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Conference on Computation Physics-2006 (I27) The propagation of a microwave in an atmospheric pressure plasma layer : 1 and 2 dimensional numerical solutions. Xiwei HU, Zhonghe JIANG, Shu ZHANG and Minghai LIU H uazhong U niversity of S cience & T echnology Wuhan, P. R. China

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Xiwei HU, Zhonghe JIANG, Shu ZHANG and Minghai LIU

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Conference on Computation Physics-2006 (I27)The propagation of a microwave in an atmospheric pressure plasma layer:1 and 2 dimensional numerical solutions

Xiwei HU, Zhonghe JIANG,

Shu ZHANG and Minghai LIU

Huazhong University of Science &Technology

Wuhan, P. R. China

August 30, 2006

IIntroductionand motivationIIOne dimensional solutionIIITwo dimensional solutionIVConclusions

IIntroduction and motivation

The classical mechanism

  • firstly, the EM wave transfer its wave energy to the quiver kinetic energy of plasma electrons through electric field action of waves.

  • Then, the electrons transfer their kinetic energy to the thermal energy of electrons, ions or neutrals in the plasmas through COLLISIONS between electrons or between electrons and other particles.

The electron fluid motion equation

  • f0 is the microwave frequency,

  • νee , νei and νe0 is the collision frequency of electron-electron, electron-ion and electron-neutral, respectively.

Pure plasma (produced by strong laser):νe=νee+νei,

Pure magnetized plasma (in magnetic confinement devices, e.g. tokamak): νe=0,

The mixing of plasma and neutral (in ionosphere or in low pressure discharge): νe=νe0.

In all of above cases:νe / f0 << 1

Taking the WKB (or ekonal) approximation

The solution of electron fluid equation is

The Appleton formula

Whenp=50 – 760 Torr

νe0≈6-466G(109) Hz,

electron density of APP

ne ≈1010– 1012 cm-3,

correspondent cut off frequency

ωc≈2- 20 GHz,


νe0 ≥or >>ωc≈2πf0.

f0 : frequency of electromagnetic wave

The goal of our work

  • Study the propagation behaviors of microwave by solving the coupled wave (Maxwell) equation and electron fluid motion equation directly in time and space domain instead of in frequency and wavevector domain.

II One dimensional case

II.1 The integral-differential equation

II.2 The numerical method, basic wave form and precision check

II.3 The comparisons with the Appleton formula

II.4 Outline of numerical results

II.1The integral-differential equation

The coupled set of equations

  • Begin with the EM wave equation

  • Coupled with the electron fluid motion equation

  • Combinewave and electron motion equations, we have got a integral-differential equation:

  • Obtain numerically the full solutions of EM wave field in space and time domain

II.2The numerical method, precision check and basic wave forms

Numerical Method

  • Compiler:

    Visual C++ 6.0

  • Algorithm:

    —average implicit difference method for differential part

    —composite Simpson integral method for integral part

Check the precision of the code

  • Compare the numerical phase shift with the analytic result inνe0 =0.

  • The analytic formula for phase shift

Bell-like electron density profile

Phase shift Δφ when νe0 =0

Waveform of Ey (x)ne = 0.5 nc,d = 2 λ0, νe0 = 0.1 ω0

Wave forms: passed plasma, passed vacuum, interference, phase = 0.5 nc,d = 2 λ0, νe0 = 1.0 ω0

The reflected plane wave E2


The comparison with the Appleton formula

Brief summary (1)

  • When n0 /nc <1, the reflected wave is weak, the Δφ andT obtained from analytic (Appleton) formula and numerical solutions are agree well.

  • When n0 /nc >1, the wave reflected strongly, the Appleton formula is no longer correct. We have to take the full solutions of time and space to describe the behaviors of a microwave passed through the APP.

II.4 Outline of numerical resultsPhase shift ΔφTransmissivity TReflectivity RAbsorptivity A


  • E0—incident electric field of EM wave,

    E1—transmitted electric field,

    E2—reflected electric field

  • Transmissivity:

    T=E1 /E0 , Tdb =-20 lg (T).

  • Reflectivity:

    R=E2 /E0 , Rdb =-20 lg (R).

  • Absorptivity: A=1 - T2 - R2

The bell-like profile

2. The trapezium profile

3. The linear profile

Three models of ne(x)∫ne{m} (x) dx =Ne=constant, m=1,2,3.

Effects of profiles are not important

The phase shift | Δφ |

Briefly summary (2)

1.\Δφ\ increases withn0andd.

2. When νe0 → 0,\Δφ\ → the maximum value in pure (collisionless) plasmas.

3. Then, \Δφ\ decreases withνe0/ω0increasing.

4. When νe0/ω0 >>1, Δφ→0 –the pure neutral gas case.

The transmissivity Tdb and The absorptivity A reach their maximum atνe0/ω0 ≈1

Briefly summary (3)

  • All four quantities Δφ, T, R, A depend on

    --the electron density ne(x),

    --the collision frequency νe0 ,

    --the plasma layer width d.

is more important than and d

  • According to the collision damping mechanism, the transferred wave energy is approximately proper to the total number of electrons, which is in the wave passed path.

  • represents the total number of electrons in a volume with unit cross-section and width d when the average linear density of electron is .

TdB seems a simple function of the product of n and d

  • Let

    TdB (nd)=F(ne , νe)

  • When νe> 1,

    F(ne , νe) = Const.

  • When νe < 1 ,

    F(ne , νe) increases slowly with ne

F(ne ,νe )

III Two dimensional case

III.1 The geometric graph and arithmetic

III.2 Comparison between one and two dimensional results in normal incident case

III.3 Outline of numerical results

III.1Geometric graph for FDTDIntegral-differential equations

When microwave obliquely incident into an APP layer

  • The propagation of wave becomes a problem at least in two dimension space.

  • Then, the incidence angleθand thepolarization (S or P mode) of incident wave will influence the attenuation and phase shift of wave.

The equations in two dimension case

  • Maxwell equation for the microwave.

  • Electron fluid motion equation for the electrons.

s-polarized p-polarized

Combine Maxwell’s and motion equationsintegral-differential equations

  • S-polarized integral-differential equations:

  • P-polarized integral-differential equations:

III.2Comparison between one and two dimensional results in normal incident case

III.3The numerical resultsabout the effects of incidence angles and polarizations

The influence of incidence angle

The effects of the density profile


1. When nmax /nc >1, the Appleton formula should be replayed by the numerical solutions.

2. The larger the microwave incidence angle is, the bigger the absorptivity of microwave is.

3. The absorptivity of P (TE) mode is generally larger than the one of S (TM) mode incidence microwave.

4. The bigger the factor is, the better the absorption of APP layer is.

5. The absorptivity reaches it maximum when .

6.The less the gradient of electron density is, the larger (smaller) the absorptivity (reflectivity) is.

Thanks !

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