Plasma-wave interaction in a helicon thruster

IEPC-2011-047

Presented at the 32

nd

International Electric Propulsion Conference,

Wiesbaden,Germany

September 11{15,2011

Daniel Mart´ınez

∗

and Eduardo Ahedo

†

Universidad Politecnica de Madrid,Madrid 28040,Spain

The absorption of large wave energy by the plasma is the ﬁrst important process in

the discharge chamber of a helicon thruster.This paper presents ﬁrst steps of ongoing

research on the development of a plasma-wave model consistent with the hydrodynamics of

the plasma inside the chamber.The standard plasma resonator model provides important

information about the energy ﬂuxes inside the plasma column for diﬀerent conditions.

There is a large axial energy ﬂux that spreads the distribution of the absorbed energy map

far from the antenna location.This implies that the boundary condition at the front end

of the vessel requires further research.Other relevant aspects for a consistent plasma-wave

model are pointed out.

Nomenclature

B

0

= Applied magnetic ﬁeld k

⊥

= transversal wavenumber component

E

= Induced electric ﬁeld k

∥

= longitudinal wavenumber component

B

= Induced magnetic ﬁeld u = Power density

r

p

= plasma column radius S = Poynting’s vector

r

a

= antenna radius f,g = generic functions

r

w

= wall radius j = Current density

k

= wavenumber vector I = Electric current

I.Introduction

Space plasma thrusters based on helicon sources are a subject of intensive current research.

1{4

One of

the principal projects,and genuinely European,is HPH.COM(Helicon Plasma Hydrazine Combined Micro),

funded by the European Union within the 7th Framework Programme and conducted by a consortium of

15 institutions from 7 European countries.The main objective of the project is to design,test,optimize,

and develop an helicon-based plasma thruster in the range 50-100 watts.In Ref.5 we presented an overview

of the research carried out by our group for this project and for another one,ﬁnanced by the Air Force

Oﬃce for Sientiﬁc Research,centered on plasma expansion and detachment in magnetic nozzles.So far,our

activity has covered the internal and external plasma dynamics inside and outside the source,except for

the plasma-wave interaction process,which is crucial for plasma generation and heating.This process is the

objective of the present paper.Section 2 details the physical processes and ﬁgures of merit of an helicon

thruster.Section 3 presents the plasma-wave interaction model.Section 4 presents and discusses results.

Conclusions are in Section 5.

MsE student,martinez.ruiz.daniel@gmail.com

y

Professor,eduardo.ahedo.upm.es (web.fmetsia.upm.es/ep2)

1

The 32

nd

International Electric Propulsion Conference,Wiesbaden,Germany

September 11{15,2011

II.Physics of a helicon thruster

The main elements of a helicon thruster are (i) a feeding system that injects gas into a cylindrical vessel,

(ii) a radiofrequency(RF) antenna system that emits radiofrequency waves in the range 1-30 Mhz,and (iii) a

set of magnetic coils surrounding the vessel.The coils create a longitudinal ﬁeld with three roles.Inside the

chamber,the magnetic ﬁeld is quasi-axial and facilitates the propagation of helicon waves within the plasma

column.

6

Wave energy is deposited into electrons via plasma-wave resonance.Heated electrons ionize the

injected gas by bombardment.The second role of the magnetic ﬁeld is to conﬁne the plasma away from the

tube wall,thus avoiding energy losses and producing a highly collimated beam.For instance,Tysk et al.

7

ﬁnd a 50 times radial decrease of the plasma density (for a 1000G ﬁeld),instead of the 1.6 times decrease in

a free-acceleration unmagnetized presheath.Strong radial decreases of n

e

are also found by Gilland et al.

8

Outside the source,the magnetic ﬁeld becomes divergent,thus creating a nozzle eﬀect on the magnetized

plasma.

Beyond gaining insight in the plasma physics,a central aspect of our research has been to assess the

propulsive capabilities of an helicon thruster.In order to compete with other plasma thrusters the helicon

thruster must oﬀer good ﬁgures for speciﬁc impulse,thrust eﬃciency,and thrust/weight ratio.The two ﬁrst

ﬁgures-of-merit are directly related to the plasma discharge.First,since the helicon thruster is basically

an electrothermal accelerator,the speciﬁc impulse is proportional to the square root of the plasma internal

energy.Therefore a high plasma temperature is needed (a condition that is not required for an helicon

source for non-propulsive applications).Then,good thrust eﬃciency requires:(i) eﬃcient wave-plasma

energy conversion from the antenna,(ii) near-total plasma ionization inside the vessel,(iii) eﬃcient plasma

heating,with small energy losses to the walls,(iv) eﬃcient conversion of internal energy into directed axial

energy,minimizing plume divergence,and (v) eﬃcient plasma detachment.

In order to study the plasma discharge in a helicon thruster,it is convenient to distinguish the diﬀerent

stages and processes.There are the production stage inside the plasma source and the acceleration stage

in the magnetic nozzle.Two types of processes take place inside the source:the resonant wave-plasma

interaction,leading to the deposition of wave energy into the plasma and the multiple phenomena governing

internal plasma dynamics.A choked plasma ﬂow is expected around the tube exit.Two other distinguished

processes take place in the magnetic nozzle:the supersonic plasma acceleration and its magnetic interaction

with the thruster,and plasma detachment from the magnetic nozzle.The four main processes are coupled

but,in order to understand the main phenomena and parameters at play,some extra assumptions are

commonly made in order to treat each process independently.Subsequent studies and numerical codes will

look for a deeper and consistent coupling of the four independent models.

The internal radial dynamics for a magnetized plasma were studied in Ref.9.It is found that,within

the regime of interest and for the bulk of the plasma,the radial proﬁle of plasma density is quasi universal

satisfying

n

e

(r,z)

n

0

= J

0

(

a

0

r

R

)

,(1)

with n

0

= n

e

(0,z) the density proﬁle at the plasma axis,R the tube radius,J

0

a Bessel function,and

a

0

≃ 2.405 the ﬁrst zero of J

0

.Eq.1 is valid except in the thin,inertial and Debye layers next to the

tube lateral wall,but these do not seem important for the analysis of the next section.The internal axial

dynamics,studied in Ref.10,determine n

e

(0,z) [see Fig.6 there].The r-averaged plasma density presents

a maximum where the plasma ﬂow changes from backwards to forwards,and it decreases by a factor of two

at the tube ends.For a high-density plasma,of potential interest in a helicon thruster,eﬀects of the induced

magnetic ﬁeld modify both the total magnetic ﬁeld in the chamber and the radial proﬁle,Eq.1,of the

plasma density.

11

This regime is not desirable since helicon waves could not propagate in the plasma.

Plasma processes in the magnetic nozzle include supersonic acceleration,thrust transmission,plasma

detachment,and hypothetical formation of double layers.We have treated these processes in great detail in

several publications,the main ones being.

12{18

The the interaction of the rf wave emitted by the antenna with the plasma is the only process we had

not treated yet,since there was much theoretical research done on it.

6,19{21

Nonetheless,once we have

acquired a reasonable understanding of the internal plasma dynamics,is necessary to attempt advancing in

the plasma-wave interaction problem.There is still much work ahead before achieving a characterization of

the plasma-wave process that can be matched consistently with the plasma ﬂuid dynamics.The rest of the

paper reports on our ﬁrst steps in that direction.

2

The 32

nd

International Electric Propulsion Conference,Wiesbaden,Germany

September 11{15,2011

O

max

k d

e

k d

e

O

trans

TG

HE

Figure 1.Dispersion relation for the inﬁnite plasma case.

III.Formulation of the plasma-wave model

The most common analysis of the plasma-wave interaction assumes a normal wave with a time dependence

∝ (iωt) and to solve the Maxwell equations

∇×E = −iωB,∇×B = µ

0

[iω

¯

¯ϵ · E +j

a

],(2)

where the dielectric tensor

¯

¯ϵ(r) carries all the plasma information through its density n

e

(r) and collision-

frequency functions,and j

a

is the current density of the rf antenna.If the guide ﬁeld is along 1

z

,the

dielectric tensor takes the form

¯

¯ϵ = ϵ

0

κ

1

−iκ

2

0

iκ

2

κ

1

0

0 0 κ

3

where the three scalar components have been normalized and their expressions are found in textbooks [for

instance,see p.111-112 of Ref.22].The ranges of frequencies of interest for helicon waves is

ω

pe

≫ω

ce

≫ω ≫ω

lh

,ν

e

,(3)

where ω

pe

=

√

e

2

n

e

/ϵ

0

m

e

is the plasma frequency,ω

ce

= eB

0

/m

e

is the electron gyrofrequency,ω

lh

≃

eB/

√

m

e

m

i

is the lower hybrid frequency,and ν

e

is the eﬀective collision frequency for electrons.

In that frequency region,both the ion motion and the displacement current are negligible and the

dispersion relation for the basic case of a uniform and innite plasma (i.e.n

e

= const) is

k

2

d

2

e

=

ω

±ω

ce

| cos θ| −(ω +iν

e

)

(4)

where k is the wavenumber,θ the angle between B and the wavenumber vector k,and d

e

= c/ω

pe

=

√

m

e

/(e

2

µ

0

n

e

) is the electron skin-depth.

Figure 1 plots Eq.(4),for ν

e

/ω = 0,in terms of the longitudinal and transversal components of the

wavenumber vector,k

∥

= k cos θ and k

⊥

= k sinθ,respectively.The general solution are two pairs of modes,

consisting of long-wavelength helicon waves and short-wavelength Trievelpiece-Gould (TG) waves.If Eq.(4)

is solved for k

⊥

d

e

(k

∥

d

e

,ω

ce

/ω,ν

e

/ω,ω

ce

/ω

pe

),one has that

(1) the two types of waves propagate for 2α < k

∥

d

e

<

√

α/(1 −α),with α = ω/ω

ce

,

(2) only TG waves propagate for k

∥

d

e

>

√

α/(1 −α),

3

The 32

nd

International Electric Propulsion Conference,Wiesbaden,Germany

September 11{15,2011

Figure 2.Resonator with single loop antenna of radius r

a

placed at the middle of the chamber z

0

= L=2.

Azimuthal modes are not present since the antenna stands azimuthal symmetry,m= 0.

(3) no wave propagates for k

∥

d

e

< 2α.

Since d

e

∝ n

−1/2

e

and ω

ce

∝ B

0

,plasma density and magnetic ﬁeld determine the propagation character-

istics of a wave of given frequency ω.The range of k

∥

d

e

where the helicon waves propagate is naturally the

propitious one for the plasma to absorb the rf emission.Since ν

e

/ω ≪1,helicon waves are weakly damped,

but TG waves are strongly damped,which makes them surface waves at the plasma edge.

A.The cylindrical plasma column

We focus now on our working case of a cylindrical,axially-uniform plasma column of length L and radius

r

p

.This column is surrounded by an antenna located at r = r

a

(r

a

> r

p

),and the two of them are encased

in a metallic cylinder of radius r

w

and length L,as sketched in Fig.2.This plasma resonator problem has

been treated extensively in the literature.The analytical studies for uniform plasma density of Shamrai and

Taranov

19,23,24

are particularly illuminating.Nonuniform plasmas have been treated by Arnush and Chen

21

and Cho and Liebermann

25

among others.Their modeling is followed here.

Let be n

e

(r) the plasma density and ν

e

(r) its collision frequency.First,the azimuthal and axial uniformity

of the plasma column allows a Fourier expansion in normal modes ∝ expi(lπz/L+mθ),for l and m integer

numbers.Second,since E

r

and E

θ

are zero at the axial ends (z = 0 and z = L) of the vessel some symmetries

apply in the l-modes.This yields the following expansions

19,25

E

r

(r,θ,z)

E

θ

(r,θ,z)

B

z

(r,θ,z)

=

∑

m,l

E

r,lm

(r)

E

θ,lm

(r)

B

z,lm

(r)

· sin(lπz/L) · exp

imθ

(5)

E

z

(r,θ,z)

B

r

(r,θ,z)

B

θ

(r,θ,z)

=

∑

m,l

E

z,lm

(r)

B

r,lm

(r)

B

θ,lm

(r)

· cos(lπz/L) · exp

imθ

(6)

The antenna current density takes the form

j

a

(r) = Iδ(r −r

a

)[1

z

s

z

(θ,z) +1

θ

s

θ

(θ,z)] (7)

where I is the current along the wire (in amperes,say) and the functions s

z

and s

θ

model the wire geometry.

These functions (expressed in m

−1

,for instance),are to be expanded in Fourier series as

s

k

=

1

L

∑

l,m

I

k,lm

I

exp[i(lπz/L+mθ)],k = z,θ.(8)

For instance for the simple loop sketched in Fig.2,one has s

z

= 0 and

s

θ

(θ,z) = δ(z −z

a

) =

1

L

+

2

L

∞

∑

l=1

[

cos

(

l

πz

L

)

cos

(

l

πz

a

L

)

+sin

(

l

πz

L

)

sin

(

l

πz

a

L

)]

(9)

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The 32

nd

International Electric Propulsion Conference,Wiesbaden,Germany

September 11{15,2011

Next,we deﬁne phase-shifted ﬁelds

E

ϕ

= −iE

θ

,B

ϕ

= iB

θ

,(10)

and dimensionless magnitudes

ˆ

I

k

=

I

k

I

ℓ = l

πc

Lω

,ˆr =

rω

c

,

ˆ

B =

Bc

µ

0

Iω

,

ˆ

E =

E

µ

0

Iω

.(11)

Observe that µ

0

Iω/c is the reference magnetic ﬁeld for an antenna current I.

Then,the Maxwell equations for each normal mode (E

lm

,B

lm

) turn into the four diﬀerential equations

d

ˆ

E

ϕ

dˆr

=

(

mκ

2

ˆrκ

1

−

1

ˆr

)

ˆ

E

ϕ

+

ℓm

ˆrκ

1

ˆ

B

ϕ

+

(

1 −

m

2

ˆr

2

κ

1

)

ˆ

B

z

,(12)

d

ˆ

E

z

dˆr

=

κ

2

ℓ

κ

1

ˆ

E

ϕ

−

(

1 −

ℓ

2

κ

1

)

ˆ

B

ϕ

−

ℓm

κ

1

ˆr

ˆ

B

z

,(13)

d

ˆ

B

ϕ

dˆr

=

mℓ

ˆr

ˆ

E

ϕ

+

(

κ

3

−

m

2

ˆr

2

)

ˆ

E

z

−

1

ˆr

ˆ

B

ϕ

,(14)

d

ˆ

B

z

dˆr

=

(

ℓ

2

+

κ

2

2

κ

1

−κ

1

)

ˆ

E

ϕ

−

mℓ

ˆr

ˆ

E

z

+ℓ

κ

2

κ

1

ˆ

B

ϕ

−

mκ

2

ˆrκ

1

ˆ

B

z

,(15)

and the algebraic linear equations

ˆ

E

r

=

κ

2

κ

1

E

ϕ

−

m

ˆrκ

1

B

z

+

ℓ

κ

1

B

ϕ

,(16)

ˆ

B

r

= −ℓ

ˆ

E

ϕ

+

m

ˆr

ˆ

E

z

,(17)

where subscripts lm have been omitted from the variables.

The above set of diﬀerential equations have boundary conditions at r = 0,r = r

a

,and r = r

w

.At r = r

w

,

the electric ﬁeld parallel to the wall is zero:

ˆ

E

ϕ

(ˆr

w

) = 0,

ˆ

E

z

(ˆr

w

) = 0.(18)

At r = r

a

,the current sheet generated by the antenna that produces jumps in the parallel magnetic ﬁeld.

From Maxwell’s equations

ˆ

B

z

(ˆr

+

a

) −

ˆ

B

z

(ˆr

−

a

) = −s

θ

c

ω

= −ˆs

θ

,(19)

ˆ

B

ϕ

(ˆr

+

a

) −

ˆ

B

ϕ

(ˆr

−

a

) = is

z

c

ω

= iˆs

z

.(20)

At r = 0,only bounded modes are acceptable.The Taylor expansion of Eqs.(12)-(15) yields that there are

only two bounded modes.For m= 0,these modes satisfy

(

ˆ

E

ϕ

,

ˆ

E

z

,

ˆ

B

ϕ

,

ˆ

B

z

) = C

1

(ˆr,0,0,2) [1 +O(ˆr)] (21)

(

ˆ

E

ϕ

,

ˆ

E

z

,

ˆ

B

ϕ

,

ˆ

B

z

) = C

2

(0,2,ˆrκ

3

,0) [1 +O(ˆr)] (22)

Applying the principle of superposition,the general solution for modes satisfying conditions at ˆr = 0 and

ˆr = ˆr

a

is

ˆ

E

z

(ˆr) = C

1

ˆ

E

z1

(ˆr) +C

2

ˆ

E

z2

(ˆr) +

ˆ

I

θ

ˆ

E

z3

(ˆr) +

ˆ

I

z

ˆ

E

z4

(ˆr) (23)

and similarly for the other ﬁve ﬁeld components.Then the boundary conditions at ˆr = ˆr

w

provide two

conditions,

A

1

[

C

1

C

2

]

= A

2

[

ˆ

I

θ

ˆ

I

z

]

,(24)

that determine (C

1

,C

2

) in terms of (

ˆ

I

θ

,

ˆ

I

z

).This procedure is followed for each (l,m)-mode.Resonance for

a particular (l,m)-mode takes place when

det(A

−1

2

A

1

) = 0.(25)

5

The 32

nd

International Electric Propulsion Conference,Wiesbaden,Germany

September 11{15,2011

0

10

20

30

40

50

-200

-100

0

100

200

l

Er

0

10

20

30

40

50

-0.05

0

0.05

l

E

0

10

20

30

40

50

-2

0

2

4

l

Ez

0

10

20

30

40

50

-1

-0.5

0

0.5

1

x 10

-6

l

Br

0

10

20

30

40

50

-2

0

2

4

x 10

-7

l

B

0

10

20

30

40

50

-5

0

5

x 10

-6

l

Bz

Figure 3.Contribution of the ﬁrst ﬁfty longitudinal modes to each of the ﬁelds’ components at B

0

= 450 G.

The full 3D solution is obtained by solving all signiﬁcant modes (l,m).For the single-loop antenna,with

m = 0 as the only longitudinal mode the solution is axilsymmetric.The relative weigh of the longitudinal

modes decays quickly after the ﬁrst 5-7 modes and becomes negligible at l ∼ 30.Furthermore,the high-l

modes tend to compensate by pairs among themselves as shown in Figure 3.Also if the antenna is centrally

located (at z

a

= L/2),even l-modes are negligible.Figure 4 plots some solutions.

B.Energy absorption

Once the electromagnetic ﬁeld are determined,we can proceed to determine the absorption of the wave

energy by the plasma.In a steady-state,one has

p

a

= ∇· S,(26)

where

p

abs

=

1

2

Re(E

∗

· j) =

1

2

Re(E

∗

·

¯

¯σ · E),S =

1

2µ

0

Re(E

∗

×B),(27)

are the absorbed energy density,p

abs

and the Poynting or energy ﬂux vector S,respectively,and the plasma

conductivity is

¯

¯σ ≃ iω

¯

¯ϵ.The ﬁelds E and B are the complex total ﬁelds,i.e.sum of all the (l,m) modes

and proportional to expiωt.

The absorbed energy density distribution is the main magnitude required by the plasma dynamics model.

Its integral over the plasma volume yields the total absorbed power,P

abs

which can also be computed from

the normal of the Poynting vector to the plasma column surface,

P

abs

= 2π

∫

L

0

dz

∫

r

p

0

drrp

abs

(r,z) ≡ −

∫

L

0

dzS

r

(r

p

,z) (28)

Figure 5 plots,for a uniform plasma,the dependence of the absorbed power with the magnetic ﬁeld and

the plasma density.Strong resonance peaks are observed when one of the two magnitudes is tuned.Notice

that the absorbed energy is proportional to the square of the antenna current,I

2

.The aim of the system

6

The 32

nd

International Electric Propulsion Conference,Wiesbaden,Germany

September 11{15,2011

0

0.005

0.01

0.015

0.02

-200

0

200

r [m]

Er

0

0.005

0.01

0.015

0.02

0

0.05

0.1

r [m]

E

0

0.005

0.01

0.015

0.02

-2

0

2

r [m]

Ez

0

0.005

0.01

0.015

0.02

-20

-10

0

10

r [m]

Br

0

0.005

0.01

0.015

0.02

-200

0

200

r [m]

B

0

0.005

0.01

0.015

0.02

0

500

1000

r [m]

Bz

Figure 4.Radial proﬁles of electromagnetic waves for B

0

(in Gauss)=260 (blue lines) and 850 (red lines).

Short wavelength waves are the TG waves,most visible at high conﬁnement.

23

Dotted vertical lines show the

interphase between plasma and vacuum and the antenna radial position.Forty longitudinal modes are used,

though even modes are zero.Resonator dimensions are L = 20cm,r

p

= 10mm,r

a

= 12mm,r

w

= 20mm.

would be to operate at one of the peaks,for instance at n

0

= 3.8· 10

18

m

−3

and B

0

≃ 300 Gauss.The ﬁgure

also plots the inﬂuence of collisional frequency in the absorbed power.

IV.Results and discussion

A.In uence of the density prole

The ﬁrst study we wanted to carry out is to assess the diﬀerences between a column with uniform density

and a non-uniform column satisfying Eq.(1),which is the one we expect in most of the plasma column.

We will compare cases with the same average density and the average density for the proﬁle of Eq.(1) is

0.44n

0

.Figure 6 shows the presence of similar resonances for the nonuniform case,although the behavior

is not so neat as in the uniform case.Further analysis of these results is required to discern whether there

is any relevant diﬀerence in behavior.Plasma collisionality is assumed proportional to plasma density (i.e.

dominated by electron-ion collisions).

Figure 7 plots the 2D distribution of the absorbed energy density (multiplied by r in order to weight

adequately cylindrical volumes) for the uniform and nonuniform cases at antiresonance regimes.Energy

absorption is more peaked (both radially and axially) near the antenna for the nonuniform plasma.The

diﬀerence in the axial proﬁles is the most surprising one.Stronger wave reﬂection at the back and front plates

are more relevant for the uniform plasma due to the higher axial transmission of energy when operating out

of resonance.

Figure 8 plots,for the two cases of Fig.7,the perpendicular component of the energy ﬂux vector S

r

,

showing that most of the energy ﬂux enters the plasma (negative ﬂux according to the surface vector) around

7

The 32

nd

International Electric Propulsion Conference,Wiesbaden,Germany

September 11{15,2011

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Magnetic field B

0

, T

Power absorption, W

= 2.2 Hz

= 0.5 Hz

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x 10

19

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Plasma density, m

-3

Power absorption, W

Figure 5.a) Power transmision for diﬀerent applied magnetic ﬁelds when I = 1 A and n

e

= 3:8 10

18

m

3

for the same geometry than previous ﬁgure.Red dashed line stands for low collisional frequency and blue

continuous line for high frequency of collisions.b) Power transmision for variation of plasma density for 350

Gauss conﬁnement and the same other parameters as in a).Dashed line at b) represents the volume integration

of power density and continuous lines refer to energy ﬂux through the control area.

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

B

0

, T

Pabs, W

Figure 6.Absorbed power versus magnetic conﬁnement for uniform (blue) and non-uniform (red) density

proﬁles.

the antenna location.The axial proﬁle of the absorbed power in sections z = const,

¯p

r,abs

(z) = 2π

∫

r

p

0

drrp

abs

(r,z) (29)

is also plotted.The diﬀerences between these magnitudes means that the absorption of the wave energy by

the plasma does not take place preferentially near the antenna,but it is distributed rather uniformly axially.

Though,integration along z axis of both yields the same result.Therefore the Poynting vector has a strong

axial component inside the plasma,which is observed in Fig.9,and as a consequence boundary conditions

at the back and front plates are very relevant.

Moreover,radial proﬁles diﬀerences of power absorption between density cases are dependent on the res-

onance or antiresonance regime.As shown on Fig.10,resonance absorption turns out to be counterintuitive.

The radially varying plasma absorbs more power at the outer volume than an homogeneous column for the

pictured cases.On the other hand,antiresonance regimes show a more intense decay of the quantity of power

absorbed in the homogeneous case than in the nonhomogeneous one.The behaviour of the homogeneous

case is probably due to the stronger damping at the edge of the column meaning less deposition in the inner

region.

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September 11{15,2011

Figure 7.2D plasma power absorption map for B

0

=580 Gauss:(a) uniformand (b) nonuniformplasma.Same

geometrical values,average density of 3:8 10

18

m

3

and antenna current of 1A are used in both cases.

B.In uence of the antenna location

The resonator model shows strong wave reﬂection at both end plates.The case we want to analyze has an

open front end and the plasma ﬂows through it.Therefore,the resonator boundary conditions at the front

end could be not adequate.This issue is quite important,taking into account the last results of the previous

subsection.Real boundary conditions ’downstream’ are still uncertain to us.Since the density proﬁle is not

uniform axially,and density decays quickly once the plasma starts to accelerate supersonically,it is likely

that waves are reﬂected back below a critical density but this aspect has not been researched yet.

For now,we decided to apply the resonator model to a plasma column,with the front plate located very

far away from the antenna,in order to assess the diﬀerences in the region near the antenna.In particular

we consider a column of length L = 1m,while we keep the antenna near the back end at z

a

= 10cm.

Radial variation of density is considered from now on.Changes such as inserting odd longitudinal modes

(m= 0,1,2...∞) because of the symmetry loss and modiﬁcation of the antenna’s current expresion through

Dirac delta series are made.Though,antenna’s current remains as 1 A for the following cases.

Wave number components from the dispersion relation of the uniform inﬁnite plasma problem,Eq.(4)

are recalled.Both propagation components,k

⊥

and k

∥

,depend on magnetic applied ﬁelds.Analysis of the

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The 32

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International Electric Propulsion Conference,Wiesbaden,Germany

September 11{15,2011

0

0.05

0.1

0.15

0.2

0

2

4

6

8

10

12

z, m

2 Pabs(r,z)r·dr & 2 r S·n

0

0.05

0.1

0.15

0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

z, m

2 Pabs(r,z)r·dr & 2 r S·n

Figure 8.Power integration along the radius versus energy ﬂow through each z section a) Homogeneous plasma

b) Radially-varying density plasma.Recall Fig.6 to notice lower absorption at (b) when using 580G.

r, m

z, m

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0

0.05

0.1

0.15

0.2

r, m

z, m

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0

0.05

0.1

0.15

0.2

Figure 9.Power absorption on the 2D plane together with Poynting’s vector direction.(a) Uniform plasma

(b) Non-uniform plasma.Notice the eﬀect of the wall reﬂection on S.

equation leads to the fact that lower magnetic conﬁnement implies more intense axial propagation of waves,

which is related to longitudinal component of the wave number,and viceversa.

Figures 11 and 12 show results for several conﬁnement strengths.The ﬁrst of them shows the previously

used longitudinal power comparison for the near-the-plate antenna.The other ﬁgure’s color diagram shows

the power density at each point of the theta-constant plane.Note that power density of the last diagrams has

not been multiplied by their corresponding radius unlike in Fig.9,in order to visualize ﬂuxes and absorption

near the axis in an easier way.However,they are not volume-proportional as explained above.

Current lines get steeper as B

0

decreases.Even so,power absorption is spread along the z axis due to

the higher longitudinal component of the wave number (k

∥

).As the magnetic regime increases,the lines

where power is deposited lean towards a longitudinal axis parallel orientation.Nonetheless,high magnetic

conﬁnement regimes concentrate the power absorption at the transversal plane,where the antenna is located.

When integrating the power deposition along the volume conformed from z = 0 to z = 0.2,this quantity

over the total power absorbed reaches a percentage of ∼ 80% as the magnetic conﬁnement is increased.See

Table 1.

At high conﬁnement regimes power absorption for far distances in axial direction decreases and thus,

reﬂected ﬂuxes become lower.Even for 0.11 Teslas (Fig.12.f),it bears a small contribution of the wall

reﬂection at z ∈ [0,0.2] and r ∈ [0,0.002] (area of interest),yet it is a negligible quantity (as radius is close

to zero).This conﬁrms the conclusions drawn from the dispersion relation equation.

Nevertheless,these regimes of high B

0

are not the most important operation points as they do not achieve

high power absorption in order to use the source for space propulsion.Regimes close to high resonances may

10

The 32

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International Electric Propulsion Conference,Wiesbaden,Germany

September 11{15,2011

0

0.002

0.004

0.006

0.008

0.01

0

5

10

15

r, m

2 Pabs(r,z)r·dz

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

-100

0

100

200

300

400

500

r, m

2 Pabs(r,z)r·dz

Nonhomogeneous

Homogeneous

Figure 10.Power absorption proﬁles in the radial direction for homogeneous (red dotted line) and nonhomo-

geneous (blue line) plasma for (a) 580G (b) resonance regimes of 241G and 316G respectively.

0

0.2

0.4

0.6

0.8

1

-0.5

0

0.5

1

1.5

2

2.5

3

z, m

2 Pabs(r,z)r·dr & 2 r S·n

0

0.2

0.4

0.6

0.8

1

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

z, m

2 Pabs(r,z)r·dr & 2 r S·n

Figure 11.Power absorption (blue continuous line) and energy ﬂux (red dotted line) along the z axis for a)

100 Gauss and b) 330 Gauss.Higher conﬁnement produces a concentration of power absorbed at the position

of the antenna due to the dominant transversal wave number.Diﬀerent ﬂuxes from the antenna have to be

noticed as the plasma also varies and thus,the energy does not get in and out of the volume through the same

regions.

be preferred for these purposes.

B

0

100

280

330

500

650

1100

%P

23.97

33.67

30.66

21.96

51.4

81.38

Table 1.Percentage of power deposited from z = 0 to z = 20cm relative to the total absorbed power.

V.Conclusions

A standard plasma resonator model has been used in order to understand the propagation and absorption

of the helicon waves inside a plasma column.Resonance peaks in terms of magnetic conﬁnement and plasma

density,known from other authors’ works,have been reproduced.The comparison of the response for a uni-

form plasma column and the nonuniform typical proﬁle of a magnetized plasma show signiﬁcant diﬀerences.

The analysis of the Poynting vector shows that,for homogeneous plasmas,energy ﬂux penetrates the plasma

around the antenna location but then it spreads longitudinally.On the other hand,nonhomogeneous proﬁles

of density enhance absorption near the antenna and reduce the axial transmission.As a consequence,energy

absorption takes rather uniformly in the axial direction,mainly at uniform cases.The axial propagation

of the Poynting vector also means that the boundary condition at the front end can inﬂuence much the

response.

The study of the magnetic conﬁnement variation conﬁrms some facts obtained from the dispersion rela-

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September 11{15,2011

z, m

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

200

400

600

800

1000

1200

1400

z, m

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

200

400

600

800

1000

1200

1400

z, m

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

200

400

600

800

1000

z, m

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

200

400

600

800

1000

1200

1400

z, m

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

100

200

300

400

500

600

700

800

900

1000

z, m

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

200

400

600

800

1000

1200

1400

1600

1800

Figure 12.Power absorption on the 2D plane together with Poynting’s vector direction for several values of

magnetic conﬁnement a)100G b)280G c) 330G d)500G e)650G f)1100G.Color accounts for level of absorption.

White arrows show the direction and strength of the Poynting vector,energy ﬂux.

tion.When front end is set at a great distance,for high B

0

ﬁelds,propagation of energy is directed in the

z-axis direction but power absorption is enhanced in the antenna plane.On the opposite side,low magnetic

ﬁelds enhance radial propagation and longitudinal absorption.Thus,good approximation of propulsion de-

vices by using a resonator are found only for large conﬁnement values which diminish the eﬀec of end plate

reﬂection.

Therefore,further research must be carried on the downstream boundary conditions before the plasma-

wave model can be matched to the ﬂuid dynamics problem.This investigation can possibly be done taking

into account the axial variation of the density proﬁle and looking for natural reﬂection of the waves below a

critical density.

There are other aspects of the problem that also merit consideration.First,wave eﬀects on ions should

be assessed.Second,the cold plasma approximation needs to be checked,since in a helicon thruster the

electron temperature is expected to be of the order of several tens of eV.Third,the time scale separation

between the wave and ﬂuid responses and nonlinear wave eﬀects need to be studied.And ﬁnally,the presence

of hot electrons,observed in some experiments

7,26,27

needs to be worked out.

Acknowledgments

This work has been supported by the Gobierno de Espa˜na (Project AYA2010-16699).

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International Electric Propulsion Conference,Wiesbaden,Germany

September 11{15,2011

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