# Wave propagation and power deposition in blue-core helicon plasma

Lei Chang1,2, Juan F. Caneses3, Saikat C. Thakur4, and Huai-Qing Zhang1∗
1State Key Laboratory of Power Transmission Equipment & System Security and
New Technology, Chongqing University, Chongqing, $400044$, China  2Institute
of Plasma Physics, Chinese Academy of Sciences, Hefei, $230031$, China
3CompX, Del Mar, CA $92014$, USA  4Department of Physics, Auburn University,
Auburn, AL $36849$, USA<EMAIL_ADDRESS>
###### Abstract

The wave propagation and power deposition in blue-core helicon plasma are
computed referring to recent experiments. It is found that the radial profile
of wave electric field peaks off-axis during the blue-core formation, and the
location of this peak is very close to that of particle transport barrier
observed in experiment; the radial profile of wave magnetic field shows
multiple radial modes inside the blue-core column, which is consistent with
the experimental observation of coherent high $m$ modes through Bessel
function. The axial profiles of wave field indicate that, once the blue-core
mode has been achieved, waves can only propagate inside the formed column with
distinct phase compared to that outside. The wave energy distribution shows a
clear and sharp boundary at the edge of blue-core column, besides which
periodic structures are observed and the axial periodicity inside is nearly
twice that outside. The dispersion relation inside the blue-core column
exhibits multiple modes, a feature of resonant cavity that selects different
modes during frequency variation, while the dispersion relation outside gives
constant wave number with changed frequency. The power deposition appears to
be off-axis in the radial direction and periodic in the axial direction, and
mostly inside the blue-core column. Analyses based on step-like function
theory and introduced blue-core constant provide consistent results. The
equivalence of blue-core column to optical fiber for electromagnetic
communication is also explored and inspires a novel application of helicon
plasma, which may be one of the most interesting findings of present work.

Keywords: helicon plasma, blue-core, wave propagation, power deposition

PACS: 52.35.Hr, 52.25.Os, 52.50.Qt, 52.80.Pi

##  1 Introduction

The underlying physics of helicon discharge, which can produce high-density
plasma with remarkable ionisation rate, have been attracting great research
interest[1, 2, 3, 4, 5, 6]. Besides further measurements of TG
(Trivelpiece–Gould) mode and energetic electrons, the blue-core phenomenon
that plasma shrinks onto axis and emits bright light in blue colour (argon)
for high input power and confining magnetic field remains mysterious and most
challenging of the field[7, 8, 9, 10, 11, 12]. It represents a general bright-
core mode of helicon discharge for other gases as well, for example nitrogen
and helium[13, 14], although in different colours, and is the highest level
for which the gas is fully ionised (further increased power does not enhance
the plasma density but the ionisation state due to ion pumping effects[15]).
Four common features have been drawn from this mode: radial electrostatic
confinement, light emission in phase axially, azimuthal instabilities driven
by radial pressure gradients, and high-beta (beta is the ratio of particle
pressure to magnetic field pressure) effects[16]. This work is devoted to
studying these features in terms of wave propagation and power deposition. A
well-benchmarked electromagnetic solver (EMS)[17] will be employed for
computations, referring to the recent experiments on CSDX (controlled shear
decorrelation experiment)[9, 18], and the step-like function theory[19] will
be utilised for physics analysis, followed by equivalent analogy of blue-core
plasma column to optical fiber for electromagnetic communications. We shall
present that the spatial features of blue-core mode observed in experiments
could be well seen from the wave propagation and power deposition
computations, which offer more details. Further exploration on the high-beta
effects, maybe in pulsed mode[12, 20], will be given in a follow-up work. The
equivalence to optical fiber may promote novel applications of blue-core
helicon plasma, an exciting and valuable finding of present work.

##  2 Numerical Scheme

###  2.1 Electromagnetic solver

The EMS, which has been used successfully to model various plasma sources[21,
22, 23, 24, 25, 26], is based on two Maxwell’s equations: Faraday’s law and
Ampere’s law[17],

$\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t},$ (1)
$\nabla\times\mathbf{B}=\mu_{0}\left(\mathbf{j}_{a}+\frac{\partial\mathbf{D}}{\partial
t}\right),$ (2)

with $\mathbf{E}$ and $\mathbf{B}$ the wave electric and magnetic fields,
respectively. The symbols of $\mu_{0}$ and $t$ are standard permeability of
vacuum and time. The system is driven by the current density $\mathbf{j}_{a}$
of external antenna. Perturbations vary in form of $\exp[i(kz+m\theta-\omega
t)]$, with $k$ the axial wave number, $m$ the azimuthal mode number and
$\omega$ the driving frequency, for a right-hand cylindrical coordinate system
$(r;\theta;z)$. The displacement vector $\mathbf{D}$ is linked to $\mathbf{E}$
via a cold-plasma dielectric tensor[27],

$\mathbf{D}=\varepsilon_{0}[\varepsilon\mathbf{E}+ig(\mathbf{E}\times\mathbf{b})+(\eta-\varepsilon)(\mathbf{E}\cdot\mathbf{b})\mathbf{b}].$
(3)

Here, $\varepsilon_{0}$ is the permittivity of vacuum and $\mathbf{b}$ is the
unit vector of external magnetic field ($\mathbf{b}=\mathbf{B_{0}}/B_{0}$).
The dielectric tensor comprises three components:

$\varepsilon=1-\sum_{\alpha}\frac{\omega+i\nu_{\alpha}}{\omega}\frac{\omega_{p\alpha}^{2}}{(\omega+i\nu_{\alpha})^{2}-\omega_{c\alpha}^{2}},$
(4)
$g=-\sum_{\alpha}\frac{\omega_{c\alpha}}{\omega}\frac{\omega_{p\alpha}^{2}}{(\omega+i\nu_{\alpha})^{2}-\omega_{c\alpha}^{2}},$
(5)
$\eta=1-\sum_{\alpha}\frac{\omega_{p\alpha}^{2}}{\omega(\omega+i\nu_{\alpha})}.$
(6)

The subscript $\alpha$ labels the species of particles, i. e. ion and
electron, and the plasma frequency
$\omega_{p\alpha}=\sqrt{n_{\alpha}q_{\alpha}^{2}/\varepsilon_{0}m_{\alpha}}$
and cyclotron frequency $\omega_{c\alpha}=q_{\alpha}B_{0}/m_{\alpha}$ are
standard definitions. The phenomenological collision frequency $\nu_{\alpha}$
accounts for collisions between electrons, ions and neutrals, where background
pressure is implemented. For the half-turn helical antenna considered below,
$\mathbf{j}_{a}$ has three components:

$\begin{array}[]{ll}j_{ar}=0,\end{array}$ (7)
$\begin{array}[]{ll}j_{a\theta}=&I_{a}\frac{e^{im\pi}-1}{2}\delta(r-R_{a})\left\\{\frac{i}{m\pi}\left[\delta(z-z_{a})+\delta(z-z_{a}-L_{a})\right]\right.\\\
\\\
&\left.+\frac{H(z-z_{a})H(z_{a}+L_{a}-z)}{L_{a}}e^{-im\pi[1-(z-z_{a})/L_{a}]}\right\\},\end{array}$
(8) $\begin{array}[]{ll}j_{az}=I_{a}\frac{e^{-im\pi[1-(z-z_{a})/L_{a}]}}{\pi
R_{a}}\frac{1-e^{im\pi}}{2}\delta(r-R_{a})\times
H(z-z_{a})H(z_{a}+L_{a}-z).\end{array}$ (9)

Here, the subscript $a$ denotes the antenna, i. e. $L_{a}$ the length, $R_{a}$
the radius, $z_{a}$ the distance to left endplate, $I_{a}$ the magnitude of
antenna current, and $H$ is the Heaviside step function. The boundary
conditions enclosing the model are formed by assuming that the tangential
components of $\mathbf{E}$ vanish on the surface of chamber walls:

$E_{\theta}(R,z)=E_{z}(R,z)=0,$ (10) $E_{r}(r,0)=E_{\theta}(r,0)=0,$ (11)
$E_{r}(r,L)=E_{\theta}(r,L)=0,$ (12)

where $R$ and $L$ are the radius and length of chamber, respectively.

###  2.2 Domain and conditions

The EMS model is solved by finite difference method based on four staggered
rectangular grids[17]. Figure 1 shows the computational domain. It is drawn as
close as possible to the recent layout of CSDX experiments[9, 18], while
taking into account the real operation of EMS. The source chamber (pyrex
glass) in length of $0.4$ m and diameter of $0.15$ m connects the diffusion
chamber (stainless steel) of length $2.8$ m and diameter $0.2$ m coaxially.
Two diagnostic ports are located at $z=1.2$ m ($P_{1}$) and $z=2$ m ($P_{2}$),
respectively, in the diffusion chamber to measure the cross-sectional
parameters.

Figure 1: Schematic of computational domain referring to CSDX[9, 18, 28].

The source plasma is generated by a half-turn helical antenna in length of
$0.2$ m and diameter of $0.15$ m, for which the current density is expressed
by Eq. (7)$\sim$Eq. (9). The antenna is driven by a radio-frequency (RF) power
supply with fixed frequency of $13.56$ MHz and maximum power of $5$ kW. Here,
we also fix the current magnitude to $12$ A throughout the paper for accurate
comparison, which is within the power capacity. The filling gas is argon and
the external magnetic field is uniform with typical strengths of $0.08$ T,
$0.12$ T, $0.14$ T and $0.16$ T. Accordingly, four normalised radial profiles
of plasma density are constructed, as shown in Fig. 2. They are fitted from
experimental data[9, 29] with expressions: $\exp[-2746.46r^{2}]$ for $0.08$ T,
$\exp[-4167.88r^{2}]$ for $0.12$ T, $\exp[-35526.8r^{2}]$ for $0.14$ T, and
$\exp[-31493.9r^{2}]$ for $0.16$ T. We can see that the plasma shrinks onto
axis with increased magnetic field, i. e. larger gradient in radius, whereas
field strength higher than $0.14$ T yields little difference. Other conditions
are also set to be the same to the CSDX experiments[9, 18], including electron
temperature of $4$ eV.

Figure 2: Normalised radial profiles of plasma density for four external
magnetic field strengths, fitted from experimental data on the CSDX[9, 29].

##  3 Computed Results

###  3.1 Wave propagation

We first investigate the wave propagation characteristics for the plasma
density profiles constructed above. Figure 3 and Fig. 4 give the radial
profiles of wave electric field and magnetic field, respectively, calculated
at the first port ($P_{1}$ in Fig. 1), which provides convenience for
experimental verification. It can be seen that the wave electric field clearly
forms a local peak around $r\approx 0.015$ m for high field strengths.
Considering that this peak expels ions away and attracts electrons together on
both sides, we thereby claim that it forms a local transport barrier and is an
evidence of radial electrostatic confinement. Further, this temporally
oscillating peak can induce ponderomotive force which may also contribute to
the formation of transport barrier[30, 31, 32]. Indeed, the location of this
peak is very close to the experimental observation of transport barrier at
$r\approx 0.02$ m during the blue-core formation[9, 18].

$\begin{array}[]{ll}(a)&(b)\\\
\includegraphics[width=201.6317pt,angle={0}]{er_gsr.eps}&\includegraphics[width=212.47617pt,angle={0}]{eth_gsr.eps}\end{array}$

Figure 3: Radial profiles of wave electric field (a, radial component; b,
azimuthal component) for four external magnetic field strengths, measured at
$z=1.2$ m ($P_{1}$).

Moreover, the wave magnetic field shows additional radial modes inside
$r\approx 0.015$ m for high field strengths. This is consistent with the
experimental observation that very coherent high-$m$ fluctuations occur inside
the blue-core column. Due to the limitation of EMS that only a single $m$ can
be considered for each run of simulation ($m=1$ throughout the paper for the
half-turn helical antenna employed here), however, we cannot see directly the
appearance of high-$m$ structure but indirectly from the formation of multiple
radial modes.

$\begin{array}[]{ll}(a)&(b)\\\
\includegraphics[width=207.26897pt,angle={0}]{br_gsr.eps}&\includegraphics[width=208.13574pt,angle={0}]{bth_gsr.eps}\end{array}$

Figure 4: Radial profiles of wave magnetic field (a, radial component; b,
azimuthal component) for four external magnetic field strengths, measured at
$z=1.2$ m ($P_{1}$).

This can be inferred from the resonant eigenmodes[33]:

$B_{r}=\frac{iU}{2Q}\left[(\beta+k)J_{m-1}(Qr)+(\beta-k)J_{m+1}(Qr)\right],$
(13)
$B_{\theta}=-\frac{U}{2Q}\left[(\beta+k)J_{m-1}(Qr)-(\beta-k)J_{m+1}(Qr)\right],$
(14)

with $U$ the amplitude constant and $Q^{2}=\beta^{2}-k^{2}$. It is the Bessel
function $J_{m}(r)$ that correlates the radial and azimuthal modes, while in
physics they are essentially determined by the spatial eigenmode resonance.
Overall, the radial profiles of wave electric field and magnetic field show
significant difference when the confining magnetic field increases from low
magnitude (before blue-core formation) to high magnitude (after blue-core
formation). These different propagation features before and after the blue-
core formation can be also observed in the axial direction. We compute the
wave magnetic field, which is easily measurable by B-dot probe in experiment,
at two radial locations, namely $r=0$ m (inside blue-core column) and $r=0.04$
m (outside blue-core column). As shown by Fig. 5, the wave propagation is more
evanescent for higher field strength with blue-core formation, and this occurs
for both radial locations. Moreover, comparing the wave field profiles at
these two locations, we can see that the wave propagation on axis surpasses
that near edge, especially after the blue-core formation.

$\begin{array}[]{ll}(a)\\\
\includegraphics[width=325.215pt,angle={0}]{bth_gsz_in.eps}\\\ (b)\\\
\includegraphics[width=329.54881pt,angle={0}]{bth_gsz_ex.eps}\end{array}$

Figure 5: Axial profiles of wave magnetic field for various confining field
strengths, measured inside (a, $r=0$ m) and outside (b, $r=0.04$ m) blue-core
column.

To show more details, we compute the phase of wave magnetic field inside and
outside the blue-core column ($B_{0}=0.16$ T), via
$\theta=\textrm{arctan}~{}\textrm{Im}[B_{\theta}]/\textrm{Re}[B_{\theta}]$.
Figure 6 presents the results. One can see that the wave propagation inside
has identical phase in the axial direction ($z\approx 1.1\sim 2.7$ m away from
antenna and endplates), which is consistent with the light emission in phase
axially observed in experiment[16], whereas there seems no regular phase
outside but periodic singularities.

$\begin{array}[]{ll}(a)\\\
\includegraphics[width=323.04477pt,angle={0}]{phase_in.eps}\\\ (b)\\\
\includegraphics[width=329.54881pt,angle={0}]{phase_ex.eps}\end{array}$

Figure 6: Phase for wave magnetic field inside (a, $r=0$ m) and outside (b,
$r=0.04$ m) blue-core column ($B_{0}=0.16$ T).

The axial profiles of wave electric field show similar features. This implies
that the existence of blue-core plasma column in a sense confines the wave
propagation inside, which is illustrated more clearly by the two-dimensional
contour plots of wave energy distribution shown in Fig. 7. We can see that,
with the field strength increased, the wave energy shrinks towards axis and
forms a sharp boundary around $r\approx 0.018$ m, close to the location of
transport barrier at $r\approx 0.02$ m observed in experiment[9, 18].
Interestingly, quasi-periodic structures are formed axially on both sides of
the boundary layer, and the periodic length inside is close to twice that
outside. This structure looks similar to the beat pattern of helicon radiation
observed earlier[23, 34], and is consistent with the whistler bouncing at
sharp plasma edge[35, 36]. Moreover, the wave energy distribution becomes off-
axis when it expands from the source region to the diffusion region.

Figure 7: Wave energy distributions for various confining magnetic field
strengths.

Actually, for these high-field cases, i. e. $B_{0}=0.14$ T and $B_{0}=0.16$ T,
there always exists an off-axis peak in the radial profile of wave energy
density, as shown in Fig. 8. The radial locations of these off-axis peaks
($r\approx 0.015$ m) are very close again to that of experimental transport
barrier ($r\approx 0.02$ m)[9, 18]. Here, integration has been performed along
the axial direction, same to the method of experimental measurement, showing
the accumulated wave energy in the cross section.

Figure 8: Integrated radial profiles of wave energy for the four external
magnetic field strengths. The inset shows normalised results.

To reveal the intrinsic physics of different wave propagation features inside
and outside the blue-core column, we focus on the case of $B_{0}=0.16$ T and
run the EMS code for frequency range of $f=\pi\sim 10\pi$ MHz. Utilising a
Fourier decomposition method to extract the dominant $k$ from the axial
profiles of wave field, we obtain the dispersion relations shown in Fig. 9.

$\begin{array}[]{ll}(a)&(b)\\\
\includegraphics[width=199.4681pt,angle={0}]{dispersion_gs_a.eps}&\includegraphics[width=199.4681pt,angle={0}]{dispersion_gs_b.eps}\end{array}$

Figure 9: Dispersion relations inside (a, $r=0$ m) and outside (b, $r=0.04$ m)
blue-core column for $B_{0}=0.16$ T (Gaussian density profile).

For the location inside blue-core column, analytical dispersion relation[37]

$k=\frac{2\pi R_{p}}{3.83}\frac{e\mu_{0}n_{e}}{B_{0}}f$ (15)

has been also plotted with $R_{p}=0.02$ m (plasma radius) and
$n_{e}=n_{e0}\times 10^{19}~{}\textrm{m}^{-3}$. Here, the number $3.83$ is the
first non-zero Bessel root of $J_{1}(r)=0$, representing the first radial
mode. The straight lines are fitted results. It seems that the formed blue-
core column behaves as a resonant cavity, which could shift and select
different modes for different ranges of driving frequency. For the location
outside blue-core column, the computed dispersion relation is not linear, thus
cannot be fitted with Eq. (15). Instead, we have drawn a horizontal line to
label the average value of $k=22.37~{}\textrm{m}^{-1}$. The corresponding wave
length is $\lambda=0.28$ m, close to the length of antenna ($0.2$ m), and
independent of frequency, whereas the wave length inside blue-core column is
much shorter and varies with frequency. This demonstrates that the wave modes
inside and outside blue-core column are essentially different. As shown in
Fig. 5 and Fig. 6, once the blue-core column has been formed, wave propagation
is mostly confined inside, similar to the light propagation in optical
fiber[38, 39]. This feature is also consistent with the experimental
observation of light emission in phase axially[16], and may inspire novel
applications of blue-core helicon plasma, which will be analysed further in
next section.

###  3.2 Power deposition

Next, we explore how the power is deposited from antenna to plasma, especially
when the blue-core column has been formed. Figure 10 shows the radial profiles
of computed power deposition, which have been integrated over the axial
direction in a similar way as for Fig. 8. One can see that the magnitude of
power deposition reduces significantly when the field strength increases from
$0.12$ T to $0.14$ T, indicating an evolution into distinct mode. This is
consistent with the experimental observation that the discharge transits from
blue-colour mode to blue-core mode[9]. The decreased power for shrunk density
profiles implies that certain magnitude of plasma density near edge is
beneficial for power coupling, which also agrees with previous studies[40, 41,
42, 43, 44]. More interestingly, we find that the power deposition is hollow
in radius for all field strengths and its peak moves closer to axis when the
field strength increases.

Figure 10: Integrated radial profiles of power deposition for the four
external magnetic field strengths. The inset shows normalised results.

A full picture of power deposition is given by Fig. 11. We can see that,
similar to Fig. 7, the distribution shrinks radially towards axis when the
field strength is increased and forms a sharp boundary after the blue-core
establishment. The radial location of this layer ($r=0.015$ m) is slightly
closer to axis than that formed by the wave energy distribution ($r=0.018$ m),
but both close to the radius of measured transport barrier ($r=0.02$ m)[9,
18]. Moreover, the axial periodic structures are observed again inside and
ouside the boundary layer; different from Fig. 7, however, the periodic
lengths are equivalent. Further, the maximum power deposition is off-axis,
especially far away from the antenna in the diffusion region.

Figure 11: Power depositions for various confining magnetic field strengths.

##  4 Theoretical Analysis

###  4.1 Step-like function theory

To reveal the underlying physics more clearly, we employ the step-like
function theory developed by Breizman and Arefiev for radially localised
helicon mode[19]. The theory comprises two equations:

$\frac{1}{r}\frac{\partial}{\partial r}\left[r\frac{\partial E}{\partial
r}\right]-\frac{m^{2}}{r^{2}}E=-\frac{m}{k^{2}r}\frac{\omega^{2}}{c^{2}}\frac{E\partial
g/\partial r}{1+(m\partial g/\partial r)/k^{2}r\eta},$ (16)
$\frac{1}{r}\frac{\partial}{\partial r}\left[\varepsilon
r\frac{\partial}{\partial r}E_{z}\right]-\frac{m}{r}\left[\frac{\partial
g}{\partial r}+\frac{\varepsilon m}{r}\right]E_{z}-k^{2}\eta E_{z}=0,$ (17)

for helicon and TG modes, respectively, with
$E=E_{z}\left[1+k^{2}r\eta/(m\partial g/\partial r)\right]$. To focus on the
radial density gradient, an artificial step-like density profile can be
constructed:

$n_{\alpha}(r)=\left\\{\begin{array}[]{ll}n_{0}{}{}{}&\textrm{for}~{}~{}r<r_{\ast},\\\\[5.0pt]
n_{\ast}{}{}{}&\textrm{for}~{}~{}r>r_{\ast}.\end{array}\right.$ (18)

The symbols of $n_{0}$ and $r_{0}$ represent the maximum density on axis and
edge radius of plasma column, respectively, while $n_{\ast}$ and $r_{\ast}$
label the magnitude and radius of density jump. Figure 12 shows an
illustration. Please note that this step-like density profile, although is
unrealistic in experiment, can manifest the effect of radial density gradient
and simplify the mathematical treatment.

Figure 12: Step-like density profile in radius, a schematic illustrating the
concept of blue-core constant below ($C_{b}=C_{n}\times
C_{r}=n_{\ast}/n_{0}\times r_{\ast}/r_{0}$).

The resulted electric field has expression[19]:

$E(r)=E_{0}\left\\{\begin{array}[]{ll}(r/r_{\ast})^{|m|}{}{}{}&\textrm{for}~{}~{}r<r_{\ast},\\\\[5.0pt]
(r/r_{\ast})^{-|m|}{}{}{}&\textrm{for}~{}~{}r>r_{\ast},\end{array}\right.$
(19)

with $E_{0}$ a constant. For the $m=1$ mode considered here, its radial
profile is displayed in Fig 13 (with $r_{\ast}=0.02$ m and $r_{0}=0.07$ m
referring to the CSDX experiments[9, 18]).

Figure 13: Radial profile of wave electric field for step-like density shown
in Fig. 12.

This resembles the computed results shown in Fig. 3 for the blue-core mode
(high field cases) that wave electric field peaks off axis and decreases
toward both axis and edge. The analytical dispersion relation is accordingly

$\omega=2\frac{m}{|m|}\frac{\omega_{ce}k^{2}c^{2}}{\omega_{pe}^{2}(n_{\ast})-\omega_{pe}^{2}(n_{0})}.$
(20)

To compare with numerical results, again we run the EMS code for various
frequencies ($f=\pi\sim 10\pi$ MHz) and employ Fourier decomposition method to
extract the dispersion relations, similar to the procedure done for Fig. 9.
Here, the employed conditions include $n_{\ast}=0.3\times n_{0}$ and
$r_{\ast}=0.3\times r_{0}$. As shown in Fig. 14, the dispersion curves from
step-like theory (Eq. (20)) and simple relation (Eq. (15)) agree with the
computed results (dots) qualitatively both inside and outside the blue-core
column. Surprisingly, the simple relation based on slab geometry and single
density value (on axis here) exhibits slightly better consistence, as observed
in previous studies[23, 26]; moreover, as long as the blue-core column has
been formed, we observe interference on the axial profiles of wave field for
both Gaussian and step-like density configurations, which indicates the
existence of multiple wave modes inside the blue-core column. Comparing Fig.
14(b) with Fig. 9(b), one could find the difference here that the wave number
outside the blue-core column is not constant but varies with driving
frequency. This may be caused by the density level outside the blue-core
column which is not vanishing here but $0.3$ of the peak value on axis.

$\begin{array}[]{ll}(a)&(b)\\\
\includegraphics[width=199.4681pt,angle={0}]{dispersion_relation_sp0.eps}&\includegraphics[width=199.4681pt,angle={0}]{dispersion_relation_sp4.eps}\end{array}$

Figure 14: Dispersion relations inside (a, $r=0$ m) and outside (b, $r=0.04$
m) the blue-core column for $B_{0}=0.16$ T (step-like density profile).

Next, to explore in detail the effects of density jump, i. e. magnitude and
location, we introduce the ratios of $C_{n}=n_{\ast}/n_{0}$ and
$C_{r}=r_{\ast}/r_{0}$, respectively, and the product of $C_{b}=C_{n}\times
C_{r}$ to quantify the shrinking feature of blue-core plasma, as shown in Fig.
12. This product can be defined in phrase of “blue-core constant” or more
generally “bright-core constant” to cover other gases as well.

$\begin{array}[]{ll}(a)&(b)\\\
\includegraphics[width=199.4681pt,angle={0}]{cn.eps}&\includegraphics[width=199.4681pt,angle={0}]{cr.eps}\end{array}$

Figure 15: Dependence of $\gamma$ ($\gamma=P_{\ast}/P_{0}$) on: (a)
$C_{n}=n_{\ast}/n_{0}$, (b) $C_{r}=r_{\ast}/r_{0}$.

We also setup the parameter of $\gamma=P_{\ast}/P_{0}$ in simulations to
measure the ratio of power deposition inside the blue-core column ($P_{\ast}$)
to that in total ($P_{0}$). The radius for $P_{\ast}$ is chosen to be $r=0.02$
m, same to the measured edge of blue-core column in experiments[9, 18]. Figure
15 shows the computed dependence of $\gamma$ on $C_{n}$ and $C_{r}$. We can
see that $\gamma$ largely increases with reduced $C_{n}$ and $C_{r}$, which is
expected because the plasma column is more shrunk for smaller $C_{n}$ and
$C_{r}$, except for $C_{n}<0.2$ the ratio drops back. This could be attributed
to the plasma density near edge which is too low to efficiently couple the
power from antenna into core. This critical role of edge density has been also
claimed by other studies[40, 41, 42, 43, 44].

###  4.2 Equivalence to optical fiber

Inspired by the finding above that wave propagation is confined mostly inside
the blue-core column and the radial density gradient is very large (close to
step-like), we propose that this blue-core helicon plasma could be used for
electromagnetic communications, similar to the optical fiber for light
communication[38, 39]. To achieve this, the condition of total reflection has
to be met, and Fig. 16 gives an illustration. This illustration is consistent
with previous findings that obliquely propagating waves are reflected by
radial density gradient before they reach the edge of plasma[45, 46],
following a zig-zap motion confined to the high density core.

Figure 16: Illustration of blue-core plasma column behaving as “optical fiber”
for electromagnetic communications.

According to the law of refraction, namely
$n_{1}\sin\phi_{1}=n_{2}\sin\phi_{2}$ with $n$ the index of refraction and
$\phi$ the angle to normal direction, we know that the threshold angle for
total reflection ($\phi_{2}=\pi/2$) is
$\phi_{1}=\textrm{arcsin}~{}(n_{2}/n_{1})$. Referring to the definition of $n$
($n=c/v_{ph}$ or $n=\lambda_{0}/\lambda$ with $c$ the speed of light, $v_{ph}$
the phase velocity, and $\lambda_{0}$ the wavelength in vacuum) and waves in
uniform magnetised plasma[27, 47], we can write $n$ in form of

$n^{2}=\frac{G\pm F}{2(\varepsilon\sin^{2}\psi+\eta\cos^{2}\psi)}$ (21)

with:

$G=(\varepsilon^{2}-g^{2})\sin^{2}\psi+\varepsilon\eta(1+\cos^{2}\psi),$ (22)
$F^{2}=\left[(\varepsilon^{2}-g^{2})-\varepsilon\eta\right]^{2}\sin^{4}\psi+4g^{2}\eta^{2}\cos^{2}\psi.$
(23)

Please note that here $\psi$ labels the angle of wave vector to the confining
magnetic field, which lies in the same direction of blue-core edge, so that we
have $\psi+\phi_{1}=\pi/2$ as shown in Fig. 16. For either parallel wave, i.
e.

$n^{2}=1-\sum_{\alpha}\frac{\omega_{p\alpha}^{2}}{\omega(\omega\pm\omega_{c\alpha})}\approx\sum_{\alpha}\frac{\omega_{p\alpha}^{2}}{\omega(\omega\pm\omega_{c\alpha})},$
(24)

or oblique wave such as whistler mode, i. e.

$n^{2}=\frac{\omega_{pe}^{2}}{\omega(\omega_{ce}\cos\psi-\omega)},$ (25)

we can all draw conclusion with high-density approximation
($\omega^{2}\ll\omega_{p\alpha}^{2}$ and
$\omega_{c\alpha}^{2}\ll\omega_{p\alpha}^{2}$) that the index of refraction is
proportional to the square root of plasma density, i. e.
$n\propto\sqrt{n_{\alpha}}$, if other conditions (frequency and field
strength) are fixed. Therefore, as long as the incident angle is bigger than
the threshold value of
$\phi_{1}=\textrm{arcsin}~{}(\sqrt{n_{\ast}}/\sqrt{n_{0}})$, the total
reflection will occur and the blue-core plasma can indeed behave as an
“optical fiber” for electromagnetic communications. Figure 17 plots the two-
dimensional wave amplitude ($B_{\theta}$) for different values of $C_{n}$
($C_{r}$ fixed to $0.3$). We can see that with increased magnitude of density
jump, i. e. decreased $C_{n}$ ($=n_{\ast}/n_{0}$), there forms a clear
boundary layer around the location of density jump ($r=0.021$ m) which
separates the cylinder radially into two regions: inner and outer. While the
bright area near the boundary layer may be caused by wave reflections from
antenna and core, the bright area near axis confirms that wave propagation is
well confined inside the inner, as long as the density jump is sufficient, a
feature same to optical fiber.

Figure 17: 2D Wave amplitude ($B_{\theta}$) for different values of $C_{n}$
($C_{r}$ fixed to $0.3$).

Similarly, the blue-core plasma could also act as a waveguide for waves of
other frequencies[48]. Experimental verification of these ideas can be
implemented on a well-defined blue-core helicon plasma with a tiny wave-
exciting antenna located inside the central column and multiple receiving
antennas placed outside. We leave it as a separate study and shall present in
the future.

##  5 Conclusion

The mechanism of blue-core phenomenon during helicon discharge has been an
enigma for the research community. Different from existing studies which
mainly employ optical camera or spectrometer to capture the formation
procedure experimentally, this work devotes itself to revealing the detailed
physics through computing the wave propagation and power deposition
characteristics, referring to the recent experiments on CSDX[9, 18]. A well-
benchmarked electromagnetic solver, based on Maxwell’s equations and a cold-
plasma dielectric tensor, is made use of. We found that: (i) the wave electric
field peaks off-axis and near the radial location of particle transport
barrier observed in experiment, an evidence of radial electrostatic
confinement, during the blue-core formation; (ii) the wave magnetic field
shows multiple radial modes inside the blue-core column, consistent with the
experimental observation of coherent high $m$ modes through Bessel function;
(iii) the axial profiles of wave field demonstrate that, once the blue-core
mode has been established, waves can only propagate inside the central column
with identical phase in the axial direction, agreeing with the early finding
of light emission in phase axially[16]; (iv) the two-dimensional distributions
of wave energy and power deposition show off-axis (or hollow) features in
radius, especially far from antenna in the diffusion region, and periodic
structures in axial direction besides the blue-core boundary layer with either
double (wave energy) or the same (power deposition) periodicity; (v) the
analysis using step-like function theory provides consistent results, in terms
of the radial profile of wave electric field and dispersion relation inside
the blue-core column; (vi) the equivalence of blue-core helicon plasma column
to optical fiber for electromagnetic communications possesses theoretical
feasibility, as long as the incident angle is larger than the threshold value,
e. g. $\phi_{1}=\textrm{arcsin}~{}(\sqrt{n_{\ast}}/\sqrt{n_{0}})$, and
inspires novel applications of helicon plasma. Future research may be devoted
to the experimental verification of this “blue-core fiber” and the physics
modelling of a more challenging topic: high-beta effects.

We appreciate Dr. Guangye Chen for providing the EMS code and many
instructions for its usage. This work was supported by the Chinese Academy of
Sciences “$100$” Talent Program (B) and the Science Foundation of Institute of
Plasma Physics (DSJJ-$2020$-$07$).

## Data Availability Statement

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

## ORCID IDs

Lei Chang: https://orcid.org/0000-0003-2400-1836

Juan F. Caneses: https://orcid.org/0000-0001-6123-2081

Saikat C. Thakur: https://orcid.org/0000-0002-8422-2705

Huai-Qing Zhang: https://orcid.org/0000-0002-2631-4740

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