# Effect of Inhomogeneous magnetic field on Plasma generation in a low
magnetic field helicon discharge

Sonu Yadav<EMAIL_ADDRESS>Institute for Plasma Research, HBNI, Bhat,
Gandhinagar 382428, India  Prabal K Chattopadhyay<EMAIL_ADDRESS>Institute for Plasma Research, HBNI, Bhat, Gandhinagar 382428, India  Kshitish
K. Barada  Department of Physics and Astronomy, University of California, Los
Angeles, CA 90095, USA  Soumen Ghosh  Department of Physics, University of
California, San Diego, CA 92093, USA  Joydeep Ghosh  Institute for Plasma
Research, HBNI, Bhat, Gandhinagar 382428, India

###### Abstract

The ionization efficiency of helicon plasma discharge is explored by changing
the low axial magnetic field gradients near the helicon antenna. The highest
plasma density is found for a most possible diverging field near the antenna
by keeping the other operating condition constant. Measurement of axial wave
number together with estimated radial wavenumber suggests the oblique mode
propagation of helicon wave along the resonance cone boundary. Propagation of
helicon wave near the resonance cone angle boundary can excite electrostatic
fluctuations which subsequently can deposit energy in the plasma. This process
has been shown to be responsible for peaking in density in low field helicon
discharges, where the helicon wave propagates at an angle with respect to the
applied uniform magnetic field. The increased efficiency can be explained on
the basis of multiple resonances for multimode excitation by the helicon
antenna due to the availability of a broad range of magnetic field values in
the near field of the antenna when a diverging magnetic field is applied in
the source.

††preprint: AIP/123-QED

##  I Introduction:

Efficient plasma sources are studied for their application as an economic
plasma source for micro-electronics industry J., D., and W. (1991), negative
ion production Mieno  _et al._ (1996), neutron generation Jung  _et al._
(2004) and space propulsion Charles (2009). The helicon plasma source is one
of the most efficient plasma source. This plasma source is well known for
their generation of high density plasma at relatively low powers and magnetic
field. Beyond the certain values of rf power and magnetic field, it is found
that the helicon wave can be absorbed in the plasma and generate the high
density. A large number of experiments have been conduct in various tube
length and antenna geometry. Normally helicon sources operate at a magnetic
field which corresponds to an electron cyclotron frequency 20 to 40 times of
source frequency. This means for normal operation of the helicon source using
13.56 MHz source 200 G magnetic field or more is required. However, helicon
sources have shown resonance abortion even around a particular low magnetic
field. Here, low magnetic field (20-30 G) means $f_{ce}\sim$ 5 times source
frequency.

Recently, low magnetic field helicon sources are proposed for space propulsion
Chen (2008); Ling  _et al._ (2010); Harle, Pottinger, and Lappas (2013) and
also can be used as sources for semiconductor applications C, C, and Boswell
(2011) as they can drastically reduce the cost of the magnet power supply and
subsequently the power consumption. Resonance absorption at low magnetic
fields W.  _et al._ (1996); Chen  _et al._ (1997); T.  _et al._ (1998); J  _et
al._ (2003); Sato, Oohara, and Hatakeyama (2007); Wang  _et al._ (2007); G.,
W., and R. (2004); S., Junghee, and Choe (2006); Lafleur, Charles, and Boswell
(2011) less than 100 G in a helicon discharge when a magnetic field is
increased contrary to its behavior of monotonic increase at higher magnetic
fields F and W (1997). All these experiments are carried out using uniform
magnetic fields near the antenna. However, Lafleur et al Lafleur, Charles, and
Boswell (2011) using a diverging magnetic field using one coil only near the
antenna have reported a ten-fold increase in density compared to the density
at zero magnetic field. While using two coils they observed that the density
does not increase as much compared to while using only the single coil. In
that reported experiment, the source coil current is varied and the current in
the exhaust coil is kept constant at one of the values between 0-5 A (in 1 A
steps). With the exhaust coil kept on, the non-uniformity of the magnetic
field near the antenna decreases. No explanations are given for higher density
production for the non-uniform magnetic field Case. Our earlier results Barada
_et al._ (2013) on the density peaking phenomena in a uniform low magnetic
field near antenna have shown that density increase is due to the resonance
behavior while wave propagating near resonance cone. Signature of excitation
of low frequency electrostatic fluctuations supports this hypothesis. As the
helicon modes (radial eigenmodes) in a bounded system are discrete, only a
single mode can undergo this phenomenon at a critical magnetic field when a
uniform magnetic field is applied near the antenna. For a non-uniform magnetic
field, there could be the resonance of different radial eigenmodes. This might
lead to higher production efficiency. Hence, study the density peaking
behavior in the nonuniform magnetic fields is useful.

In the present work, plasma production efficiency is studied by applying
different diverging magnetic field configuration near the source (antenna).
The plasma density increases with increasing the magnetic field non-uniformity
near the antenna. It is observed that the plasma density is highest at
particular non-uniform magnetic field configuration at low B in our
experiments. These results may be thought to be due multimode cyclotron
resonance due to the nonuniform magnetic fields near the antenna.

This paper is organized as follows. The experimental setup and diagnostics are
described in section II followed by experimental results and discussion in
section III and summary is presented in section IV.

##  II  Experimental Setup and Diagnostics:

The experiments are performed in the linear helicon plasma experimental device
( Fig. 1), which has been described before Barada  _et al._ (2012); Yadav  _et
al._ (2018). The vacuum system consists of a source and expansion chamber. The
source chamber is made of borosilicate glass tube and has 9.5 cm inner
diameter and 70 cm length and the left end is terminated by a insulating
plate. The other end of the source tube is attached to a 51 cm long stainless
steel (SS) expansion chamber of 21 cm inner diameter. The whole chamber is
evacuated to a base pressure of $1\times 10^{-6}$ mbar by a diffusion pump
connected to the expansion chamber. The Argon is used as the working gas and
experiments are carried out at pressure $1\times 10^{-3}$ mbar. An 18 cm long,
m = +1 right helicon antenna, placed around the source chamber is used to
produce plasma by means of a 13.56 MHz rf power generator through an L-type
impedance matching network. The reflected power is kept less than 2$\%$ for
all the experiments. The axial location of the antenna center is defined as z
= 0 and all other axial locations are with reference to this antenna center as
shown in Fig. 1.

Figure 1: Schematic diagram of the Helicon plasma experimental setup

Five water-cooled electromagnet coils, as shown in Fig. 1, are used to
generate an axial magnetic field ($B_{0}$). Four different magnetic field
configurations are used in our experiments, produced by passing direct current
(DC) through different electromagnet combinations. Fig. 2a shows the axial
variation of the simulated magnetic field when 1A DC current is passed through
different electromagnet combinations to produce four different magnetic field
configurations. Fig. 2b shows the axial variation of normalized magnetic field
with respect to field value at antenna center. Fig. 2b noticeably indicates
the magnetic field homogeneity or inhomogeneity in each configurations. The
leftmost coil as shown in the Fig. 1, is the first coil and the rightmost coil
is the fifth coil. Case A corresponds to a magnetic field profile when 1 A
current is passed through all 5 coils (coil 1-5). The Case B magnetic field is
produced by taking out the first coil and keeping the other 4 (coil 2-5). The
Case C magnetic field is obtained by removing the first and second coils from
left and keeping the other 3 coils (coil 3-5). Case D magnetic field
corresponds to 1 A current in the fourth and last coil by removing the first
three coils from the left.

Figure 2: Axial variation of (a) magnetic field strength and (b) Normalized
magnetic field with respect to field value at antenna center for Cases of A-D
when 1 A current is passed in the coils

Two separate single rf compensated Langmuir probes Yadav  _et al._ (2018), one
straight, inserted from the top radial port in the source chamber at z = 31 cm
and other L shaped, inserted off-axially from the end flange of the expansion
chamber at z = 50 cm, are used to measure the plasma density, $n_{0}$. These
locations also correspond to before ($z_{before}$ = 31 cm) and after
($z_{after}$ = 50 cm) both in the magnetic and geometric expansion Fig. 1. The
probe tips of nearly equal collection area, are made up of cylindrical
tungsten wire of 1 mm diameter and 4 mm length. The plasma density determined
from the ion saturation current by considering the sheath expansion I .

The wave detection is performed by measuring the amplitude and phase of axial
wave magnetic field ($B_{z}$), by positioning the high frequency B-dot probe
about 10 cm downstream from the antenna center and pulling it back in 2 cm
increments. The B-dot probe is a single loop of 6 mm diameter made from the
shielded coaxial cable of 1.8 mm outer diameter. At each position, the
amplitude is measured as the peak-to-peak value averaged over ten shots. To
measure the axial phase variation, the antenna current measured simultaneously
using a high frequency Rogowski coil, acts as a phase reference. Both signals
(antenna current and B-dot probe) are fed to separate channels of a digital
oscilloscope.

Figure 3: Helicon discharge mode transitions. Langmuir probe measurement of
center plasma density at (a) z = 31cm, (b) z = 50cm and (c) Value of load
capacitor ($C_{L}$) and (d) Antenna rms current as function of rf power,
measured at $1\times 10^{-3}$ mbar and $B_{0}$ = 160G ($I_{B}$ = 87A). Dash
lines show the capacitive (E) to inductive (H) and inductive to helicon wave
(W) mode transitions. Figure 4: On-axis variation of plasma density with
magnetic field at antenna center for the Case B in the (a) source chamber at z
= 31cm and (b) expansion chamber at z = 50cm, for different values of rf
powers, 100W solid diamonds, 300W solid triangles, 400W solid circle, 500W
solid square and 600W solid star.

##  III  Experimental Results and Discussion

It is well know that the helicon discharges are quite versatile and depending
on operating parameters, the discharge can be operated in capacitive (E),
inductive (H) or Helicon wave (W) mode J., D., and W. (1991); W.  _et al._
(1996); Kyeong, Sheridan, and Boswell (1999). To determine the different
discharge mode in our experiment, we measured the center plasma density in
both source chamber at z = 31cm (Fig. 3a) and expansion chambers at z = 50cm
(Fig. 3b). The $n_{0}$ is measured in quasi-uniform magnetic field
configuration, i.e. Case B as function of rf input power at $1\times 10^{-3}$
mbar pressure and B0 $\sim$ 160G ($I_{B}$ = 87A) applied magnetic field. Fig.
3 shows the gradual and abrupt changes in center plasma density that can be
identified as E-H and H-W transitions near 50 and 700W respectively. The
discharge mode transitions are also associated with discrete change in values
of load capacitor ($C_{L}$), Fig. 3c with optimized reflected power less than
2$\%$ by keeping the fixed value of tune capacitor ($C_{T}$) in matching
network Barada  _et al._ (2012). The measured values of rms antenna current
($I_{ant}$) as function of rf power is shown Fig. 3d, it also reflects the
mode transitions near 50 and 700W. It is found that discharge mode transition
from H to W is clearly observed in source and expansion plasma density for
power 700W. The density jump at B0 $\sim$ 160G is associated with the helicon
wave abortion Kyeong, Sheridan, and Boswell (1999).

In our previous experimental work Barada  _et al._ (2013) it has been shown
that the helicon wave is also absorbed even at low magnetic field ($<$ 100G).
The multiple density peaks at low magnetic fields has already been observed in
our Helicon Experimental device Barada  _et al._ (2012) when the uniform
magnetic field is applied near the antenna in the source chamber. In the
present experiments, the phenomenon of density peaking at the low magnetic
field is explored by changing the axial magnetic field gradient at the antenna
location. The $n_{0}$ measurement is performed in the four different axial
magnetic field configurations. These different magnetic field configurations
are entitled by the Case A, B, C and D as shown in Fig. 2. The $n_{0}$ is
obtained using the rf compensated Langmuir probe at two different axial
locations, z = 31 cm and 50 cm. For all present Cases (A, B, C, and D) the
values of the magnetic field at abscissa are taken at antenna center i.e. z =
0 cm.

Fig. 4a and Fig. 4b show the $n_{0}$ as function of magnetic field for the
Case B for different rf powers at z = 31cm and z = 50 cm respectively. As the
magnetic field is slowly increased from 0 G, the density rises monotonically
till 15 G (at 300 W) to 25 G (at 600W), before falling sharply to about half
of its peak value near 50-60 G in the source chamber, at z = 31 cm (Fig. 4a).
The effect of low magnetic field phenomena is also present in the bigger
expansion chamber, Fig. 3b. The density rise in the expansion chamber (far
away from the antenna) is monotonic till 25-40 G but fall is rather gradual
than source chamber. The fall of density with magnetic field is not same in
both the chambers may be due different diffusion process involved. At low rf
power (100W) the very weak density peak is present in both the source and
expansion chambers. The magnitude of the peak density is higher and the
profile becomes broader for increased rf powers. It is also noted that with
increasing rf power the density peak shifted to larger magnetic field values.
The similar observations can also be found in the previous experiments G., W.,
and R. (2004); T., C., and W. (2010); Y.  _et al._ (2015).

It has already been established that the density peaking at the low magnetic
field is the results of wave coupling rather than capacitive or inductive
coupling Lafleur, Charles, and Boswell (2011); Barada  _et al._ (2013). In
reference 19 it was shown that the wave propagation near the resonance cone
surface causing the density peaking. To establish the existence of helicon
wave, the measurement of phase and amplitude of axial wave magnetic field
$B_{z}$, are carried out using the B-dot probe. Fig. 5 shows the axial
variation of the phase and amplitude of $B_{z}$ at 400W rf power in the Case
B. At 22 G, the axial variation of wave amplitude, Bz (Fig. 5a) has a spatial
modulation whereas, phase difference (Fig. 5b) shows the traveling wave
character. This type of amplitude modulation can be explained by the beating
of different helicon wave modes corresponding to fundamental and higher order
radial modes, simultaneously, excited by the antenna T.  _et al._ (1998);
Chen, Sudit, and Light (1996); Max  _et al._ (1995). These different helicon
modes are propagating with different angles of the axial magnetic field.
Depending on the magnetic field strength near the antenna, one of these modes
can predominantly take part in the ionization process by resonant absorption.
An effective traveling wavelength can be found by the slope $\frac{d\phi}{dz}$
of fitted straight line to axial phase variation (Fig. 5b), where $\phi$ is
the phase difference. At 22 G, in the near field of the antenna the effective
traveling wavelength, ${\lambda_{\parallel}}$ = 360$(\frac{d\phi}{dz})^{-1}$ ,
is about 34 cm. However, for the high magnetic field at 50 and 100 G, nearly
constant phase variation (Fig. 5b) suggested the barely existence of helicon
wave. The amplitudes also does not indicate any wave pattern, (Fig. 5a). This
implies that the non-existence of a helicon wave at 50 and 100G in our
experimental conditions in the case B.

The angle of obliquely propagating helicon wave with the axial magnetic field
can be found from the relation, $cos\theta=\frac{k_{\parallel}}{k}$, where
$k=\sqrt{k_{\parallel}^{2}+k_{\perp}^{2}}$, $k_{\parallel}$ and $k_{\perp}$
are the total, parallel and perpendicular wavenumbers, respectively. The
calculated value of wave propagation angle is comes about 76.40∘ by using the
measured values of parallel wavenumber , i.e. $k_{\parallel}$ = 18.4 $m^{-1}$
and estimated perpendicular wavenumber $k_{\perp}{{}_{1}}$. The value of
$k_{\perp}{{}_{1}}$ = $\frac{3.83}{a}$ = 76.6 $m^{-1}$ can be found from the
$1^{st}$ root of the first order Bessel function $J_{1}(k_{\perp}a)$ , by
considering the plasma radius $a$ = 5 cm ref . The calculated value of
resonance cone angle Barada  _et al._ (2013),
$cos\theta=\frac{\omega}{\omega_{ce}}$, for 22 G at which the density peak
occurs (Fig. 4a), is comes about 77.30∘. It is found that the helicon wave
propagation angle well matches with the resonance cone angle. The details of
the relation between resonance cone angle and wave propagation angle is
described in reference 19.

Figure 5: Axial variation of (a) amplitude and (b) phase of an axial component
of wave magnetic field ($B_{z}$) at 400W rf power and $1\times 10^{-3}$ mbar
for the Case B at magnetic field value 22 G, solid triangles, 50 G solid
circle and 100 G solid square.

The absorption of helicon wave near the resonance cone surface is correlated
with the excitation of electrostatic fluctuation Boswell . In the present
experiment, the measurements of density fluctuation are performed at the
magnetic field values where helicon wave present and absent. For this, an
electrostatic probe (rf compensation Langmuir probe) biased in the ion
saturation region, is placed in the source chamber at $r,z=0,31$ cm. A 14-bit
PXI based data acquisition system is used to acquire the time series signal
with low pass filter of 1.9 MHz with 50k record length at the sampling rate of
100 kS/s. The frequency spectrum (FFT) of density fluctuation at 400 W rf
power and $1\times 10^{-3}$ mbar fill pressure for various magnetic field
strength in the Case B is shown in Fig. 6. The strong peaks are observed
between 10 to 20 kHz frequency at 22 G of magnetic field value, above this
there is no significant peak in the frequency spectrum. The presence of
electrostatic fluctuations at low magnetic field ($\sim$ 25G) are associated
with the density peaking (Fig. 4) where helicon wave also exist. The magnetic
field $>$ 25 G, the non-existence electrostatic fluctuations is consistent
with non-propagation of helicon wave (Fig. 5) and hence this can be correlated
to density fall Fig. 4.

Figure 6: The frequency spectrum of plasma density fluctuation obtain at the
different magnetic field with rf compensated Langmuir probe in Case B at on-
axis location of z = 31cm, 400W rf power and $1\times 10^{-3}$ mbar fill
pressure.

Fig. 7 shows the variation of density fluctuation level with an applied
magnetic field for two different powers at pressure $1\times 10^{-3}$ mbar. It
is observed that level of fluctuation increases significantly at those
magnetic fields where the density peaks occur. Therefore in our experiments,
the oblique helicon wave propagation near the resonance cone boundary excites
the non-convective electrostatic wave Barada  _et al._ (2013); Boswell ;
Krämer  _et al._ (2007); Akhiezer, Mikhailenko, and Stepanov (1998) which
transfers the energy from helicon wave to the plasma creating density peaks at
specific magnetic field. This process is responsible for density peaking at
the low magnetic field in our experiments.

Figure 7: Amplitude variation of density fluctuation with the external
magnetic field as a function of rf powers in Case B at the location of
$r,z=0,31$cm and $1\times 10^{-3}$ mbar fill pressure. 200 W solid triangles
and 400 W solid square. Figure 8: Variation of plasma density in the source
chamber at z = 31cm with the applied magnetic field at antenna center for
Cases A-D; at pressure $1\times 10^{-3}$ mbar and rf powers (a) 600W and (b)
300W.

After establishing the presence of the helicon wave when density peaked at low
magnetic field in the quasi-uniform magnetic field configuration, the
measurement of plasma density in the source chamber is made for other
different magnetic field configuration as shown in Fig. 2 by Case A, B, C, and
D. In all the Cases the magnetic field is varied and density is measured at
the location, z = 31 cm at $1\times 10^{-3}$ mbar pressure at 600 W (Fig. 8a)
and 300 W (Fig. 8b). To get the same magnetic field at the antenna center (z =
0), different current is passed for different coil configurations. As the
coils are removed, it can be seen from Fig. 2, that field non-uniformity is
increases from Case A to Case D. For Cases A, B and C the density is peaked
around 17-22 G and the peak value increases as non-uniformity increases for
the Case A to C. With removing the one more coil from the case C, the magnetic
field near the antenna become more non-uniform or diverging in the Case D. In
the case D, the density peaked around 50-60 G and increases about five times
than the uniform magnetic field case at the same magnetic field value. Fig. 9
shows the density at 50G, increases as magnetic field non-uniformity increases
from case A to D. The plasma production efficiency, Shunjiro and Takao (2004)
$\frac{N_{e}}{p_{rf}}$ which is the ratio of number of total electrons to
input power is also increases with the magnetic field inhomogeneity. Fig. 10
shows the production efficiency estimated at on-axis at z = 31 cm in all four
cases. To estimate the production efficiency, the production region is
important, i.e. plasma core instead of plasma edge. It is seen from above
discussion that the inhomogeneous magnetic field near the antenna location
improves the plasma production significantly.

Our results show a systematic variation of plasma density with magnetic field
inhomogeneity. Plasma production is more when the magnetic field is more
diverging near the antenna. This kind of behavior of increase of production
efficiency with magnetic field divergence may be attributed to multimode
oblique cyclotron resonance. In the direction of the magnetic field, the m =
+1 right helicon antenna is supposed to excite different axial and radial
eigenmodes Max  _et al._ (1995); V. and G. (1996). Depending upon the magnetic
field near the antenna all these different modes can simultaneously couple to
different helicon modes and couple power in the plasma. In our earlier
experiments Barada  _et al._ (2013) it was shown that multiple modes can be
excited at different uniform magnetic fields near the antenna. These modes can
have oblique cyclotron resonance for a critical magnetic field which satisfies
$cos\theta=\frac{\omega}{\omega_{ce}}=\frac{k_{\parallel}}{k}=\frac{k_{\parallel}}{\sqrt{k_{\parallel}^{2}+k_{\perp}^{2}}}$.
If multiple modes are present, they will have multiple resonances at different
critical magnetic fields. With a uniform magnetic field, only a single
resonance is possible for a mode. When a non-uniform magnetic field (because
of the divergence) is applied, different magnetic field values are available
near the antenna near field for different eigenmodes resonance conditions.
Among all the possible excited modes some modes can have resonances. This will
lead to more power absorption and hence rise in the efficiency of the source
as observed in the present experiment. Another interesting feature to be noted
here is that the width of the peaks of density increases as the non-uniformity
in the magnetic field is increased. This is indicative of the presence of
different resonating modes. As low field helicon sources are proposed recently
for space propulsion Chen (2008); Ling  _et al._ (2010), using a highly
diverging magnetic field near the antenna may be better suited in terms of
efficiency of the source. It will reduce the weight of the thrusters as well
as the power consumption.

Figure 9: Density as function of magnetic field non-uniformity in different
Cases at 50G and $1\times 10^{-3}$ mbar pressure. Figure 10: On-axis plasma
production efficiency at z = 31cm, in different Cases at 50G, 600W and
$1\times 10^{-3}$ mbar pressure.  Figure 11: On-axis variation of plasma
density with the magnetic field at 400 W RF power and $1\times 10^{-3}$ mbar
fill pressure. Case B (solid triangle) and Case D (solid square). Figure 12:
Axial variation of amplitude and phase of $B_{z}$ for Case B and D magnetic
field configuration for 22 G (a and b) and 50 G (c and d) at 400 W rf power
and $1\times 10^{-3}$ mbar fill pressure. Figure 13: Variation of on-axis
plasma density at z = 31cm with 87 A DC current for the Cases of B (solid
triangle), C (solid circle) and D (solid square) at pressure $1\times 10^{-3}$
mbar.

Fig. 11 shows the on-axis plasma density for the Case B and D at 400 W rf
power. The plasma density is peaked around 40-50 G in the Case D and it is
nearly 6 times higher than the Case B at that magnetic field value. To
investigate the presence of multimode in the inhomogeneous magnetic field
configuration, the axial profiles of the wave amplitude and phase of $B_{z}$
are measured in the Case D and comparisons are made with the quasi-uniform
magnetic field configuration, i.e. Case B. Fig. 11 shows the density variation
with magnetic field for Case B and D at which wave amplitude and phase of
$B_{z}$ are measured. Fig. 12 shows the axial variation of the amplitude (Fig.
12a and c) and the phase difference (Fig. 12b and d) of $B_{z}$ at two
different magnetic field values, 22 G ($\frac{\omega_{ce}}{\omega_{rf}}$ =
4.5) and 50 G ($\frac{\omega_{ce}}{\omega_{rf}}$ = 10), respectively for the
Case B (solid triangle) and D (solid square) at 400 W rf power and $1\times
10^{-3}$ mbar fill in pressure. At 22 G, for both cases B and D, the phase
difference variations (Fig. 12a) show the traveling wave character and
amplitude (Fig. 12b) modulations show the beating wave pattern. The beating
pattern T.  _et al._ (1998); Chen, Sudit, and Light (1996); Max  _et al._
(1995) of amplitude in presence of propagating wave indicates the existence of
more than one (radial or axial) mode in our experiment. At 50 G, for Case B
(where density has 5 times lower magnitude from the Case D, Fig. 11), the
phase difference has no significant axial variation (Fig. 12d, solid
triangles) i.e. wave propagation may ceases at this magnetic field value. The
contrasting results are observed in the Case D at 50 G (where the density is
peaked), where $B_{z}$ has amplitude modulation (Fig. 12c, solid square) and
the phase oscillates (Fig. 12d, solid square) substantially with the position.
It seems that, in the Case D, helicon wave exist along the entire axial
distance including the expansion chamber and has partially traveling and
partially standing wave structure. This implies that waves can have the
interference of multiple radial or axial modes. This results in a standing
wave field structure Max  _et al._ (1995); M.  _et al._ (2004); Takahashi,
Chiba, and Ando (2014) that gives rise to the nearly constant phase signal at
25 $<$ z $<$ 45 cm in the Case D, Fig. 12d. The multiple modes exist due to
presence of different exciting condition corresponding to availability of
broad range of magnetic field near the antenna, Case D Fig. 2.

Fig. 13 shows the variation of on-axis plasma density with RF powers for the
Cases of B, C, and D at $1\times 10^{-3}$ mbar fill pressure. The density in
the Cases B, C and D is obtain by keeping the fixed DC current, 87 A, which
corresponds to magnetic field values at the antenna center about 150, 90 and
34 G, respectively. It is clear from the Fig. 13 that, above 300W rf powers,
in the Case D, the density is five times higher than the Case B. The density
at 1000W, in case B is about $8\times 10^{16}$ $m^{-3}$ which can be
achievable at 200W in the Case D. These results successfully demonstrate that
the higher ionization can be achieved by careful placing of electromagnet or
permanent magnet near the antenna. This will effectively reduce the coast of
the magnetic power supply and power consumption which in turn enhanced helicon
discharge efficiency

##  IV  summary and Conclusion:

Four different magnetic field configurations near the helicon antenna are used
to look for the effect of magnetic field inhomogeneity. It is shown that
plasma production efficiency increases with increasing the magnetic field
inhomogeneity. The results are explained on the basis of multimode excitation
due to the presence of a wide range of magnetic fields nearby the antenna. The
widths of the peaks in density increase with non-uniformity of the applied
magnetic field. This is indicative of multimode resonance absorption leading
to higher efficiency. As the less coil is more efficient, the cost of the
source will be reduced drastically.

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