The propagation of a microwave in an atmospheric pressure plasma layer: 1 and 2 dimensional numerical solutions 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

I II III Two dimensional solution IVConclusions I Introduction and motivation II One dimensional solution III Two dimensional solution IV Conclusions

I Introduction 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. 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. 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 f 0 is the microwave frequency, f 0 is the microwave frequency, ν ee, ν ei and ν e0 is the collision frequency of electron-electron, electron- ion and electron-neutral, respectively. ν ee, ν ei and ν e0 is the collision frequency of electron-electron, electron- ion and electron-neutral, respectively.

Pure plasma (produced by strong laser) : ν e =ν ee ＋ ν ei, Pure plasma (produced by strong laser) : ν e =ν ee ＋ ν ei, Pure magnetized plasma (in magnetic confinement devices, e.g. tokamak): ν e =0, Pure magnetized plasma (in magnetic confinement devices, e.g. tokamak): ν e =0, The mixing of plasma and neutral (in ionosphere or in low pressure discharge): ν e =ν e0. The mixing of plasma and neutral (in ionosphere or in low pressure discharge): ν e =ν e0. In all of above cases: ν e / f 0 << 1 In all of above cases: ν e / f 0 << 1 Taking the WKB (or ekonal) approximation Taking the WKB (or ekonal) approximation The solution of electron fluid equation is The solution of electron fluid equation is

The Appleton formula

 When p ＝ 50 – 760 Torr ν e0 ≈ ６－４ 66 G(10 9 ) Hz ν e0 ≈ ６－４ 66 G(10 9 ) Hz,  electron density of APP n e ≈10 10 – cm -3 n e ≈10 10 – cm -3, correspondent cut off frequency ω c ≈ ２ - 20 GHz ω c ≈ ２ - 20 GHz,  so ν e0 ≥or >>ω c ≈2πf 0 ν e0 ≥or >>ω c ≈2πf 0. f 0 : frequency of electromagnetic wave f 0 : 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. 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.1 The integral-differential equation

The coupled set of equations Begin with the EM wave equation Begin with the EM wave equation Coupled with the electron fluid motion equation Coupled with the electron fluid motion equation

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

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

Numerical Method Numerical Method Compiler: Compiler: Visual C Visual C Algorithm: Algorithm: — average implicit difference method for differential part — composite Simpson integral method for integral part

Check the precision of the code ν e 0 =0. Compare the numerical phase shift with the analytic result in ν e 0 =0. The analytic formula for phase shift

Bell-like electron density profile

Phase shift Δφ when ν e0 =0 n e / n c Δφ calcul (degr.)Δφ theor ( degr.) Relative Error （％）

Waveform of E y (x) Waveform of E y (x) n e = 0.5 n c ， d = 2 λ 0, ν e0 = 0.1 ω 0

Wave forms: passed plasma, passed vacuum, interference, phase shift. Wave forms: passed plasma, passed vacuum, interference, phase shift. n e = 0.5 n c ， d = 2 λ 0, ν e0 = 1.0 ω 0

The reflected plane wave E 2

II.3 The comparison with the Appleton formula

Brief summary (1) When n 0 /n c <1, the reflected wave is weak, the Δφ T obtained from analytic (Appleton) formula and numerical solutions are agree well. When n 0 /n c <1, the reflected wave is weak, the Δφ and T obtained from analytic (Appleton) formula and numerical solutions are agree well. When n 0 /n c >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. When n 0 /n c >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 results Phase shift Δφ Transmissivity T Reflectivity R Absorptivity A

Determination E 0 — incident electric field of EM wave, E 0 — incident electric field of EM wave, E 1 — transmitted electric field, E 1 — transmitted electric field, E 2 — reflected electric field E 2 — reflected electric field Transmissivity: Transmissivity: T=E 1 /E 0, T db =-20 lg (T). Reflectivity: Reflectivity: R=E 2 /E 0, R db =-20 lg (R).  Absorptivity: A=1 - T 2 - R 2

Three models of n e (x) ∫n e {m} (x) dx =N e =constant, m=1,2,3. 1.The bell-like profile 2. The trapezium profile 3. The linear profile

Effects of profiles are not important

The phase shift | Δφ |

1. \Δφ\ increases with n 0 and d. 2. When ν e0 → 0 ， \Δφ\ → the maximum value in pure (collisionless) plasmas. 3. Then, \Δφ\ decreases with ν e0 /ω 0 increasing. 4. When ν e0 / ω 0 >>1, Δφ → 0 – the pure neutral gas case. Briefly summary (2)

The transmissivity T db and The absorptivity A reach their maximum at ν e0 /ω 0 ≈1

Briefly summary (3) All four quantities Δφ, T, R, A depend on All four quantities Δφ, T, R, A depend on --the electron density n e (x), --the electron density n e (x), --the collision frequency ν e0, --the collision frequency ν e0, --the plasma layer width d. --the plasma layer width d.

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

T dB seems a simple function of the product of n and d Let Let T dB (nd)= F(n e, ν e ) T dB (nd)= F(n e, ν e ) When ν e > 1, When ν e > 1, F(n e, ν e ) = Const. F(n e, ν e ) = Const. When ν e < 1, When ν e < 1, F(n e, ν e ) increases slowly with n e F(n e, ν e ) increases slowly with n e

F(n e,ν e )

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.1 Geometric graph for FDTD Integral-differential equations

When microwave obliquely incident into an APP layer The propagation of wave becomes a problem at least in two dimension space. The propagation of wave becomes a problem at least in two dimension space. Then, the incidence angleθ and the polarization (S or P mode) of incident wave will influence the attenuation and phase shift of wave. Then, the incidence angleθ and the polarization (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. Maxwell equation for the microwave. Electron fluid motion equation for the electrons. Electron fluid motion equation for the electrons.

s-polarized p-polarized

Combine Maxwell’s Combine Maxwell’s and motion equations integral-differential equations S-polarized equations: S-polarized integral-differential equations: P-polarized equations: P-polarized integral-differential equations:

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

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

The influence of incidence angle

density profile The effects of the density profile

IV Conclusion

1. When n max /n c >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.