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