This document describes research on modeling the effects of a plasma plume from a Hall thruster on radio frequency signals. It summarizes previous simulation work using a ray tracing code and outlines the author's new aperture phase method approach. The aperture phase method models the plume's phase distortions using Zernike polynomials. It allows the effects to be scaled to different frequencies, amplitudes, aperture sizes, and as a time-varying model. The method provides a more portable way to reproduce the plume's impacts on communication signals compared to previous simulation methods.

1. Channel Modeling of aChannel Modeling of a
Plasma PlumePlasma Plume
By
Christian D. Zuniga
The University of Texas at Austin
Wednesday August 24, 2005

2. Presentation OutlinePresentation Outline
Project Description
Previous Research
– Ray Tracing codes (BeamServer)
My Research
– Aperture Phase Method using Zernike
Polynomials
– Generalization to system parameters
Conclusions

3. Project DescriptionProject Description
Hall ThrusterHall Thruster
Provides thrust by
electrically
accelerating ions.
Electrons emitted
from a cathode ionize
a gas.
Emitted electrons
also neutralize the
departing ions.
• Picture from “http://ctr-
sgi1.stanford.edu./CITS/hall_main.html

4. Project DescriptionProject Description
Problem: Plasma Plasma plume from
Hall Thruster can interfere with
communication signals.

5. Plasma Plume ModelPlasma Plume Model
 Collisionless, cold, and
unmagnetized plasma.
11 2
2
<−==
ω
ωp
pv
c
N
αθ−
= erarne )/()( 2
1
e
e
p
m
rne
0
2
2 )(
ε
ω =

6. Plasma Plume InstabilitiesPlasma Plume Instabilities
Coherent well-defined waves through
broadband turbulence.
Ionization instability.
)2cos(2 krmftaM −−= φπ
)()1(),( rnMtrn ee +=

7. Previous ResearchPrevious Research
Propagation in Plasma
– Reentry vehicle
– Ionosphere
– Plasma Slab
Propagation in Plasma Plume
– Experiments
– Simulations (BeamServer)

8. BeamServerBeamServer
 Fully configurable in 3D.
 Sophisticated ray-tracing
code.
 Aperture field integration.
 Verified with experiments
and analytical results

9. Exit PlaneExit Plane
Contains point of
intersection, final ray
tube area, and the
Electric Field of the
rays.
ffi
Sjk
A eNNDFeEE e
ˆ/0
0
−
=


10. Farfield PatternFarfield Pattern
Farfield Pattern obtained by integrating
over the fields in the exit plane.
)ˆˆ)(/(),,( 0
φϑφϑ φϑ AArerE rjk
+= −

''ˆ)'()'(
ˆ
ˆ
2
1
2
'0 0
dydxtrHrEe
jk
A
A
AA
rrjk
⋅




×









+×




−
=





∫∫
⋅ 



φ
θ
η
ϑ
φ
πφ
ϑ

11. Farfield PatternFarfield Pattern
Add contribution of each ray tube
∫∫
∑
⋅
−⋅
≈=
=
⋅×−×=
=
dxdye
Au
uJ
uS
uSeAT
tHEB
TB
jk
A
B
ar
rkj
f
PPkj
f
AA
i
i



1)(
2)(
)(
ˆ))ˆ(ˆ(
)(
4
1
)(
0
0
ηθφ
π
θ
θθ

12. Amplitude ResultsAmplitude Results
 No Plasma With Plasma
θsin
/)(
0
1
ku
uuJ
=

13. Phase ResultsPhase Results
 No Plume  With Plume

14. BeamServer SummaryBeamServer Summary
Simulates oscillatory antenna-plume system
with arbitrary configurations.
Time-consuming
Not portable to communication system
models

15. My ResearchMy Research
Project objective: Develop a channel
model of the plasma plume that:
 Reproduces its effects on a signal.
 Is a function of system parameters.
 Is more portable.
Friis Free Space Equation 2
2
)4( d
GGP
E rtt
f
π
λ
=

16. Farfield Magnitude vFarfield Magnitude v
FrequencyFrequency

17. I Reproducing TransferI Reproducing Transfer
CharacteristicsCharacteristics
Configuration is given.
Aperture Synthesis Methods – Given a
farfield pattern, find an equivalent aperture.
– Woodward Lawson Method
– IFT Method

18. Woodward-Lawson MethodWoodward-Lawson Method
Form pattern from
composing
functions
)/2(
)sin()/(
])sin()/sin[
anFb
na
na
baF
dn
m
mn
n
π
πθλπ
πθλπ
=
−
−
= ∑−=

19. Disadvantages of WoodwardDisadvantages of Woodward
Lawson MethodLawson Method
Dependence on antenna-plume parameters
has to be curve-fitted
Does not reproduce sidelobes very well
Does not reproduce pattern at arbitrary φ
angles

20. Aperture Phase ErrorsAperture Phase Errors
Aperture phase errors produce
significant distortion.
– Linear phase errors shift the overall pattern
– Quadratic phase errors raise the sidelobe levels
Plasma plume produces similar
distortions.

21. Aperture Phase MethodAperture Phase Method
Plume mainly alters
the phase.
Phase expanded in
Zernike Polynomials.
Antenna-plasma
system replaced by
original aperture
with an added phase
error.
∫−=
s
e dssNLS
0
)(),( ϕρ

22. Aperture Phase ModelAperture Phase Model

23. Zernike PolynomialsZernike Polynomials
Radial Polynomials.
– Orthogonal in the unit interval (0<ρ<1).
– Highest power is n, lowest power is m, n-|m| is
even.
– Satisfy the relation
∫ +
−
−=
1
0
12
)(
)1()()(
u
uJ
duJR n
mn
m
m
n ρρρρ

24. Zernike Polynomial ExpansionZernike Polynomial Expansion
1. Obtain from BeamServer
2. Interpolate phase to a circular grid
3. Find Zernike coefficients anm and bnm
∫ ∫
∫ ∫
∑ ∑
=
=
+=
∞
=
≥
−=
1
0
2
0
1
0
2
0
0
0
2,
sin)(),(
cos)(),(
)sincos)((),(
π
π
ϕρρϕρϕρ
ϕρρϕρϕρ
ϕϕρϕρ
ddmRSb
ddmRSa
mbmaRaS
m
nenm
m
nenm
n
m
nnm
nmnm
m
ne

25. Zernike Polynomial ExpansionZernike Polynomial Expansion
# Even
Polynomials
IE (10-3
)
5 0.564
10 0.2959
15 0.2144
20 0.1954
25 0.1593
30 0.1334
∫∫ −= dASSIE ese ||

26. DistortionsDistortions
n m Value (x λ
=0.075m
Zernike
Polynomial
Aberration Name
0 0 0.06 1 Piston
1 1 -0.0078 Distortion
2 2 0.0078 Astigmatism
2 0 -0.0070 Curvature of Field
4 2 -0.0021 Astigmatism
θρ cos
)12(3 2
−ρ
θρ 2cos6 2
θρρ cos)23(8 3
−

27. Farfield Pattern SynthesisFarfield Pattern Synthesis
Modified aperture field is
Farfield Pattern is
Similar to an aberrated optical system
A
Sjk
A EeE e ),(0
' ϕρ
=
ϕρρφθ
π
ϕφθρϕρ
ddeEeE
a
jk
A
Sjk
f
e
∫ ∫
−
=
0
2
0
)cos(sin),( 00
),(

28. Synthesized v. Actual FarfieldSynthesized v. Actual Farfield
PatternsPatterns

29. II Dependence on SystemII Dependence on System
ParametersParameters
 Frequency
 Amplitude
 Aperture Size
 Dominant Oscillation at 25 KHz

30. Frequency ScalingFrequency Scaling
 Plasma index of refraction expanded in a binomial
series
 Phase error depends inversely on frequency
 g computed at a low frequency ω0 to compute the
patterns at higher frequencies ω
2
2
2
1
ω
ωp
N −≈
ω
ϕρ
ω
ϕρω
ωϕρ
c
gg
c
kSe
),(),(
),,( 2
=≈
),,(),,( 0
2
0
ωϕρ
ω
ω
ωϕρ ee SS 





=

31. Actual and SynthesizedActual and Synthesized
Coefficient FrequencyCoefficient Frequency
DependenceDependence

32. Farfield Patterns at DifferentFarfield Patterns at Different
FrequenciesFrequencies

33. Amplitude ScalingAmplitude Scaling
 Use binomial expansion of plasma index of
refraction.
 Phase error depends linearly on a1.
 g computed at a weak plume amplitude to
compute patterns at higher plume amplitudes.
αθ−
= erarne )/()( 2
1 2
2
2
1
ω
ωp
N −≈
2
1
1
),(
),,,(
ω
ϕρ
ωϕρ
ga
aSe ≈
),,(),,( 10
1
10
1 aS
a
a
aS ee ϕρϕρ 





=

34. Farfield Patterns at DifferentFarfield Patterns at Different
AmplitudesAmplitudes

35. Aperture Size DependenceAperture Size Dependence
Expand at a large
aperture size a0.
Integrate only up to
a1.
∫ ∫
−
=
1
00
0
2
0
)cos(sin),(
),(
a
jk
A
Sjk
f dAeEeE e
π
ϕφθρϕρ
φθ

36. Aperture Size DependenceAperture Size Dependence

37. Boresight ModelBoresight Model
Intensity proportional to wavefront
deformation
General Model
)(1|| 2222
><−><−= eef SSkE
∑ ∑
∞
=
≥
−=
+−=
1
0
,..2,
22
22
2
12
1||
n
m
nnm
nmnmf ba
c
a
E
ω

38. Boresight ModelBoresight Model

39. Time-Varying ModelTime-Varying Model
 Oscillations in the plasma make the exit plane
phase vary with time.
 Quasi-static approximation (transit time <<
oscillation period).
 25 KHz rotating spoke oscillations.
 Exit plane phase expanded in a Fourier Time
Series.
)sin(),()cos(),(),(),,(
1
0 tftfftS nbn
n
nane ωϕρωϕρϕρϕρ ++= ∑
∞
=

40. Time-Varying ModelTime-Varying Model
 can be expanded in terms of
Zernike Polynomials
 approximated as the original time-
invariant phase error
 can be calculated by running the
code at t=0, and t=T/4
eoeee SandSS ,,0
0eS
eoee SandS ,
)sin),(cos),((),(),,(
))cos(1)((),(
20
2
tStSaStS
krmtarntrn
eoeeee
ee
Ω+Ω+=
−−Ω+=
ϕρϕρϕρϕρ
φ

41. Actual v. Synthesized PatternsActual v. Synthesized Patterns
at Different Timesat Different Times

42. Actual and SynthesizedActual and Synthesized
Farfield PhaseFarfield Phase

43. Boresight ModelBoresight Model
Intensity proportional to wavefront
deviation
Farfield phase linearly related to wavefront
variation
2
1
2
0
22
σσ +=><−>< SS
))sin()cos(()( '
200
'
100
21
tata
c
aa
t pfpf Ω+Ω=
ω
α

44. Aperture Phase ModelAperture Phase Model
AdvantagesAdvantages
Parameter
Change
BeamServer Zernike
Model
Geometry
(except concentric
apertures)
Rerun (~1hr) Recalculate
(1hr)
Frequency Rerun (~1hr) Scalable
Amplitude Rerun (~1hr) Scalable
Time-
variation
Rerun (~2 days) Scalable

45. Model LimitationsModel Limitations

46. Effects on CommunicationsEffects on Communications
Loss in SNR Synchronization








=
0
2
N
E
QP b
b

47. ConclusionsConclusions
Aperture Phase Method using Zernike
Polynomials can model antenna-plume
system.
Model scalable to higher frequencies, plume
amplitudes and aperture sizes.
Model expandable in time.