the
electron temperature and the plasma density slightly increases the correlation
and reduces the time-lag with the light fluctuations. The experimental set-up
and the electrostatic probe are first introduced in Section II. The high speed
imaging optical system is then described: it consists of a high speed camera
recording the light emitted by the plasma and filtered either around 488 nm
(ArII line), 750 nm or 810 nm (ArI lines), and optical and geometrical
corrections are taken into account for the image analysis. Based on excitation
cross sections extracted from the bibliography, a model is proposed in Section
III to take into account the dependence of high speed images with the electron
temperature. Then using the time-averaged plasma parameter profiles, this
model is compared to the light intensity radial profiles in Section IV, and
shows that spatial variations of the electron temperature cannot be ignored.
The correlations between the instantaneous plasma parameters and the
simultaneously recorded light intensity are then explored in Section V,
highlighting that the fluctuations of electron temperature is a key physical
parameter to understand the temporal fluctuations of radiated light intensity.
Finally, a first expansion of the proposed model, which clarifies the
comparison between the camera recordings and the probe measurements, is
discussed in Section VI. Conclusions are drawn in Section VII.

##  II Experimental setup, electrostatic and optical diagnostics

###  II.1 Experimental set-up

The experimental setup consists of a magnetized plasma column, described in
details elsewhere Plihon  _et al._ (2015) and sketched in Fig. 1. The plasma
source is an inductive 3-turns helicoidal coil wrapped around a $20$ cm long,
11 cm inner diameter borosilicate tube, fed by a 3.3 kW, 13.56 MHz radio-
frequency power supply through a manual L-type matching box. The source tube
ends in a 11 cm diameter BN disk. The plasma expands in a $20$ cm diameter,
$80$ cm long grounded stainless steel cylindrical chamber, ending in a DN 200
borosilicate optical window, from which the light emitted by the plasma
emitted is recorded by a high-speed camera. Three coils located along the
chamber create a confining axial magnetic field. The magnetic field is not
perfectly homogeneous along the $z$ direction and the values referred in the
remaining of this article are taken as the mean values along the axis, ranging
from $160$ G to $640$ G. The inhomogeneity of the magnetic field has to be
taken into account when comparing probe measurements and camera imaging;
geometrical corrections are introduced in Subsection II.4. Electrostatic
probes can be inserted along two diameters along the $x$ axis, at positions
$L_{1}=16$ cm and $L_{2}=49$ cm along $z$ (the origin $z=0$ taken at the
junction between the source and the chamber). All results presented in this
article were obtained for 1 kW of radio frequency power operated in Argon gas
at a pressure of 1 mTorr. To keep the whole apparatus in a steady thermal
state, the operation of the plasma is pulsed: the plasma is sustained over
typically 5 seconds, during which data are acquired, with a repetition period
of typically 30 s. The experiment is fully automated to allow high
repeatability and reproducibility of the plasma. The level of shot to shot
reproducibility was $\pm 0.6\%$ for the ion saturation current of a Langmuir
probe, with a standard deviation of $0.2\%$ (estimated from a series of 40
shots). Diametrical scans of the plasma parameters measured by the probe where
acquired sequentially: each spatial point has been acquired during one plasma-
pulse, and the probe is translated between two pulses.

Figure 1: Sketch of the experimental set-up. Locations of probes are shown as
dot-dashed lines. See text for details.

###  II.2 Probe measurements

Since the goal of the present article is to show that electron temperature
fluctuations cannot be ignored when analyzing high speed camera imaging, we
used a probe giving access to the instantaneous values of the electron
temperature, plasma density and floating potential, know as a five tips probe
Tsui  _et al._ (1992); Theiler  _et al._ (2011), very similar to a triple
probe Chen and Sekiguchi (1965). A photograph of the probe and an electrical
sketch are provided in Fig. 2. The probe consists of five 1 mm long, 0.2 mm in
diameter Tungsten wires spaced by 0.8 mm, housed in a 3 mm diameter BN
cylinder. The plasma density $n$ (assuming electroneutrality $n=n_{e}=n_{i}$,
where $n_{e}$ is the electron density and $n_{i}$ the ion density), electron
temperature $T_{e}$ and floating potential $V_{f}$ are then computed from
currents and voltages measured independently from the five tips. This probe
also provides a measure of the turbulent transport, not detailed in this work.
The raw electrical signals were conditioned using a home-made integrated
electronics (current/voltage conversion and galvanic isolation using opto-
couplers) and were digitized using a PXI-e 6368 data acquisition module at a
frame rate of 200 kHz.

The floating potential $V_{f}$ is measured from the voltage of the external
tips 1 and 5 as $V_{f}=(V_{1}+V_{5})/2$. The plasma density and electron
temperatures are deduced from the currents and voltage measurements of tips 1,
2 and 4. Tips 2 and 4 are biased negatively relative to the central tip 3
using a constant voltage $\Delta V=42$ V fed from an external Li-Ion battery.
Currents $I_{2}$ and $I_{4}$ reach the ion saturation current
$I_{i}=(I_{2}+I_{4})/2$ when the biasing $\Delta V$ is sufficiently large. Tip
3 collects an electron current (equal to $-2I_{i}$) and the electron
temperature is deduced from $V_{f}$ and $V_{3}$ as
$T_{e}=(V_{3}-V_{f})/\ln(3)$ (note that the factor $\ln(2)$ for a usual triple
probe is replaced here by $\ln(3)$ due to a current collection on tip 3 which
is twice the opposite of the ion saturation current). The plasma density is
computed from both $I_{i}$ and $T_{e}$ using $n=I_{i}/2\pi R_{p}ec_{s}\alpha$,
with $R_{p}$ the tips radius, $e$ the electron charge,
$c_{s}=\sqrt{eT_{e}/M_{i}}$ the Bohm velocity and $\alpha$ a parameter
depending on the ratio of the probe tip radius over the Debye length that
accounts for the tips cylindrical geometry Chen, Evans, and Arnush (2002). The
electrical measurements from each tips can be confidently considered
independent since their spacing (0.8 mm) is much larger than the Debye length
($\approx$15 $\mu$m). It is worth noting that the spacing between tips is of
the order of the pixel resolution for the high-speed camera imaging (0.7
mm/px): all tips of the probe will thus be considered at the same location
when comparing the probe measurements to imaging.

Figure 2: Left : photograph of the five-tips probe used for the measure of
$I_{i}$, $V_{f}$ and $T_{e}$. Right : electrical sketch of the probe.

Radial profiles of the plasma density and electron temperature are shown in
Fig. 3 for values of the confining magnetic field ranging from $160$ G to
$640$ G. The profiles of the standard deviation of temporal fluctuations are
displayed as shaded colors. Time-averaged values and standard deviations were
computed from 2 seconds measurements. The plasma density increases at the
center of the plasma column, while it decreases at the edge of the plasma
column, as the value of $B$ increases111 The local decrease of density at the
center for $B=480$ G and $B=640$ G is expected to be due to the probe
intrusiveness.. The electron temperature peaks around $x=4.5$ cm at the lowest
value of $B=160$ G, which can be understood as a signature of the inductive
heating occurring over the penetration depth close to the source tube. This
peak, although less marked, is still present up to the highest value of $B$. A
last interesting feature is the strong increase of $T_{e}$ (both the time-
averaged value and the amplitude of the fluctuations) at the edge of the
column for $B=640$ G, which is well reproducible and not fully understood yet.

Figure 3: Radial profiles of the plasma density $n_{e}$ and electron
temperature $T_{e}$, at $z=L_{2}$.

We underline that the probe provides simultaneous measurements of the temporal
fluctuations of the plasma parameters. In the remainder of this article, light
intensity recordings are systematically compared to models based on the
dependence of emitted light intensity with $n$ and $T_{e}$, using collisional
cross sections from the literature. Temporal fluctuations of light emission is
also compared to the rough measurements of $I_{i}(t)$, $V_{f}(t)$ and
$T_{e}(t)$ from the probe – this choice of $I_{i}$ instead of $n$ was
motivated by the fact that time series extracted from high speed camera
imaging is usually compared to the ion saturation current fluctuations of
Langmuir probe, used as a proxy for the plasma density fluctuations Antar, Yu,
and Tynan (2007); Oldenbürger  _et al._ (2010); Brandt  _et al._ (2011); Cui
_et al._ (2015).

###  II.3 Visible light camera imaging

Simultaneously to probe measurements, the light emitted by the plasma column
is high-speed recorded. The light radiated by the plasma is filtered using
interferometric filters centered around three specific wavelengths, namely
$488$ nm, $750$ nm and $810$ nm. Figure 4 shows the spectra of the light
emitted by the plasma column at its center, for increasing values of $B$,
recorded using an OceanOptics USB 2000+ Spectrometer. The optical fiber
collected the light along the $z$-axis (from the center of the optical
borosilicate window). Peaks in light intensity are observed for each of the
three central wavelengths of the filters, see bottom panel of Fig. 4, where
the dark dashed lines show the limits of the Full Half Width Maximum
transmission of the filters as provided by the manufacturer. The electronic
transitions at the origin of these spectral lines are provided in Table 1. The
line emitted at $488$ nm is an ArII line, and we will neglect the contribution
of the light emitted around $484$ nm when analyzing the images filtered around
$488$ nm. All other transitions are ArI lines. The knowledge of the upper
levels of the radiated light is used later on for the determination of the
dependence of the light intensity with $n$ and $T_{e}$. Note finally that as
the magnetic field increases, the intensity of the ArII line slightly
increases whereas it decreases for the neutral lines.

Figure 4: Spectra of the light naturally radiated by the plasma column at its center, for values of the magnetic field raging from $160$ G to $640$ G. Top : full spectra. Bottom : zoom in the value ranges of the optical filters that are used in this article for the camera measurements, around $488$ nm, $750$ nm and $810$ nm. Line (nm) | Upper level | Lower level | Species
---|---|---|---
487.99 | $3s^{2}3p^{4}(^{3}\textrm{P})4p$ | ${}^{2}\textrm{D}^{\circ}$ | $5/2$ | $3s^{2}3p^{4}(^{3}\textrm{P})4s$ | ${}^{2}\textrm{P}$ | $3/2$ | ArII
750.39 | $3s^{2}3p^{5}(^{2}\textrm{P}^{0}_{1/2})4p$ | ${}^{2}[1/2]$ | $0$ | $3s^{2}3p^{5}(^{2}\textrm{P}^{0}_{1/2})4s$ | ${}^{2}[1/2]^{\circ}$ | $1$ | ArI
751.47 | $3s^{2}3p^{5}(^{2}\textrm{P}^{0}_{3/2})4p$ | ${}^{2}[1/2]$ | $0$ | $3s^{2}3p^{5}(^{2}\textrm{P}^{0}_{3/2})4s$ | ${}^{2}[3/2]^{\circ}$ | $1$
810.36 | $3s^{2}3p^{5}(^{2}\textrm{P}^{0}_{3/2})4p$ | ${}^{2}[3/2]$ | $1$ | $3s^{2}3p^{5}(^{2}\textrm{P}^{0}_{3/2})4s$ | ${}^{2}[3/2]^{\circ}$ | $1$ | ArI
811.53 | $3s^{2}3p^{5}(^{2}\textrm{P}^{0}_{3/2})4p$ | ${}^{2}[5/2]$ | $3$ | $3s^{2}3p^{5}(^{2}\textrm{P}^{0}_{3/2})4s$ | ${}^{2}[3/2]^{\circ}$ | $2$
Table 1: Upper and lower energy levels of the dominant spontaneous radiated
light transitions analyzed in this article, from the NIST database Team
(2020).

Filtered light is recorded using a Phantom v2511 camera facing the $z=80$ cm
transparent chamber end. The central line of sight corresponds to the
$z$-axis. In order to minimize the parallax effect, the distance between the
camera and the chamber window is artificially increased by a set of two
mirrors, reaching $3.5$ m in the present configuration. In spite of this
arrangement, parallax cannot be neglected: a $10$ cm in diameter circle at the
back of the chamber is seen on the images $22\%$ smaller than a $10$ cm in
diameter circle at the front of the chamber. The images are recorded at a
frame rate of $200$ kfps, with an exposure time of $4.5$ $\mu s$, and by time
series of $100$ ms. The camera trigger was also acquired on the data
acquisition module, in order to get simultaneous measurements with the probe.
The spatial resolution is $256\times 256$ px2 for a field of view of
approximately $20$ cm, i.e. capturing fully the inside of the chamber. The
focus is set at the position $z=L_{2}$. However the depth of field $DoF$ with
the optical lenses used here (aperture f/4, focal length $135$ mm) is of the
order of the chamber length (considering a very strict circle of confusion of
the pixel resolution size as a lower limit, gives $DoF\approx 54$ cm). Hence
we have to consider that the recorded light is the result of an integration
over $z$, the implication of which are detailed in the next subsection.

The resulting image of the mean intensity $I_{cam}$ and the fluctuations
standard deviation $\sigma(\tilde{I}_{cam})$ are plotted in Fig. 5, for the
lowest value of the magnetic field ($B=160$ G) and using the $750$ nm filter.
Note that the camera images are displayed in a frame $(x^{*},y^{*})$, whose
definition is given below. The first observation is that, as expected, the
emitted light is axisymmetric. We thus discuss the features according to the
radius $r^{*}=\sqrt{x^{*2}+y^{*2}}$ in the image plane. The largest intensity
is observed around a radius of $\sim 2.8$ cm, with a sharp gradient outward,
which is maximal at $\sim 4.5$ cm. The amplitude of the fluctuations is of the
order of 10 $\%$ of the total amplitude. The fluctuation pattern also presents
a ring of large amplitude around $\sim 3.5$ cm. Similar features are observed
on the images recorded with the other filters around $488$ nm and $810$ nm
(see appendix Appendix A : Camera imaging radial profiles). Surprisingly the
light intensity profile shape only slightly evolves with the magnetic field
amplitude whereas the plasma parameters ($n$, $T_{e}$, $V_{p}$) radial
profiles shapes vary as the magnetic field amplitude increases (see Fig. 3).
This discrepancy in the radial profiles evolution between camera imaging and
probe measurements is addressed in more details in section IV. Note finally
that the constant light intensity level for $r\gtrsim 5$ cm, visible on radial
mean profiles displayed in appendix Appendix A : Camera imaging radial
profiles, does not correspond to light directly emitted by the plasma, which
density and temperature strongly decrease at the column edge. Light collected
at the edge is likely due to reflections on the cylindrical chamber and BN
end, and therefore constitutes noise of the camera imaging data. This noise
level, computed over the radial locations $r^{*}\leq 5$ cm and the four
magnetic field values, corresponds on average to $\sim 50$ $\%$ of the signal
for the light intensity data filtered around 488 nm, $\sim 25$ $\%$ for 750 nm
and $\sim 35$ $\%$ for 810 nm. For the comparison with probe profiles, $95$
$\%$ of the noise level is subtracted to each mean profile of the camera
imaging data in order to keep positive values for the light intensity. An
example of a resulting profile of $I_{cam}$ is shown in Fig. 7.

Figure 5: Images of the light recorded by the camera at $B=160$ G, filtered
around $750$ nm: time averaged intensity (left) and the intensity fluctuations
standard deviation (right).

###  II.4 Geometrical corrections

It is important to note that a direct comparison between probe and camera
light intensity radial profiles is not relevant. While probe measurements are
done at a precise location on the z-axis, $I_{cam}$ is the result from a light
integration over z, and because of the parallax effect and the inhomogeneity
of the axial magnetic field the camera lines of sight scan locations with
different values of the plasma parameters. In order make the data comparison
possible, a transformation is applied to the probe profiles, modeling the
light integration process along the z-axis.

The first assumption is that of cylindrical symmetry of the plasma column,
supported by the images shown in Fig. 5. All the following computation are
therefore performed in the 2D plane ($z,r>0$). The magnetic field spatial
configuration is deduced from a numerical computation taking into account the
real geometry of the coils, and is combined with the parallax effect computed
from calibration images, to deduce the magnetic lines configuration as seen
from the camera. The integration along the $z$-axis also requires a second
assumption: frozen plasma parameters along magnetic field lines. This was
checked by comparing radial profiles of $n$ and $T_{e}$ measured at $z=L_{1}$
and $z=L_{2}$, while the radial axis was transformed following the computed
magnetic field lines, for the four set of $B\in[160;640]$ G. The results show
a very reasonable match with a discrepancy of the profiles shape along the
radius of less than $\sim 20\%$ for both $n$ and $T_{e}$. Under the last
assumption, the plasma parameters for the full plasma column can be
reconstructed from a single radial scan with the probe. An illustration of
this process is given in Fig. 6, which shows a two-dimensional map of the
electron temperature, reconstructed from a profile measured at $z=L_{2}$ for
$B=160$ G. Note that the map is plotted with respect to ($z,r^{*}$), the
variables of the distorted space in which the camera lines of sight are
parallel.

Figure 6: Spatial map of the electron temperature, reconstructed from the five
tips probe profile performed at $z=L_{2}$ (dark dot-dashed line). The spatial
domain inside the cylindrical chamber and the source is distorted so as to
make the camera lines of sight parallel (red dashed lines), which makes the
integration along $z$ easier. $B=160$ G.

Integrated quantities may then be computed from the reconstructed time-
averaged plasma parameters profiles in the $(r^{*},z)$ plane as
$X^{*}(r^{*})=\int_{z}X(r^{*},z)dz$. The results of this process are shown in
Fig. 7, for $n^{*}$ and $T_{e}^{*}$ at $B=160$ G. Figure 7 shows the profiles
of the plasma density $n$ and electron temperature $T_{e}$ measured as a
function of $r$ in the $z=L_{2}$ plane, and the integrated profiles $n^{*}$
and $T_{e}^{*}$ as a function of $r^{*}$. The integrated profiles $n^{*}$ and
$T_{e}^{*}$ are computed only up to $r^{*}=6.9$ cm, which corresponds to the
first camera line of sight away from the center, that crosses the limiting
magnetic field line which intercepts the chamber wall at $z=L_{2}$ (see Fig.
6). For $r^{*}>6.9$ cm, the single probe measurements at $L_{2}$ is not enough
to reconstruct the plasma parameters on all field lines crossing the lines of
sight. For comparison a radial profile of $I_{cam}$ is plotted in Fig. 7. It
is computed as the average along $\theta$ of the mean camera image. The
profile of the electron temperature $T_{e}$ measured at $z=L_{2}$ is peaked at
$r\approx 4.5$ cm, while $T_{e}^{*}$ is peaked at $r^{*}\approx 3$ cm, which
falls very close to the peak observed for the light intensity at 750 nm: this
suggests the importance of $T_{e}$ for light emission recorded by the camera.
Moreover, it confirms that the time averaged light intensity $I_{cam}$ is not
simply proportional to the time averaged plasma density for our experimental
conditions. Similar conclusions hold for lines at 488 nm and 810 nm.

Figure 7: Radial profiles of light intensity $I_{cam}$ recorded by camera with
the $750$ nm filter, to be compared to the density $n$ and the electron
temperature $T_{e}$ measured by probe at $z=L_{2}$, as well as to their
"image" profiles $n^{*}$ and $T_{e}^{*}$, for $B=160$ G.

##  III Time averaged light intensity strongly depends on $T_{e}$

The dependence on $T_{e}$ has to be considered in order to understand the
profile of $I_{cam}$. In the present plasma conditions of low density ($n\sim
10^{18}$ m-3), low degree of ionization ($5\textnormal{-}20$ $\%$), ion
temperature of the order of a few tenths of an eV and low pressure ($p_{0}\sim
1$ mTorr) implying $T_{e}\gg T_{i}\gg T_{n}$, allowing the application of the
Corona model Fantz (2006). As a consequence the intensity naturally radiated
at a given wavelength only depends on the electron density and on the ground
state density on the one hand, and, on the other hand, on the excitation rate
constant $K_{ex}$ from the ground state to the radiative state (given as the
"upper" states in Table 1).

The light intensity for each of the transitions identified in Table 1 are thus
expected to be proportional either to the functional form $f_{n}=n\,K_{ex}$
for the ArI lines or the functional form $f_{n^{2}}=n^{2}\,K_{ex}$ for the
ArII line - this point will be discussed further below. For each of the
excited state, the excitation rate constant $K_{ex}$ strongly depends on
$T_{e}$ 222 The rate constant $K_{ex}=\langle\sigma_{ex}v\rangle_{v}$ can be
expressed as $K_{ex}=\left(\frac{m_{e}}{2\pi
eT_{e}}\right)^{3/2}\displaystyle\int_{0}^{\infty}\sigma_{ex}(v)ve^{-\frac{mv^{2}}{2kT_{e}}}4\pi
v^{2}dv$ and recast as $K_{ex}=\left(\frac{e}{2\pi
m_{e}T_{e}}\right)^{1/2}\int_{0}^{\infty}\sigma_{ex}(\varepsilon)\,\varepsilon\,e^{-\frac{\varepsilon}{T_{e}}}d\varepsilon$
, and a non uniform radial profile of $K_{ex}$ is expected from the density
and electron temperature profiles shown in Fig. 3. Assuming a Maxwellian
electron distribution function, $K_{ex}$ can be computed from the cross
section $\sigma_{ex}$. Finally, $K_{ex}(T_{e})$ is fitted numerically in the
range [1, 5] eV (corresponding to the temperature of our experimental
conditions) using the functional form
$K_{ex}(T_{e})=K_{ex}^{0}e^{-\epsilon_{ex}/T_{e}}$ with $\varepsilon_{ex}$ a
fitting parameter equivalent to an averaged excitation energy Lieberman and
Lichtenberg (2005).

Figure 8 (left) shows the cross sections of excitation processes
$\textrm{Ar}\rightarrow\textrm{Ar}(4p[1/2]0)$ and
$\textrm{Ar}^{+}\rightarrow\textrm{Ar}^{+}(4p\,^{2}\textrm{D}^{\circ}\,5/2)$
(the energy state ${Ar}^{+}(4p\,^{2}\textrm{D}^{\circ}\,5/2)$ will be denoted
$Ar^{+*}$ for simplicity in the following), along with an example for
$T_{e}=4$ eV of the integrand $\varepsilon e^{-\varepsilon/T_{e}}$. The
computed evolution of $K_{ex}(T_{e})$ as a function of $T_{e}\in[1,10]$ eV is
plotted in Fig. 8 (right, open blue symbol) for the excitation process
$\textrm{Ar}\rightarrow\textrm{Ar}(4p[1/2]0)$. The values of
$\sigma_{ex}(T_{e})$ used for these computations are taken from the
literature. For the ArI lines a recent and complete set of Argon cross
sections was used Alves (2014). Note that for the light filtered around $810$
nm, the average of the two equally contributing radiative states identified in
Table 1 is used to compute $K_{ex}$. This is a rather strong assumption given
that the line at 811.53 is known to be mostly populated from a metastable
state and not from the ground state. However, we stress here that our goal is
to demonstrate that the importance of the temperature fluctuations when
interpreting high-speed camera imaging and that precise plasma spectroscopy is
beyound the scope of the present article. While database providing cross
sections for neutral lines are widely available Alves (2014) the dataset for
ArII lines at low electron temperature are very seldom in the literature. The
set of cross sections $\sigma_{ex}$ for the $Ar^{+}\rightarrow Ar^{+*}$
excitation process was extracted from recent simulations Dipti and Srivastava
(2016) and are shown in Fig. 8a), dashed line. The average energy found for
$\epsilon_{ex}^{Ar^{+}\rightarrow Ar^{+*}}=20$ eV. Since the ground state for
this excitation process is Argon ion, the resulting light intensity is
expected to follow the functional form $f_{n^{2}}$. A second excitation
process has been identified in the literature for the $Ar^{+*}$ state, from
direct ionization and excitation from the neutral Ar atom ground level, and a
second set of cross section was extracted from an experimental study Strinic
_et al._ (2004). An average excitation energy $\epsilon_{ex}^{Ar\rightarrow
Ar^{+*}}=35.4$ eV is found, at the value close to the sum of the excitation
energy $\epsilon_{ex}^{Ar^{+}\rightarrow Ar^{+*}}$ and the 15.8 eV ionization
energy of Argon atom. Note that the light intensity form this physical process
is expected to follow the functional from $f_{n}$.

For all the major processes involved in populating the upper states referenced
in 1, the results of the average excitation energies computed from the
numerical fits of $K_{ex}$ as a simple Arrhenius law
$K_{ex}(T_{e})=K_{ex}^{0}e^{-\epsilon_{ex}/T_{e}}$, as well as the functional
form expected in terms of dependence with the density $n$, are summarized in
Table 2. The Arrhenius law fits very well the rate constant dependencies in
$T_{e}$, with a root mean square errors associated with the fits ranging from
$0.7.10^{-2}$ to $3.8.10^{-2}$. All the fits are plotted in appendix Appendix
B : Fits of the rate constants.

Figure 8: Left : Maxwellian distribution function example for $T_{e}=4$ eV, plotted along with the excitation cross-sections of level $4p[1/2]0$ from Ar atom ground level (curve (a)) and of $4p^{2}D^{\circ}_{5/2}$ from $\textrm{Ar}^{+}$ ground level (curve (b)). Right : computation of $K_{ex}(T_{e})$ using $\sigma_{ex}^{(a)}$ (blue dots), and fit in the range $T_{e}\in[1;5]$ eV with $f(T_{e})=K_{ex,0}e^{-\varepsilon_{ex}/T_{e}}$ (red curve). Filter (nm) | Exc. process |  $\varepsilon_{ex}$ (eV) | $f$
---|---|---|---
$488\pm 5$ | $\textrm{Ar}^{+}\rightarrow\textrm{Ar}^{+}(4p\,^{2}\textrm{D}^{\circ}\,5/2)$ | 20.0 | $f_{n^{2}}$
$\textrm{Ar}\rightarrow\textrm{Ar}^{+}(4p\,^{2}\textrm{D}^{\circ}\,5/2)$ | 35.4 | $f_{n}$
$750\pm 5$ | $\textrm{Ar}\rightarrow\textrm{Ar}(4p[1/2]0)$ | 14.4 | $f_{n}$
$810\pm 5$ | $\textrm{Ar}\rightarrow\textrm{Ar}(4p[3/2]1)$ | 15.0 | $f_{n}$
$\textrm{Ar}\rightarrow\textrm{Ar}(4p[5/2]3)$
Table 2: Excitation processes for light radiation at $488\pm 5$ nm, $750\pm 5$
nm and $810\pm 5$ nm. The excitation rate was fitted as
$K_{ex}(T_{e})=K_{ex}^{0}e^{-\varepsilon_{ex}/T_{e}}$ and the light intensity
is expected to follow the functional form $f_{n}$ or $f_{n^{2}}$ (see text for
details).

Note that the absolute spectroscopic calibration of the full optical
acquisition chain is beyond the scope this work. Thus sections IV and V focus
on the comparison between normalized profiles of the light intensity and the
simultaneously acquired plasma parameters. Section IV addresses the comparison
for time-averaged profiles, while section V focuses on temporal fluctuations.
As a consequence, the comparison between normalized radial profiles allows to
discriminate the accuracy of the following models :

$\displaystyle f_{n}$ $\displaystyle=ne^{-\varepsilon_{ex}/T_{e}}$ (2)
$\displaystyle f_{n^{2}}$ $\displaystyle=n^{2}e^{-\varepsilon_{ex}/T_{e}}$

##  IV Mean profiles

In this section, the time-averaged experimental profiles recorded by the high-
speed camera are compared to predictions using for the excitation rate
$K_{ex}(T_{e})\propto e^{-\epsilon_{ex}/T_{e}}$ and the time-averaged
experimental radial profiles of the plasma parameters. Using the axisymmetric
reconstruction of the plasma parameters in the $(r^{*},z)$ plane introduced in
subsection II.4, a time-averaged synthetic light radiation is computed as :

${f_{p}}^{*}(r^{*},\varepsilon_{ex})=\int_{z}f_{p}(n(r^{*},z),\,T_{e}(r^{*},z),\,\varepsilon_{ex})dz,$
(3)

where $f_{p}$ stands for the functional dependency $f_{n}$ or $f_{n^{2}}$
introduced in the previous section, and, for clarity, the time averaged
notation $\langle\cdot\rangle$ has been dropped. For each value of
$\varepsilon_{ex}$ in the range $[0;40]$ eV, the synthetic radial profile is
compared to the profile of light intensity measured with the camera and
averaged along $\theta$. The rms of the difference between the experimental
and the synthetic profiles are displayed in the right panels of Fig. 9 for
$B=320$ G. The synthetic profiles which lower the difference with the
experimental profiles are shown in the left panels of Fig. 9. For the sake of
comparison both models $f_{n}^{*}$ and $f_{n^{2}}^{*}$ are systematically
compared to the data.

The excitation energy that minimizes the error for the neutral $750$ nm line
with the model $f_{n}^{*}$, of $\varepsilon_{ex}^{opt}=12$ eV, is close to the
theoretical value $14.4$ eV computed from the numerical fit (Table 2). The
agreement is slightly lower for the $488$ nm line with the model
$f_{n^{2}}^{*}$, with an optimal value of $\varepsilon_{ex}^{opt}=14$ eV to be
compared to $20.0$ eV from the numerical fit. Note that a very good match is
also found with the $488$ nm data for a dependence of power law 1 in $n$
(model $f_{n}^{*}$), indicating that the parameter mostly responsible for the
shape the radial profiles of light emission is the electron temperature
through the $e^{-\varepsilon_{ex}/T_{e}}$ term. As for the neutral $810$ nm
line a very good match is found in the comparison with the model $f_{n}^{*}$,
but for a value of $\varepsilon_{ex}^{opt}=5$ eV three times lower than the
one expected from the numerical fit. And again in this case no much difference
is found between the models $f_{n}^{*}$ and $f_{n^{2}}^{*}$ : the driving
parameter for the reconstruction of $\langle I_{cam}\rangle$ is the average
electron temperature.

As a result of the same analysis very satisfactory results of profile
reconstruction (not shown here) are found for $B=160$ G. Note however that at
this lower magnetic field the values of the average rate constant minimizing
the error between $\langle I_{cam}\rangle$ and the model are relatively far
from the theoretical ones computed in section III, with
$\varepsilon_{ex}^{opt}=8.5$ and $\varepsilon_{ex}^{opt}=3.9$ for $750$ nm and
$810$ nm respectively with model $f_{n}$, and $\varepsilon_{ex}^{opt}=11.6$
for $488$ nm respectively with model $f_{n^{2}}$.

Finally the comparison is performed at $B=480$ G and $B=640$ G. At these
higher magnetic fields the models do not match well anymore the light
intensity profiles from the camera imaging. This is due to the systematic
decrease of light intensity observed at the center of the plasma column, that
cannot be reproduced by the $T_{e}$ profiles at these magnetic field values
(see profiles for $B\geq 480$ G in Fig. 3). The discrepancy between the time-
averaged emitted light and the models at high values of $B$ could be the
signature of a peak in the electron temperature close to the source radius, in
the source region, which is not measured in the electron temperature profiles
measured at $z=L_{2}$.

Figure 9: Comparison of radial profiles between the measure of $I_{cam}$, and
its modelling from the plasma parameters using the models $f_{n}$ and
$f_{n^{2}}$, for the filter at $488$ nm, $750$ nm and $810$ nm, at $B=320$ G.
The respective optimization processes of $\varepsilon_{ex}$ (bottom) are
plotted with the theoretical value coming from the numerical fit performed in
section III

##  V Temporal fluctuations

We now address the comparison between simultaneous measurements achieved with
the five-tips probe and the camera, an important task for the interpretation
of high-speed imaging to resolve the plasma dynamical behavior. The probe was
inserted along the $x$-axis and the results presented here were acquired at a
location $x_{p}=4$ cm of the probe tip (at $z=L_{2}$). But due to the magnetic
field ripple and the optical parallax, as previously discussed, the radial
location $r$ at which the probe samples the signal corresponds to a location
$r^{*}$ on the camera images closer to the center $r^{*}<r$. The
transformation process described in subsection II.4 is applied to a test
radial profile composed of a single peak at $r=4$ cm and it is found that the
peak is mostly translated at $r^{*}\sim 2.6$ cm. Hence, even if this
transformation is not reversible, the best area on the camera images to
account for what is measured at $(z=L_{2},x_{p}=4$ cm) is found around
$x_{p}^{*}=2.6$ cm . The light intensity was therefore taken as the average of
a $10\times 10$ pixels box around $x_{p}^{*}=2.6$ cm and the simultaneous
acquisition lasts 40 ms.

We recall here that the probe gives access to the instantaneous ion saturation
current $I_{i}(t)$, from which the instantaneous plasma density $n(t)$ is
computed as $n(t)\propto I_{i}(t)/\sqrt{T_{e}(t)}$. While the plasma density
appears explicitly as a control parameter for the excitation processes, in
this section we chose to present direct comparisons between $I_{i}$ and the
light intensity, in order to provide a similar analysis to the results
published in the literature using the ion saturation current from Langmuir
probes as a proxy to the density. Note that, for the regimes reported in this
study, the temperature fluctuations are a few percent, and the difference
between the normalized fluctuations of $I_{i}$ and $n$ is of the order of
$0.5\,\tilde{T}_{e}/\langle T_{e}\rangle$ and can be most of the time
neglected. The spectra of all the simultaneously measured signals are shown in
Fig. 10.

Except for the small peak at around 67 kHz for $B=[320,480,640]$ G, all the
spectral features of the light intensity fluctuations correspond to features
observed in the fluctuations of the ion saturation current, the electron
temperature or the floating potential. At low magnetic field ($B=160$ G) the
spectra are peaked around 70 kHz, and show very similar features for all the
parameters. The waves observed at this low value of the magnetic field
correspond to ion acoustic waves propagating azimuthally at the Bohm speed,
and on the outer edge of the plasma column. For higher values of $B$,
fluctuations are observed in the kHz range, and correspond once again to waves
propagating in the azimuthal direction. These waves share features with
unstable Kelvin-Helmholtz modes. The spectra for the fluctuations of the light
intensity $I_{cam}$ are very similar to the spectra of the ion saturation
current $I_{i}$. However this rough comparison in the frequency domain is not
sufficient to state which parameter controls the fluctuations of $I_{cam}$
fluctuations and a further analysis is detailed below.

Figure 10: Spectra of the simultaneous camera ($I_{cam}$) and probe ($I_{i}$,
$V_{f}$, $T_{e}$) measurements, with the $750$ nm optical filter, for
$B=[160;320;480;640]$ G.

Figure 11 shows simultaneous time series of the normalized fluctuations of
$I_{i}$, $T_{e}$ and $I_{cam}$ (filtered around $750$ nm) for $I_{B}=320$ G.
The correlations between $\tilde{I}_{cam}$ and the plasma parameters
fluctuations, as well as $\tilde{I}_{cam}$ autocorrelation, are displayed in
Fig. 12. From the time series, a strong correlation is observed between
$\tilde{T}_{e}$ and $\tilde{I}_{cam}$, which is confirmed by the high value of
the maximum correlation $\max(X_{corr}(\tilde{I}_{cam},\tilde{T_{e}}))=0.87$.
The correlation between the light intensity and the ion saturation current is
lower, with $\max(X_{corr}(\tilde{I}_{cam},\tilde{I_{i}}))=0.68$ (though this
is already a significant value, which is comparable to what was reported in
similar plasma conditions Oldenbürger  _et al._ (2010); Cui  _et al._ (2015)).
However, previous studies Oldenbürger  _et al._ (2010); Cui  _et al._ (2015)
reported measurements for which $\tilde{I}_{cam}$ and $\tilde{I}_{i}$ were
observed to evolve in phase, whereas we observe here a significant time-lag
between $\tilde{I}_{cam}$ and $\tilde{I}_{i}$. On the contrary, here, no delay
is observed between $\tilde{I}_{cam}$ and $\tilde{T}_{e}$. Finally for the
sake of readability, the time series of $\tilde{V}_{f}$ are not shown, but the
correlation with $\tilde{I}_{cam}$ shows a delay around $\pi$ (Fig. 12). This
anti-correlation between $\tilde{I}_{cam}$ and $\tilde{V}_{f}$ confirms an
observation that was recently made Light, Thakur, and Tynan (2019) for both
the emission lines at $750$ nm and $488$ nm.

Figure 11: Simultaneous measurement of $\tilde{I}_{cam}$ with a filter at
$750$ nm and $\tilde{I}_{i}$, $\tilde{T}_{e}$ by probe, for $B=320$ G. Figure
12: Correlation between the simultaneous measure of $\tilde{I}_{cam}$ with a
filter at $750$ nm and $\tilde{I}_{i}$, $\tilde{T}_{e}$, $\tilde{V}_{f}$, for
$B=320$ G.

We now compare in a systematic way the fluctuations of $I_{cam}$ with both
models $f_{n}$ and $f_{n^{2}}$. Similarly to the study reported for the time-
averaged profiles, for each value of the excitation energy $\epsilon_{ex}$ in
the range $[0;40]$ eV, synthetic fluctuating series are computed according to
the models (using the times series of the plasma parameters measured with the
probe) and compared to $\tilde{I}_{cam}$. The results are shown in Fig. 13 for
the same data set as presented in Fig. 11 and Fig. 12, and for the three
filters.

For the $750$ nm line, we observe an increased correlation with
$\tilde{f}_{n}$, reaching 0.91 for an average excitation energy of
$\varepsilon_{ex}=17\pm 3$ eV, close to the value computed in Table 2
$\varepsilon_{ex}=14.4$ eV. It is also interesting to note that the maximum of
correlation between $\tilde{I}_{cam}$ and $\tilde{f}_{n}$ corresponds to a
zero time-delay, strengthening the validity of the results. For the sake of
comparison, the other model $f_{n^{2}}$ is also tested : the correlation
between $\tilde{I}_{cam}$ and $\tilde{f}_{n^{2}}$ is displayed in Fig. 13. A
correlation up to 0.90, also associated with a zero delay between the signal,
is observed for $\varepsilon_{ex}=32\pm 3$ eV. Hence with this model the
correlation values are also increased with respect to that obtained with mere
plasma parameters, but the optimal value found for $\varepsilon_{ex}$ is far
from the reference value in Table 2. This shows that, unlike with the mean
profiles comparison presented in section IV, it is here essential to include
the correct dependence in $n$ to recover the expected theoretical value of
$\varepsilon_{ex}$.

Similarly, the time series of light fluctuations $I_{cam}$, measured at $488$
nm, are compared to the models. The maximum of correlation with the
$f_{n^{2}}$ model, which reaches 0.85, is observed for
$\varepsilon_{ex}=17.5\pm 4.5$ eV, close to the value of
$\varepsilon_{ex}=20.0$ eV reported in Table 2. This shows that the model
$f_{n^{2}}$ matches very well the light intensity at 488 nm. The maximum of
correlation with the $f_{n}$ model is observed for $\varepsilon_{ex}=8\pm 2$
eV, which is far below $\varepsilon_{ex}=35.4$ eV for the direct ionization
plus excitation from the neutral Ar atom, presented in Table 2. This result
rules out the probability of this process for light emission at 488 nm.

Finally the results for the light emission at $810$ nm show that using both
models the fluctuations correlation increases and the phase shift cancels out
for the optimal $\varepsilon$ values. We get $\max(X_{corr})>0.75$ for
$\varepsilon_{ex}=11.5\pm 3$ eV with model $f_{n}$, and $\max(X_{corr})>0.75$
for $\varepsilon_{ex}=23\pm 5$ eV with $f_{n^{2}}$. Even if the conclusion
here cannot be as unequivocal as with the two other lines at 750 nm and 488
nm, the model $f_{n}$ that is expected to better match the data at 810 nm
provides indeed a closer result to the theoretical value of
$\varepsilon_{ex}=15$ eV (Table 2).

Figure 13: Maximal values (top) and associated time shifts (bottom) of the
correlations computed between the time fluctuations signals of the light
intensity $I_{cam}$ on one hand, and $I_{i}$, $V_{f}$, $T_{e}$ measured by
probe, and the models $f_{n}$ and $f_{n^{2}}$ on the other hand, with
$\varepsilon_{ex}$ in $[0;40]$ eV. The results are presented for the optical
lines around 488 nm, 750 nm and 810 nm, and for $B=320$ G. The time shifts are
normalized to the period $T$ of the main spectral component of $I_{cam}$.

The same comparison is performed as well for $B=160$ G, $B=480$ G, $B=640$ G,
but it is not described in details here. A sum-up of maximum correlation
values and associated time shifts between $\tilde{I}_{cam}$ and
$\tilde{I}_{i}$, $-\tilde{V}_{f}$, $\tilde{T}_{e}$ and the most accurate model
($\tilde{f}_{n}$ for the neutral lines at 750 and 810 nm, $\tilde{f}_{n^{2}}$
for the 488 nm ion line) is presented in Fig. 14, for $B$ ranging from $160$ G
to $640$ G. Here the average excitation energies used in the models are taken
equal to their theoretical values computed in Table 2. Among plasma parameters
the higher correlation is observed with $\tilde{T}_{e}$ for $I_{B}\leq 480$ G.
For $B=640$ G, $\tilde{I}_{cam}$ is highly correlated with $\tilde{I}_{i}$ and
$\tilde{T}_{e}$. We can also see that $\tilde{I}_{cam}$ is delayed from
$\tilde{I}_{i}$ by $\pi$, and very close to be in phase with $\tilde{T}_{e}$,
in almost all cases. As for the models, apart from $B=640$ G, they clearly
improve the correlation and reduce the time shift with the light intensity
fluctuations. However the gain with respect to the mere electron temperature
is not so significant, and in the end one might be tempted to consider more
simply $\tilde{I}_{cam}$ to be a proxy for $\tilde{T}_{e}$ as a first order
approximation. This is discussed in more details in the next section.

Figure 14: Summary of the maximal values (left) and associated time-shifts
(right) of the correlations between the temporal fluctuations of $I_{cam}$ and
$I_{i}$, $V_{f}$, $T_{e}$, $f_{p}(I_{i},T_{e},\varepsilon_{ex})$, for the 488
nm line ($\varepsilon_{ex}=20.0$ eV and $f_{p}=f_{n^{2}}$), the $750$ nm line
(with $\varepsilon_{ex}=14.4$ eV and $f_{p}=f_{n}$) and the $810$ nm line
($\varepsilon_{ex}=15.0$ eV and $f_{p}=f_{n}$), as a function of the magnetic
field $B$. For each value of B, the time shifts are normalized to the period
$T$ of the main spectral component of $I_{cam}$.

##  VI Discussion

The features shown in Fig. 10 can be much better understood with the spatio-
temporal visualization of the plasma provided by fast camera imaging. At low
value of the magnetic filed (160 G), the spectral components at 61 kHz and 72
kHz are identified without ambiguity, thanks to camera imaging, as modes
$m=5$, $m=6$ of IAW waves, and the peaks at 5 kHz, 67 kHz and 77 kHz as the
result of the weak non-linear interactions between them. At higher magnetic
fields, the spectra maxima visible around 3 kHz and 5 kHz correspond to modes
$m=2$ and $m=3$ of low frequency waves of the type Kelvin-Helmholtz, Rayleigh-
Taylor of Drift-Waves, such as observed typically in linear devices Burin  _et
al._ (2005); Brochard, Gravier, and Bonhomme (2005). These results will be
reported in forthcoming publications. These examples show the importance of
camera imaging diagnostic for the investigation of non-linear evolution of
instabilities and waves in magnetized plasma columns. In this context it is
essential to have a fine understanding of what the camera images represent.

The results reported in section V show that the $f_{n}$ and $f_{n^{2}}$ models
accurately link fluctuations of the plasma parameters with the fluctuations of
light emission, and we now discuss our results from first order expansions of
these models :

$\displaystyle f_{n}\,\textrm{ model}$ $\displaystyle:\ \
\frac{\tilde{I}_{cam}}{\langle I_{cam}\rangle}=\frac{\tilde{n}}{\langle
n\rangle}+\xi\frac{\tilde{T}_{e}}{\langle T_{e}\rangle}$ (4) $\displaystyle
f_{n^{2}}\,\textrm{ model}$ $\displaystyle:\ \ \frac{\tilde{I}_{cam}}{\langle
I_{cam}\rangle}=2\frac{\tilde{n}}{\langle
n\rangle}+\xi\frac{\tilde{T}_{e}}{\langle T_{e}\rangle}$

with $\xi=\varepsilon_{ex}/\langle T_{e}\rangle$. These expressions show that
the balance between the normalized density and electron temperature
fluctuations on the normalized light intensity fluctuations is uniquely set by
the ratio $\xi=\varepsilon_{ex}/\langle T_{e}\rangle$. It is also interesting
to note that this balance therefore depends on the location when the electron
temperature is inhomogeneous. For an average electron temperature of the order
of $3-4$ eV, $\xi$ reaches $\sim 5$. Hence for normalized density and electron
temperature fluctuations of comparable orders of magnitude, the $\xi$ term
drives $\xi\tilde{T}_{e}/\langle T_{e}\rangle$ to be the dominant terms in Eq.
(4). Towards the edge of the plasma column, where $\langle T_{e}\rangle$
strongly decreases for $B$ lower than 480 G (see Fig. 3), $\xi$ reaches values
of the order of 10. Thus, at the edge of the plasma column, even though
density fluctuations are twice larger than temperature fluctuations, the light
intensity fluctuations are mainly driven by the fluctuations of electron
temperature.

Data from the simultaneous camera and probe measurements (at $r=4$ cm, see
section V) are analyzed in Fig. 15, for increasing values of the magnetic
field, where the amplitude of all the terms of the right hand sides of Eq. (4)
are computed for all magnetic fields. The comparison are done here for the ion
line at 750 nm and the neutral line at 488 nm. Note that $\xi_{i,n}$ are
introduced as $\xi_{i,n}=\varepsilon_{ex}^{i,n}/\langle T_{e}\rangle$ with
$\varepsilon_{ex}^{i}=14.4$ eV and $\varepsilon_{ex}^{n}=20$ eV for the ion
and neutral lines respectively. Though the plasma density exhibits higher
normalized fluctuations (red dots) than the electron temperature (green
triangles), the large value of $\xi$ leads to a dominant contribution of
$\xi\,\tilde{T}_{e}/\langle T_{e}\rangle$ in Eq. (4) (blue triangles). The
term in $\tilde{T}_{e}$ is then twice larger than the term in $\tilde{n}$, for
both lines and all $B$ field values : this explains well why $\tilde{I}_{cam}$
is globally better correlated to $\tilde{T_{e}}$ than to $\tilde{n}$ as found
in section V.

Figure 15: Comparison of the terms responsible for the light intensity
fluctuations as expressed in Eq. (4).

Let us now compare the amplitude of $\tilde{I}_{cam}/\langle I_{cam}\rangle$
to the prediction of Eq. (4), in which the phase between $\tilde{n}$ and
$\tilde{T}_{e}$ has to be taken into account. Using the five-tips probe
measurements of $n$ and $T_{e}$, the quantity $\frac{\tilde{n}}{\langle
n\rangle}+\xi\frac{\tilde{T}_{e}}{\langle T_{e}\rangle}$ is reconstructed for
each value of the radius in the range $0\leq r\leq 6$ cm, and its standard
deviation is computed, yielding a radial profile along $r$. This profile is
then transformed by the integration process described in subsection II.4 so as
to be expressed along $r^{*}$. It is compared to radial profiles of the
standard deviation of $\tilde{I}_{cam}/\langle I_{cam}\rangle$ in Fig. 16,
with the light filtered at $488$ nm and $750$ nm, and $I_{B}=[160;320;480]$ G.
Let us focus on the case $B=320$ G (full lines in Fig. 16). At $488$ nm the
match is very good inside the plasma column and relatively satisfactory at the
edge. At $750$ nm, the match is extremely satisfactory for $3.5\leq r^{*}\leq
6$, and slightly less towards the center. Overall, the reconstruction process
of the light intensity normalized fluctuations form probe measurements, using
a first order approximation of the model given in Eq. (2), is satisfactory.
The order of magnitude of $\tilde{I}_{cam}/\langle I_{cam}\rangle$ can be
recovered from the plasma parameters.

Figure 16: Comparison between the amplitudes of the normalized light intensity
fluctuations, and its reconstruction with five-tips probe measurements of $n$
and $T_{e}$ in the range $r\in[0:6]$ cm, for wavelengths $488$ nm (left) and
$750$ nm (right). $\xi_{i,n}=\varepsilon_{ex}^{i,n}/\langle T_{e}\rangle$ with
$\varepsilon_{ex}^{i}=14.4$ eV and $\varepsilon_{ex}^{n}=20$ eV. The values of
$B$ presented are $160$ G (dotted lines), $320$ G (full lines) and $480$ G
(dash-dotted lines).

##  VII Conclusion

Visible light naturally radiated by an Argon weakly magnetized plasma column
was compared to plasma parameters acquired from an electrostatic probe. Using
a Corona model, the origin of the light was interpreted in terms of simple
excitation processes that were identified, and whose dependence on both the
density and the electron temperature was modelled following a modified
Arrhenius laws $n^{a}e^{-\varepsilon_{ex}/T_{e}}$ with $a=1$ for the neutral
lines, and $a=2$ for the ion one. The average excitation energy
$\varepsilon_{ex}$ was deduced from numerical fits, using cross sections from
the literature. The model was then compared to camera recorded light for
values of the magnetic field up to $640$ G. This was done for mean radial
profiles, where geometrical corrections have to be taken into account, and for
temporal fluctuations for which simultaneous measurements of camera and probe
were performed.

More specifically, the following points were highlighted:

  * •

The influence of electron temperature fluctuations cannot be ignored for a
correct interpretation of the filtered light fluctuations, over a wide range
of plasma parameters and diverse plasma fluctuations.

  * •

The influence of the plasma density on light intensity (i.e. whether light
emitted scales as $n$ or $n^{2}$) is nearly indistinguishable for the time-
averaged profiles. Contrarily, it is very important for light fluctuations
intensity.

  * •

For most of the reported regimes, the correlation between the electron
temperature and the light fluctuations is larger than the correlation between
the plasma density and the light fluctuations. A first order approximation of
the modified Arrhenius law shows that this is expected when the energy
excitation greatly exceeds the electron temperature. It also gives an
efficient estimation whether $\tilde{I}_{cam}$ should rather be considered as
reflecting the density or electron temperature fluctuations, if not a
combination of both.

Our systematic study demonstrated the importance of considering the influence
of the electron temperature for the correct interpretation of high speed
imaging of plasma emitted light, which cannot be simply considered as a proxy
for the plasma density fluctuations.

##  VIII Acknowledgements

This work was partly supported by French National Research Agency under
contract ANR- 13-JS04-0003-01. We acknowledge support from the CNRS for the
acquisition of the high-speed camera, and useful discussions with V.
Désangles, G. Bousselin and E. De Giorgio.

## Appendix A : Camera imaging radial profiles

The radial profiles of the light intensity mean values and fluctuation
amplitude are presented in Fig. 17. These profiles show a relatively poor
signal/noise ratio for the filter around 488 nm, ranging from 1.2 at the
plasma column center for $B=160$ G, to a value of 3 when the light collected
is maximal around $r=1.5$ cm for $B=640$ G. For the filter at 750 nm this
ratio is significantly improved, ranging from 2.7 ($r=0$ cm, $B=160$ G) to 6.2
($r=2.5$ cm, $B=640$ G). At 810 nm the signal/noise ratio varies between 2.8
at the plasma center for $B=160$ G, to 5.4 at $r=2$ cm and $B=640$ G.

Figure 17: Radial profiles of the mean amplitude and fluctuations standard
deviation of the light intensity, for $B=[160;320;480;640]$ G.

## Appendix B : Fits of the rate constants

Figure 18: Cross sections of all the processes reported in table 2, plotted
along with an example of the $K_{ex}$ integrand $\varepsilon
e^{\varepsilon/T_{e}}$, for $T_{e}=4$ eV.

Figure 19: Fits of $K_{ex}(T_{e})$ as a function of the form
$K_{ex}^{0}e^{-\varepsilon_{ex}/T_{e}}$, yielding the averaged excitation
energies $\varepsilon_{ex}$ reported in table 2.

## Appendix C : Integration reverse process

Figure 20: Test of the integration process described in subsection II.4,
applied to an Heaviside function centred around $r=4$ cm.

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