the propagation of a microwave in an atmospheric pressure plasma layer: 1 and 2 dimensional...
TRANSCRIPT
Conference on Computation Physics-2006 (I27)
The propagation of a microwave The propagation of a microwave
in an atmospheric pressure in an atmospheric pressure plasma layerplasma layer::
1 and 2 dimensional numerical solutions1 and 2 dimensional numerical solutionsXiwei HU, Zhonghe JIANG, Xiwei HU, Zhonghe JIANG,
Shu ZHANG and Minghai LIUShu ZHANG and Minghai LIU
HHuazhong uazhong UUniversity of niversity of SScience &cience & TTechnologyechnology
Wuhan, P. R. ChinaWuhan, P. R. China
August 30, 2006August 30, 2006
II Introduction and
motivation
IIII One dimensional
solution
IIIIII Two dimensional solutionTwo dimensional solution
IVIV Conclusions Conclusions
II
Introduction and motivationIntroduction and motivation
The classical mechanismThe classical mechanism
firstlyfirstly, the EM wave transfer its wave , the EM wave transfer its wave energy to the quiver kinetic energy of energy to the quiver kinetic energy of plasma electrons through electric field plasma electrons through electric field action of waves. action of waves.
ThenThen, the electrons transfer their , the electrons transfer their kinetic energy to the thermal energy of kinetic energy to the thermal energy of electrons, ions or neutrals in the electrons, ions or neutrals in the plasmas through plasmas through COLLISIONSCOLLISIONS between between electrons or between electrons and electrons or between electrons and other particles.other particles.
The electron fluid motion The electron fluid motion equationequation
ff0 0 is the microwave frequency,is the microwave frequency,
ννee ee , , ννeiei and and ννe0e0 is the collision frequenc is the collision frequency of y of electron-electronelectron-electron, , electron-ionelectron-ion and and eelectron-neutrallectron-neutral, respectively. , respectively.
( , )( , ) ( , ).e
e e e e
u x tm eE x t m u x t
t
0.e ee ei e 0( , ) ( )exp( 2 ),E x t E x i f
Pure plasma (produced by strong laser)Pure plasma (produced by strong laser) :: ννee=ν=νeeee++ννeiei ,,
Pure magnetized plasma (in magnetic confinemePure magnetized plasma (in magnetic confinement devices, e.g. tokamak): nt devices, e.g. tokamak): ννee=0=0,,
The mixing of plasma and neutral (in ionosphere The mixing of plasma and neutral (in ionosphere or in low pressure discharge): or in low pressure discharge): ννee==ννe0e0 ..
In all of above cases:In all of above cases: ννe e / f/ f0 0 << 1<< 1 Taking the WKB (or ekonal) approximationTaking the WKB (or ekonal) approximation
The solution of electron fluid equation isThe solution of electron fluid equation is
},)(exp{],,[),](,,[0x
yyy dsskitiJuEtxJuE
1
e e
eu i E
m i
The Appleton formula The Appleton formula
.))((
,))((
0 0
0 0
dxxkkA
dxxkk
d
ii
d
rr
.112
2
22
22
22
2
22
2
2
22
c
pec
c
pe
c
per c
k
.112
2
22
22
22
2
22
2
2
22
c
pec
c
pe
c
pei c
k
( ) ( ) ( )r ik x k x k x
},)(exp{],,[),](,,[0x
yyy dsskitiJuEtxJuE
WhenWhen pp == 50 – 760 50 – 760 TorrTorr ννe0e0≈≈ 6-46-4 6666 G(10G(1099) Hz) Hz,
electron density of APPelectron density of APPnne e ≈10≈101010 – 10 – 1012 12 cmcm-3-3 ,
correspondent cut off frequency ωωcc≈≈ 22 - 20- 20 GHz GHz,
sosoννe0e0 ≥or >> ≥or >>ωωcc ≈2 ≈2ππff00.
ff00 : frequency of electromagnetic wave : frequency of electromagnetic wave
The goal of our workThe goal of our work
Study the propagation behaviors Study the propagation behaviors of microwave by solving the of microwave by solving the coupled wave (Maxwell) coupled wave (Maxwell) equation and electron fluid equation and electron fluid motion equation directly motion equation directly in time in time and space domainand space domain instead of in instead of in frequency and wavevector frequency and wavevector domain.domain.
II One dimensional caseII One dimensional case
II.1II.1 The integral-differential equation The integral-differential equation
II.2II.2 The numerical method, basic wave The numerical method, basic wave form and precision checkform and precision check
II.3 II.3 The comparisons with the Appleton The comparisons with the Appleton formulaformula
II.4II.4 Outline of numerical results Outline of numerical results
II.1II.1
The integral-differential The integral-differential equationequation
The coupled set of The coupled set of equationsequations
Begin with the EM Begin with the EM wave equationwave equation
Coupled with the Coupled with the electronelectron fluid fluid motion motion equationequation
),(),(),(
txuvtxEm
e
t
txueo
e
0),(4),(1),(
22
2
22
2
t
txJ
ct
txE
cx
txE
),()(),( txeuxntxJ e
CombineCombine wave and electron motion equations, we have got a integral-differential equation:
Obtain Obtain numericallynumerically the the full solutionsfull solutions of of
EM wave field in space and time domainEM wave field in space and time domain
),()(),(1),(
2
2
2
2
22
2
txEc
xw
t
txE
cx
txE pe
0),()(
0
)(2
2
dssxEevc
xw t tsvc
pe eo
II.2II.2
The numerical method, The numerical method,
precision check precision check
and and
basic wave forms basic wave forms
Numerical MethodNumerical Method CompilerCompiler: : Visual C++ 6.0Visual C++ 6.0 AlgorithmAlgorithm::— — average implicit difference average implicit difference
method for differential partmethod for differential part— — composite Simpson integral composite Simpson integral
method for integral partmethod for integral part
Check the precision of the Check the precision of the codecode
Compare the numerical phase shift with the analytic result in ννee00 =0 =0..
The analytic formula for phase shift
,)](1[
2
00 l
dxxn .
)(1)(
2
1
c
e
n
xnxn
Bell-like electron density Bell-like electron density profileprofile
22
0 12
1)(
d
xnxn e ,0 dx
Phase shift Δφ when Phase shift Δφ when ννe0 e0 =0=0
ne / nc Δφcalcul (degr.) Δφtheor ( degr.) Relative Error (%)
0.1 19.50 19.58 0.39
0.2 39.85 40.00 0.37
0.3 61.23 61.40 0.26
0.4 83.75 83.93 0.21
0.5 107.78 107.85 0.06
0.6 133.60 133.50 0.08
0.7 162.25 161.42 0.52
0.8 194.45 192.57 0.97
0.9 233.07 229.02 1.77
Waveform of EWaveform of Ey y (x)(x) ne = 0.5 nc , d = 2 λ0, νe0 = 0.1 ω0
Wave forms: passed plasma, passed vacuuWave forms: passed plasma, passed vacuum, m, interferenceinterference, , phase shiftphase shift..
ne = 0.5 nc , d = 2 λ0, νe0 = 1.0 ω0
The reflected plane wave EThe reflected plane wave E22
II.3II.3
The comparison with the The comparison with the
Appleton formulaAppleton formula
Brief summary (1)Brief summary (1)
When When nn00 /n /ncc <1 <1, the reflected wave is , the reflected wave is weak, the weak, the Δφ Δφ and TT obtained from obtained from analytic (Appleton) formula and analytic (Appleton) formula and numerical solutions are agree well.numerical solutions are agree well.
When When nn00 /n /ncc >1 >1, the wave reflected , the wave reflected strongly, strongly, the Appleton formula is no the Appleton formula is no longer correctlonger correct. We have to take the full . We have to take the full solutions of time and space to describe solutions of time and space to describe the behaviors of a microwave passed the behaviors of a microwave passed through the APP.through the APP.
II.4 II.4 Outline of numerical Outline of numerical
resultsresults
Phase shift Phase shift ΔφΔφTransmissivity Transmissivity TT
Reflectivity Reflectivity RRAbsorptivity Absorptivity AA
DeterminationDetermination
EE00—incident electric field of EM wave,—incident electric field of EM wave,
EE11—transmitted electric field,—transmitted electric field,
EE22—reflected electric field—reflected electric field TransmissivityTransmissivity: :
T=ET=E11 /E /E00 , T , Tdbdb =-20 =-20 lglg (T). (T). ReflectivityReflectivity: :
R=ER=E22 /E /E00 , R , Rdbdb =-20 =-20 lglg (R). (R). AbsorptivityAbsorptivity: A=1 - T: A=1 - T2 2 - R- R22
Three models of nThree models of nee(x)(x)∫n∫nee
{{mm} } (x) dx =N(x) dx =Nee=constant, =constant, m=1,2,3.m=1,2,3.
1.1. The bell-like The bell-like profileprofile
2.2. The trapezium The trapezium profileprofile
3.3. The linear profile The linear profile
22
0 12
1)(
d
xnxn
Effects of profiles are not importantEffects of profiles are not important
The phase shift | The phase shift | ΔφΔφ | |
1.1. \\ΔφΔφ\ \ increases withincreases with nn00 andand dd..
2. When 2. When ννe0 e0 → 0→ 0 ,, \\ΔφΔφ\ → \ → the maximum valthe maximum value in pure (collisionless) plasmas. ue in pure (collisionless) plasmas.
3. Then, 3. Then, \\ΔφΔφ\ \ decreases withdecreases with ννe0e0/ω/ω00 increasiincreasing.ng.
4. When 4. When ννe0e0// ωω0 0 >>1>>1, , ΔφΔφ→0 →0 – – the pure neutrathe pure neutral gas casel gas case. .
Briefly summary (2)Briefly summary (2)
The transmissivity The transmissivity TTdbdb and The and The absorptivity absorptivity AA reach their maximum reach their maximum
atat ννe0e0/ω/ω0 0 ≈1≈1
Briefly summary (3)Briefly summary (3)
All four quantities All four quantities ΔφΔφ, , TT, , RR, , AA depend on depend on
--the electron density --the electron density nnee(x)(x), ,
--the collision frequency --the collision frequency ννe0 e0 , ,
--the plasma layer width --the plasma layer width dd. .
is more important than is more important than and and dd
According to the collision damping According to the collision damping mechanism, the transferred wave energy is mechanism, the transferred wave energy is approximately proper to the total number approximately proper to the total number of electrons, which is in the wave passed of electrons, which is in the wave passed path.path.
represents the represents the total number of total number of
electronselectrons in a volume with unit cross- in a volume with unit cross-section and width section and width dd when the average when the average linear density of electron islinear density of electron is . .
en d
en d
en
en
TTdBdB seems a simple seems a simple function of the function of the
product of product of nn and and dd Let Let
TTdBdB (nd)= (nd)= F(nF(ne e , ν, νee))
When When ννee > 1> 1, ,
F(nF(ne e , ν, νee) = Const.) = Const.
When When ννee < 1 < 1 , ,
F(nF(ne e , ν, νee) increases ) increases
slowly with nslowly with nee
en d
F(nF(ne e ,ν,νe e ))
III III Two dimensional caseTwo dimensional case
III.1III.1 The geometric graph and arithmetic The geometric graph and arithmetic
III.2III.2 Comparison between one and two Comparison between one and two dimensional results in normal incident dimensional results in normal incident casecase
III.3III.3 Outline of numerical results Outline of numerical results
III.1III.1
Geometric graph for FDTDGeometric graph for FDTD
Integral-differential equations Integral-differential equations
When microwave obliquely incident When microwave obliquely incident into an APP layerinto an APP layer
The propagation of The propagation of wave becomes a prowave becomes a problem at least in blem at least in two two dimension spacedimension space..
Then, the Then, the incidence incidence angleangleθθ and theand the polarpolarizationization ( (SS or or PP mode) mode) of incident wave will of incident wave will influence the attenuinfluence the attenuation and phase shift ation and phase shift of wave. of wave.
.Z
Y
X
Plasma LayerAbsorbing Boundary
Connecting Boundary
Incident Wave
Reflected Wave
Outputting Bundary
The equations in two dimension The equations in two dimension casecase
Maxwell equation for the microwave.Maxwell equation for the microwave.
Electron fluid motion equation for the Electron fluid motion equation for the electrons.electrons.
0 0, .E H
H J Et t
0, .ee e e e e e
uJ en u m eE m u
t
s-polarized s-polarized p-p-polarizedpolarized
z e zJ en u
yzHE
x t
xz HE
y t
y x zz
H H EJ
x y t
0z
z e ze
u eE u
t m
yzy
EHJ
x t
y x zE E H
x y t
xzx
EHJ
y t
i e iJ en u
0i
i e i
u eE u
t m
Combine Maxwell’sCombine Maxwell’s and motion equations
integral-differential equations S-polarized S-polarized integral-differential
equations:equations:
P-polarized P-polarized integral-differential equations:equations:
0
0
yzHE
x t
0
0
xz HE
y t
( )0 0
00 0
( , )eoty v s tx ez
zc
H H nEe E x s ds
x y t n
( )0 0
00 0
( , )eot v s tx ez
xc
E nHe E x s ds
y t n
0
0
y x zE E E
x y t
( )0 0
00 0
( , )eoty v s tez
yc
E nHe E x s ds
x t n
III.2III.2
ComparisonComparison between one and two between one and two dimensional results in dimensional results in normal incident casenormal incident case
0.01 0.1 1 10 1000.0
0.2
0.4
0.6
0.8
1.0
e0/
one-dimension two-dimension
Abso
rbtiv
ity
0.1 1 10 100
0
50
100
150
200
250
300
5,00, 0.5nc
e0/f
P
ha
se
sh
ift
(0 )
one-dimension two-dimension
0.1 1 10 100
0
5
10
15
20
5,00, 0.5nc
e0/f
one-dimension two-dimension
Tra
ns
mis
siv
ity
(d
B)
0.1 1 10 100
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
5,00, 0.5nc
Re
fle
cti
vit
y
e0/f
one-dimension two-dimension
III.3III.3
The numerical results about the effects of about the effects of
incidence anglesincidence angles and and polarizationspolarizations
The influence of The influence of incidence incidence angleangle
0.01 0.1 1 10 100
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
e0/
Absorb
tivity
400
300
200
100
00
0.01 0.1 1 10 100
0
20
40
60
80
e0/
400
300
200
100
00
Phaseshift (
0)
0.01 0.1 1 10 1000
10
20
30
40
50
e0/
400
300
200
100
00
Reflectivity (dB
)
0.01 0.1 1 10 100
0
1
2
3
4
5
6
e0/
400
300
200
100
00
Tra
nsm
itiv
ity (dB)
0.1 1 10 100
0
20
40
60
80
100
120
140
S,1nc,
p,1nc,
S,1nc,
p,1nc,
s,1nc,
p,1nc,
Ph
ases
hif
t (0 )
e0
/f0.1 1 10 100
0
50
100
150
200
250
300
350 s,2n
c,
p,2nc,
s,2nc,
p,2nc,
s,2nc,
p,2nc,
Ph
as
es
hif
t(0 )
e0/f
0.1 1 10 1000
10
20
30
40
50 S,1n
c,
p,1nc,
S,1nc,
p,1nc,
s,1nc,
p,1nc,
Re
fle
cti
vit
y (
dB
)
e0
/f0.1 1 10 100
0
1
2
3
4
5
6
7
8
9
10 S,1n
c,
p,1nc,
S,1nc,
p,1nc,
s,1nc,
p,1nc,
Tra
ns
mis
siv
ity(
dB
)
e0
/f
0.1 1 10 100
0
10
20
30
40
50
60 s, bell-like profile, 2n
c, 450, 1
p, bell-like profile, 2nc, 450, 1
s, exponential profile,1.5358nc, 450, 1
p, exponential profile, 1.5358nc, 450, 1
Refle
ctiv
ity (d
B)
e0/f
0.1 1 10 100
0
20
40
60
80
100
120
140 s, bell-like profile, 2nc, 450, 1
p, bell-like profile, 2nc, 450, 1
s, exponential profile,1.5358nc, 450, 1
p, exponential profile, 1.5358nc, 450, 1
Phas
eshif
t (0 )
e0/f
The effects of the density profiledensity profile
IVIV
ConclusionConclusion
1. When 1. When nnmax max /n/nc c >1>1, the Appleton , the Appleton formula should be replayed by formula should be replayed by the numerical solutions.the numerical solutions.
2. The larger the microwave 2. The larger the microwave incidence angleincidence angle is, the bigger the is, the bigger the absorptivity of microwave is.absorptivity of microwave is.
3. The absorptivity of 3. The absorptivity of P (TE) modeP (TE) mode is is generally larger than the one of S generally larger than the one of S (TM) mode incidence microwave.(TM) mode incidence microwave.
4. The bigger the factor is, the 4. The bigger the factor is, the
better the absorption of APP layer is.better the absorption of APP layer is.
5. The absorptivity reaches it 5. The absorptivity reaches it
maximummaximum when . when .
6.The less the 6.The less the gradientgradient of electron of electron
density is, the larger (smaller) the density is, the larger (smaller) the
absorptivity (reflectivity) is.absorptivity (reflectivity) is.
en d
0 0 02e f