Opt Quant Electron (2007) 39:1245–1252 DOI 10.1007/s11082-008-9195-8
FDTD analysis on the effect of plasma parameters on the reflection coefficient of the electromagnetic wave H. W. Yang · R. S. Chen
Received: 11 May 2007 / Accepted: 19 February 2008 / Published online: 18 March 2008 © Springer Science+Business Media, LLC. 2008
Abstract In this paper, the dielectric constant of dispersive medium is written as rational polynomial function, and the relationship between D and E is derived in time-domain. It is named shift operator FDTD (SO-FDTD) method. Compared to the analytical solution, the high accuracy and efficiency of this method is verified by calculating the reflection coefficient of the electromagnetic wave through a cold plasma slab. The effect on reflection coefficient is calculated by using the SO-FDTD method. The result shows that some factors effect on reflection coefficient. They are as follows: plasma thickness, electron density, electron distribution and incident frequency. And on most conditions, parabola distribution helps reduce reflection coefficient more effectively than homogeneous distribution. Keywords Plasma · Electromagnetic wave · Finite-different time-domain (FDTD) method
1 Introduction The time-domain methods have been widely used to simulate the transient solutions of the electromagnetic wave propagation in the dispersive medium. Among these methods, the finite-different time-domain method (FDTD) (Yee 1966) is comparatively easy and effective. Over the last decade, there have been numerous investigations of FDTD dispersive medium formulations. These include the recursive convolution (RC) methods (Luebbers et al. 1991), auxiliary differential equation (ADE) methods (Young 1995; Nickisch and Franke 1992), frequency-dependent Z transform methods (Sullivan 1992), piecewise linear recursive convolution (PLRC) method (Kelley and Luebbers 1996), current density convolution (JEC) method (Chen et al. 1998) and the piecewise linear current density recursive convolution
H. W. Yang (B) Department of Physics, Nanjing Agricultural University, Nanjing 210095, P.R. China e-mail:
[email protected] R. S. Chen Department of Communication Engineering, Nanjing University of Science and Technology, Nanjing 210094, P.R. China
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(PLJERC) method (Liu et al. 2004), and frequency-dependent finite-difference time domain (FDFDTD) method (Qian et al. 2005a,b,c) etc. The general constitutive relationship in the dispersive medium is in frequency-domain. Due to the frequency-independent dielectric constant of dispersive medium, the relationship is transformed into the convolution relationship in time-domain, which makes difficult to apply FDTD directly. From the definition of Z-transition, the transition relationships among time-domain, frequency-domain and dispersive time-domain, and the convolution theory are discussed. The recursive relationship between D and E in the dispersive medium has been deduced in the reference Sullivan 2000. In this paper, with the introduction of shift operators in the dispersive time-domain by the difference approximation method, the arithmetic operator intergradations between time-domain and dispersive time-domain are obtained. The result around analytical solution is calculated without using the Z-transition method. When the constitutive relationship in frequency-domain can be written as rational fractional functions, it is transformed into time-domain, and then into dispersive time-domain. And the recursive formulations from D to E can be deduced. The formulations can be used for the FDTD calculation in dispersive medium. In this paper, with the shift operator method, the plasma reflection coefficient is calculated. After comparison with the analytical solution, some relationships are calculated, compared and analyzed. They are as follows: the relationship between the reflection coefficient and the electron density magnitude, the electron density distribution, the plasma layers thickness and the incident frequency. The regularity of relationship and its reasons are discussed.
2 SO-FDTD method With collisional cold plasma in the dispersive medium, Maxwell’s equations and the related equations are given as follows: ∂D =∇ ×H ∂t
(1)
1 ∂H =− ∇ ×E ∂t µ0
(2)
D(ω) = ε0 εr (ω)E(ω)
(3)
As to one-dimension simulation, H can be calculated from E, and then D formula can be also calculated by using the FDTD method. t 1 1 n+1/2 n+1/2 k Dn+1 (k) = D (k) − k + k − H − H (4) y y x x Z 2 2 n+1/2
Hy
k+
1 2
n−1/2
= Hy
k+
1 2
−
t n Ex (k + 1) − Enx (k) µ0 Z
(5)
Assuming the dielectric constant εr (ω) in Eq. 3 can be written as a rational fractional function: N pn ( jω)n εr (ω) = n=0 (6) N n n=0 qn ( jω) where ω is the angular frequency.
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With the transition relationship from frequency-domain to time-domain: jω → ∂/∂t, substitute Eq. 6 into Eq. 3 ∂ D(t) = ε0 εr E(t) (7) ∂t If a function in time-domain has the form of y(t) = (n + 0.5)tcan approximately be given as:
∂ f (t) ∂t ,
its central difference at
y n+1 + y n f n+1 − f n = 2 t
(8)
Now let z t be the shift arithmetic operators (Sullivan 2000) z t f n = f n+1
(9)
Combing Eqs. 8 and 9, we have yn =
2 zt − 1 t z t + 1
fn
(10)
After comparison ∂ → ∂t
2 zt − 1 t z t + 1
(11)
By substituting Eqs. 6 and 11 into Eq. 7, the constitutive relationship of dispersive timedomain can be given N
N
2 z t − 1 l 2 z t − 1 l n ql pl (12) D = ε0 En t z t + 1 t z t + 1 l=0
l=0
According to (9), if N of Eq. 12 is taken as 2, Eq. 12 can be written in a form of n+1 n n−1 D D D 1 n+1 n n−1 = E a0 + a1 + a2 − b1 E − b2 E b0 ε0 ε0 ε0
(13)
with a0 = q 0 + q 1
b0 = p0 + p1
2 + q2 t
2 + p2 t
2 t
2 t
2 ; a1 =2q0 − 2q2 2
; b1 =2 p0 − 2 p2
2 t 2 t
2 ; a2 =q0 − q1
2 + q2 t
2 ; b2 = p 0 − p 1
2 + p2 t
2 t
2
2 t
2
For a non-magnetized cold plasma, the relative dielectric constant is given by (Sullivan 2000): εr (ω) = 1 +
ω2p ω( jνc − ω)
(14)
where ω p is the plasma frequency, νc is the average value of the electron collision frequency. From Eqs. 6 and 14, one can obtain p0 = ω2p , p1 = νc , p2 = 1, q0 = 0, q1 = νc and q2 = 1. With Eqs. 13, 4 and 5, the E can be calculated by iterative process.
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Fig. 1 Comparison of the reflection coefficient (a) and transmission coefficient (b)for plasma of 1.5 cm by analytical solution and SO-FDTD
3 Validity and precision of SO-FDTD method In order to check the accuracy of this method, the reflection coefficient is calculated for a non-magnetized plasma with thickness of 1.5 cm at ω p = 2π × 28.7 × 109 rad/s and νc = 20 × 109 rad/s. The incident wave used in SO-FDTD method is the derivative of the Gaussian pulse. The computational domain is subdivided into 1,000 cells, and the plasma occupies 200 cells. As shown in Fig. 1, the results of analytical solution (Ginzburg 1970) and SO-FDTD agree very well. In the reflection coefficient calculated, we defined reflection coefficient: Er R = 20 log (15) Ei where, E i and Er are amplitudes of incident and reflected wave.
4 Effects of the plasma distribution through the slab on the reflection coefficient 4.1 Effects of the plasma thickness on the reflection coefficient, when electron density distributed as parabola With different plasma thickness and the electron density, the effect of the incident frequency on the reflection coefficient is studied by using SO-FDTD method. Now, let the electron density distributes as parabola in the plasma slab. And the electron density is 1015 /m3 on the two ends of plasma thickness and maintains unchanged. It distributes as parabola in the central part symmetry and its maximums (Nem) are 1015 , 1016 , 1017 , 1018 and 1019 / m3 respectively in the center of the plasma thickness. With different plasma thicknesses of 1.5 cm and 3.0 cm, the reflection coefficient with different incident frequency are shown in the Figs. 2 and 3. As shown in the figures, with constant values of the thickness, the reflection coefficient increases with the increase of the maximums (Nem) of electron density at the same incident frequency. And the reflection coefficient increases quickly within the low frequency range. On the contrary, within the high frequency range, the reflection coefficient increases in approximate linearity with small electron density (for example of 1015 / m3 in Figs. 2 and 3). With the increase of the incident frequency and electron density (for example of 1016 / m3 in Figs. 2 and 3), the reflection coefficient decreases firstly, and then increases in approximate linearity. But within the low frequency range and the plasma thickness of 1.5 cm or 3.0 cm,
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Fig. 2 The relationship between incident frequency and reflection coefficient with parabola distribution of the electron density, when plasma thickness d = 1.5 cm
Fig. 3 The relationship between incident frequency and reflection coefficient with parabola distribution of the electron density, when plasma thickness d = 3.0 cm
the reflection coefficient decreases steadily with accordance with the continual increase of the electron density. And with increase of the incident frequency, the reflection coefficient vibration decreases. Especially with the plasma thickness of 3.0 cm, the reflection coefficient vibrates sharply. 4.2 Comparison of relationships between the incident frequency and the reflection coefficient, when the electron density distributes as parabola and homogeneous In order to compare the relationship between the incident frequency and the reflection coefficient, in this paper, two plasma thicknesses of 1.5 cm and 3.0 cm, and maximum value Nem = 1017 /m3 in the parabola distribution and Ne0 = 6.7 × 1016 /m3 in the homogeneous distribution are calculated. The results are shown in Figs. 4 and 5. The maximum value Nem = 1019 /m3 in the parabola distribution and Ne0 = 6.7 × 1018 /m3 in the homogeneous distribution are calculated. The results are shown in Figs. 6 and 7. In order to compare it simply, every figure has the same total average value of the electron density among the plasma thickness. As shown in Figs. 4 and 5, the relationships between the incident frequency and the reflection coefficient are obtained in the approximately same curves with the plasma thickness d = 1.5 cm and d = 3.0 cm. Especially within the frequency ranges of 0–3 GHz and 50–100 GHz, the two curves almost superpose. But within the frequency range of 3–50 GHz, whatever the plasma thickness is 1.5 cm or 3.0 cm, the reflection coefficient decreases with the increase of incident frequency. And with increase of the plasma thickness, the vibration
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Fig. 4 The relationships between the incident frequency and the reflection coefficient with same total average electron density Ne0 = 6.7 × 1016 /m3 and thickness d = 1.5 cm
Fig. 5 The relationships between the incident frequency and the reflection coefficient with same total average electron density Ne0 = 6.7 × 1016 /m3 and thickness d = 3.0 cm
Fig. 6 The relationships between the incident frequency and the reflection coefficient with same total average electron density Ne0 = 6.7 × 1018 /m3 and thickness d = 1.5 cm
frequency increases, while its amplitude decreases. Within the frequency range of 3–50 GHz, smaller reflection coefficient is obtained in the parabola than homogeneous distribution. With maximum Nem = 1019 /m3 in the parabola distribution and Ne0 = 6.7 × 1018 /m3 in the homogeneous distribution, the relationships between the incident frequency and the reflection coefficient are shown in Figs. 6 and 7. Within the incident frequency range of 0–20 GHz, there’s almost no effect on reflection coefficient with different distributions and thicknesses. But within the incident frequency range of 20–100 GHz, its regularity is the same almost with that within the frequency range of 3–50 GHz shown in Figs. 4 and 5.
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Fig. 7 The relationships between the incident frequency and the reflection coefficient with same total average electron density Ne0 = 6.7 × 1018 /m3 and thickness d = 3.0 cm
With same plasma thicknesses and different distributions in Figs. 4 and 6, the reflection coefficient increases with the increase of the electron density and the incident frequency complexly. The relationship curve vibrates sharply, while the frequency range broadens. And two curves in two different distributions apart from each other gradually. After comparison, the parabola distribution helps decrease the reflection coefficient more effectively. The same regulation can also be gotten in Figs. 5 and 7, the plasma thickness d = 3.0 cm. After comparison of the results from Figs. 4 to 7, there are obvious effects on the reflection coefficient with different electron densities. And after comparison, the parabola distribution helps decrease the reflection coefficient more effectively than the homogeneous distribution. These studies can also help us use the proper plasma distribution when the plasma is finity. These have great meaning in aircraft stealthy technology.
5 Conclusion (1)
(2)
(3)
(4)
In this paper, the dielectric constant of dispersive medium is written as rational polynomial function, and the relationship between D and E is derived in time-domain. And its feasibility is verified. With two different distributions of the electron density parabola and homogeneous in plasma, the relationships among reflection coefficient and the electron density, the plasma thickness and the incident frequency are calculated. From the calculation, we know, the reflection coefficients have some important effect factors that are the plasma thickness, the electron density value, the electron density distribution and the incident frequency. With small electron density, whatever high or low frequency, there’s almost no effect of distribution on the reflection coefficient. But there is more effect with the increase of the electron density. Within the low incident frequency range, the reflection coefficient changes with the change of electron density. The reflection coefficient decreases with the decrease of electron density. When the incident frequency increases, the relationship among incident frequency, electron density and the reflection coefficient is compactness. These show that the change of electron density and incident frequency effects on the reflection coefficient greater. But, while the incident frequency achieves threshold, there has less effect of electron density on reflection coefficients. With smaller average electron density, there is less effect of distribution on the reflection coefficient. As shown in Figs. 2 and 3 with Ne = 1015 /m3 , almost two same curves are obtained. But with the increase of average value of the electron density, there is
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(5)
(6)
(7)
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much effect, while within the higher frequency there are less effect of distribution on reflection coefficient. From these figures, we can understand, if the values of the electron density, the incident frequency and the distributions of the electron density are suitable, there is an optimal value of the reflection coefficient. On some conditions, there is a minimum value. Within the low frequency range, the reflection coefficient increases with the increase of the electron density. And there’s no effect of distribution on the reflection coefficient. But within the high frequency range, the reflection coefficient vibration decreases and the vibration relates with the distribution a lot. With the same average value of the electron density, the parabola distribution helps decrease the reflection coefficient more effectively than the homogeneous distribution. There is obvious effect with high electron density, which helps do research on the distribution and stealth.
Acknowledgements This work is partially supported by National Natural Science Foundation of China under contract number 60431010, and National Science Foundation for Distinguished Young Scholars of China under contract number 60325103.
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